3.2.77 \(\int x^2 (a+b \tanh ^{-1}(\frac {c}{x^2}))^2 \, dx\) [177]

Optimal. Leaf size=1172 \[ \frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{6} b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{6} b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right ) \]

[Out]

4/3*a*b*c*x-2/3*a*b*c^(3/2)*arctan(x/c^(1/2))-2/3*b^2*c*x*ln(1-c/x^2)+1/3*b^2*c^(3/2)*arctan(x/c^(1/2))*ln(1-c
/x^2)-1/3*b*c^(3/2)*arctanh(x/c^(1/2))*(2*a-b*ln(1-c/x^2))+2/3*b^2*c*x*ln(1+c/x^2)+1/3*a*b*x^3*ln(1+c/x^2)-1/3
*b^2*c^(3/2)*arctan(x/c^(1/2))*ln(1+c/x^2)-1/3*b^2*c^(3/2)*arctanh(x/c^(1/2))*ln(1+c/x^2)-1/6*b^2*x^3*ln(1-c/x
^2)*ln(1+c/x^2)+2/3*b^2*c^(3/2)*arctan(x/c^(1/2))*ln(2*c^(1/2)/(-I*x+c^(1/2)))-1/3*b^2*c^(3/2)*arctan(x/c^(1/2
))*ln((1+I)*(-x+c^(1/2))/(-I*x+c^(1/2)))-2/3*b^2*c^(3/2)*arctanh(x/c^(1/2))*ln(2*c^(1/2)/(x+c^(1/2)))+1/3*b^2*
c^(3/2)*arctanh(x/c^(1/2))*ln(2*(-x+(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(x+c^(1/2)))-1/3*b^2*c^(3/2)*arct
an(x/c^(1/2))*ln((1-I)*(x+c^(1/2))/(-I*x+c^(1/2)))+1/3*b^2*c^(3/2)*arctanh(x/c^(1/2))*ln(2*(x+(-c)^(1/2))*c^(1
/2)/(x+c^(1/2))/((-c)^(1/2)+c^(1/2)))-2/3*b^2*c^(3/2)*arctan(x/c^(1/2))*ln(2-2*c^(1/2)/(-I*x+c^(1/2)))+2/3*b^2
*c^(3/2)*arctanh(x/c^(1/2))*ln(2-2*c^(1/2)/(x+c^(1/2)))+1/3*I*b^2*c^(3/2)*polylog(2,-I*x/c^(1/2))+1/3*I*b^2*c^
(3/2)*polylog(2,-1+2*c^(1/2)/(-I*x+c^(1/2)))+1/6*I*b^2*c^(3/2)*polylog(2,1-(1+I)*(-x+c^(1/2))/(-I*x+c^(1/2)))+
1/6*I*b^2*c^(3/2)*polylog(2,1+(-1+I)*(x+c^(1/2))/(-I*x+c^(1/2)))-1/3*I*b^2*c^(3/2)*polylog(2,I*x/c^(1/2))-1/3*
I*b^2*c^(3/2)*polylog(2,1-2*c^(1/2)/(-I*x+c^(1/2)))+1/3*I*b^2*c^(3/2)*arctan(x/c^(1/2))^2+1/12*x^3*(2*a-b*ln(1
-c/x^2))^2+4/3*b^2*c^(3/2)*arctan(x/c^(1/2))-4/3*b^2*c^(3/2)*arctanh(x/c^(1/2))+1/3*b^2*c^(3/2)*arctanh(x/c^(1
/2))^2+1/12*b^2*x^3*ln(1+c/x^2)^2+1/3*b^2*c^(3/2)*polylog(2,-x/c^(1/2))-1/3*b^2*c^(3/2)*polylog(2,x/c^(1/2))+1
/3*b^2*c^(3/2)*polylog(2,1-2*c^(1/2)/(x+c^(1/2)))-1/3*b^2*c^(3/2)*polylog(2,-1+2*c^(1/2)/(x+c^(1/2)))-1/6*b^2*
c^(3/2)*polylog(2,1-2*(-x+(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(x+c^(1/2)))-1/6*b^2*c^(3/2)*polylog(2,1-2*
(x+(-c)^(1/2))*c^(1/2)/(x+c^(1/2))/((-c)^(1/2)+c^(1/2)))

________________________________________________________________________________________

Rubi [A]
time = 1.56, antiderivative size = 1172, normalized size of antiderivative = 1.00, number of steps used = 80, number of rules used = 34, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.125, Rules used = {6045, 6042, 2507, 2521, 2498, 269, 213, 2520, 12, 266, 6820, 6135, 6079, 2497, 2505, 199, 327, 6874, 209, 30, 2637, 6139, 6031, 6057, 2449, 2352, 2526, 210, 5048, 4940, 2438, 4966, 5044, 4988} \begin {gather*} \frac {1}{12} \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2 x^3+\frac {1}{12} b^2 \log ^2\left (\frac {c}{x^2}+1\right ) x^3+\frac {1}{3} a b \log \left (\frac {c}{x^2}+1\right ) x^3-\frac {1}{6} b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right ) x^3+\frac {4}{3} a b c x-\frac {2}{3} b^2 c \log \left (1-\frac {c}{x^2}\right ) x+\frac {2}{3} b^2 c \log \left (\frac {c}{x^2}+1\right ) x+\frac {1}{3} i b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {4}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{3} a b c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )+\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}-1\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {2 \sqrt {c}}{x+\sqrt {c}}-1\right )-\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcTanh[c/x^2])^2,x]

[Out]

(4*a*b*c*x)/3 - (2*a*b*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (4*b^2*c^(3/2)*ArcTan[x/Sqrt[c]])/3 + (I/3)*b^2*c^(3/2)*
ArcTan[x/Sqrt[c]]^2 - (4*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]^2)/3 - (2*b^2*c^(
3/2)*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)])/3 - (2*b^2*c*x*Log[1 - c/x^2])/3 + (b^2*c^(3/2)*A
rcTan[x/Sqrt[c]]*Log[1 - c/x^2])/3 - (b*c^(3/2)*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/3 + (x^3*(2*a - b
*Log[1 - c/x^2])^2)/12 + (2*b^2*c*x*Log[1 + c/x^2])/3 + (a*b*x^3*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*ArcTan[x/Sqr
t[c]]*Log[1 + c/x^2])/3 - (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[1 + c/x^2])/3 - (b^2*x^3*Log[1 - c/x^2]*Log[1 +
c/x^2])/6 + (b^2*x^3*Log[1 + c/x^2]^2)/12 + (2*b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] - I*x)])
/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)])/3 - (2*b^2*c^(3/2)*ArcTanh[x
/Sqrt[c]]*Log[(2*Sqrt[c])/(Sqrt[c] + x)])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] - x))/(
(Sqrt[-c] - Sqrt[c])*(Sqrt[c] + x))])/3 + (b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[(2*Sqrt[c]*(Sqrt[-c] + x))/((Sqr
t[-c] + Sqrt[c])*(Sqrt[c] + x))])/3 - (b^2*c^(3/2)*ArcTan[x/Sqrt[c]]*Log[((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*
x)])/3 + (2*b^2*c^(3/2)*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (I/3)*b^2*c^(3/2)*PolyLog[2
, 1 - (2*Sqrt[c])/(Sqrt[c] - I*x)] + (I/3)*b^2*c^(3/2)*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)] + (I/6)*b^
2*c^(3/2)*PolyLog[2, 1 - ((1 + I)*(Sqrt[c] - x))/(Sqrt[c] - I*x)] + (b^2*c^(3/2)*PolyLog[2, -(x/Sqrt[c])])/3 +
(I/3)*b^2*c^(3/2)*PolyLog[2, ((-I)*x)/Sqrt[c]] - (I/3)*b^2*c^(3/2)*PolyLog[2, (I*x)/Sqrt[c]] - (b^2*c^(3/2)*P
olyLog[2, x/Sqrt[c]])/3 + (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (b^2*c^(3/2)*PolyLog[2,
-1 + (2*Sqrt[c])/(Sqrt[c] + x)])/3 - (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] - x))/((Sqrt[-c] - Sqrt[
c])*(Sqrt[c] + x))])/6 - (b^2*c^(3/2)*PolyLog[2, 1 - (2*Sqrt[c]*(Sqrt[-c] + x))/((Sqrt[-c] + Sqrt[c])*(Sqrt[c]
+ x))])/6 + (I/6)*b^2*c^(3/2)*PolyLog[2, 1 - ((1 - I)*(Sqrt[c] + x))/(Sqrt[c] - I*x)]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2507

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)
^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q/(f*(m + 1))), x] - Dist[b*e*n*p*(q/(f^n*(m + 1))), Int[(f*x)^(m + n)*
((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && IGtQ[q, 1]
&& IntegerQ[n] && NeQ[m, -1]

Rule 2520

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[u*(x^(n - 1)/(d + e*x^n)
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 2521

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol]
:> With[{t = ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, (f + g*x^s)^r, x]}, Int[t, x] /; SumQ[t]] /; Free
Q[{a, b, c, d, e, f, g, n, p, q, r, s}, x] && IntegerQ[n] && IGtQ[q, 0] && IntegerQ[r] && IntegerQ[s] && (EqQ[
q, 1] || (GtQ[r, 0] && GtQ[s, 1]) || (LtQ[s, 0] && LtQ[r, 0]))

Rule 2526

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2637

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[z*Log[w]*(D[v, x]/v), x], x] - Int[SimplifyIntegrand[z*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rule 4966

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x]))*(Log[2/(1
- I*c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((
d + e*x)/((c*d + I*e)*(1 - I*c*x)))]/(1 + c^2*x^2), x], x] + Simp[(a + b*ArcTan[c*x])*(Log[2*c*((d + e*x)/((c*
d + I*e)*(1 - I*c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 4988

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTan[c*x])
^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))
]/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 5044

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(-I)*((a + b*ArcT
an[c*x])^(p + 1)/(b*d*(p + 1))), x] + Dist[I/d, Int[(a + b*ArcTan[c*x])^p/(x*(I + c*x)), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 0]

Rule 5048

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

Rule 6042

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Int[ExpandIntegrand[x^m*(a + b*(Log
[1 + 1/(x^n*c)]/2) - b*(Log[1 - 1/(x^n*c)]/2))^p, x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1] && IGtQ[n, 0] &&
IntegerQ[m]

Rule 6045

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Int[x^m*(a + b*ArcCoth[1/(x^n*c)])^
p, x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p, 1] && ILtQ[n, 0]

Rule 6057

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcTanh[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 6079

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[(a + b*ArcTanh[c*x
])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Dist[b*c*(p/d), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/
d))]/(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 6135

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]

Rule 6139

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2-\frac {1}{2} b x^2 \left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x^2 \log ^2\left (1+\frac {c}{x^2}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2 \, dx-\frac {1}{2} b \int x^2 \left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{4} b^2 \int x^2 \log ^2\left (1+\frac {c}{x^2}\right ) \, dx\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b \int \left (-2 a x^2 \log \left (1+\frac {c}{x^2}\right )+b x^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )\right ) \, dx+\frac {1}{3} (b c) \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{1-\frac {c}{x^2}} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{1+\frac {c}{x^2}} \, dx\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+(a b) \int x^2 \log \left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{2} b^2 \int x^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{3} (b c) \int \left (2 a-b \log \left (1-\frac {c}{x^2}\right )+\frac {c \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{-c+x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \left (\log \left (1+\frac {c}{x^2}\right )-\frac {c \log \left (1+\frac {c}{x^2}\right )}{c+x^2}\right ) \, dx\\ &=\frac {2}{3} a b c x+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} b^2 \int \frac {2 c x^2 \log \left (1-\frac {c}{x^2}\right )}{3 \left (-c-x^2\right )} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+\frac {c}{x^2}\right )}{3 \left (1-\frac {c}{x^2}\right )} \, dx+\frac {1}{3} (2 a b c) \int \frac {1}{1+\frac {c}{x^2}} \, dx-\frac {1}{3} \left (b^2 c\right ) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{3} \left (b c^2\right ) \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{-c+x^2} \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{c+x^2} \, dx\\ &=\frac {2}{3} a b c x-\frac {1}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{3} (2 a b c) \int \frac {x^2}{c+x^2} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {x^2 \log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{1-\frac {c}{x^2}} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{3} a b c x-\frac {1}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{3} \left (b^2 c\right ) \int \left (-\log \left (1-\frac {c}{x^2}\right )-\frac {c \log \left (1-\frac {c}{x^2}\right )}{-c-x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \left (\log \left (1+\frac {c}{x^2}\right )+\frac {c \log \left (1+\frac {c}{x^2}\right )}{-c+x^2}\right ) \, dx-\frac {1}{3} \left (2 a b c^2\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{-c+x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{c+x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{3} \left (b^2 c\right ) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \log \left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{3} \left (2 i b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (i+\frac {x}{\sqrt {c}}\right )} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (1+\frac {x}{\sqrt {c}}\right )} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (2 b^2 c\right ) \int \frac {\log \left (2-\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx-\frac {1}{3} \left (2 b^2 c\right ) \int \frac {\log \left (2-\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{-c+x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{c+x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \left (-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c-x^2\right )}\right ) \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \left (\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c x}-\frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c \left (c+x^2\right )}\right ) \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c-x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c+x^2} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (i b^2 c^{3/2}\right ) \int \frac {\log \left (1-\frac {i x}{\sqrt {c}}\right )}{x} \, dx-\frac {1}{3} \left (i b^2 c^{3/2}\right ) \int \frac {\log \left (1+\frac {i x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \left (\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}-x\right )}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {c}+x\right )}\right ) \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \left (-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}-x\right )}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{2 \left (\sqrt {-c}+x\right )}\right ) \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx-\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}+x} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-2 \left (\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx\right )+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c} \left (1-\frac {i x}{\sqrt {c}}\right )}\right )}{1+\frac {x^2}{c}} \, dx+2 \left (\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx\right )-\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {2 \left (\sqrt {-c}-x\right )}{\left (-1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx-\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {2 \left (\sqrt {-c}+x\right )}{\left (1+\frac {\sqrt {-c}}{\sqrt {c}}\right ) \sqrt {c} \left (1+\frac {x}{\sqrt {c}}\right )}\right )}{1-\frac {x^2}{c}} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )-2 \left (\frac {1}{3} \left (i b^2 c^{3/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i x}{\sqrt {c}}}\right )\right )+2 \left (\frac {1}{3} \left (b^2 c^{3/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {c}}}\right )\right )\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )\\ \end {align*}

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Mathematica [F]
time = 7.33, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^2*(a + b*ArcTanh[c/x^2])^2,x]

[Out]

Integrate[x^2*(a + b*ArcTanh[c/x^2])^2, x]

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*arctanh(c/x^2))^2,x)

[Out]

int(x^2*(a+b*arctanh(c/x^2))^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x^2))^2,x, algorithm="maxima")

[Out]

1/3*a^2*x^3 + 1/3*(2*x^3*arctanh(c/x^2) - (2*sqrt(c)*arctan(x/sqrt(c)) - sqrt(c)*log((x - sqrt(c))/(x + sqrt(c
))) - 4*x)*c)*a*b + 1/12*(x^3*log(x^2 - c)^2 - 3*integrate(-1/3*(3*(x^4 - c*x^2)*log(x^2 + c)^2 - 2*(2*x^4 + 3
*(x^4 - c*x^2)*log(x^2 + c))*log(x^2 - c))/(x^2 - c), x))*b^2

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x^2))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^2*arctanh(c/x^2)^2 + 2*a*b*x^2*arctanh(c/x^2) + a^2*x^2, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*atanh(c/x**2))**2,x)

[Out]

Integral(x**2*(a + b*atanh(c/x**2))**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*arctanh(c/x^2))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c/x^2) + a)^2*x^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*atanh(c/x^2))^2,x)

[Out]

int(x^2*(a + b*atanh(c/x^2))^2, x)

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