3.1.71 \(\int x^4 (a+b \tanh ^{-1}(c x^2))^2 \, dx\) [71]

Optimal. Leaf size=1173 \[ \frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \text {ArcTan}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \text {ArcTan}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \text {PolyLog}\left (2,1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{10 c^{5/2}}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}} \]

[Out]

1/5*I*b^2*polylog(2,1-2/(1-I*x*c^(1/2)))/c^(5/2)+1/5*I*b^2*polylog(2,1-2/(1+I*x*c^(1/2)))/c^(5/2)+1/5*I*b^2*ar
ctan(x*c^(1/2))^2/c^(5/2)+2/5*a*b*arctan(x*c^(1/2))/c^(5/2)+8/15*b^2*x/c^2-2/25*a*b*x^5+1/20*x^5*(2*a-b*ln(-c*
x^2+1))^2-1/15*b^2*x^3*ln(-c*x^2+1)/c-1/5*b^2*arctan(x*c^(1/2))*ln(-c*x^2+1)/c^(5/2)+1/15*b*x^3*(2*a-b*ln(-c*x
^2+1))/c-1/5*b*arctanh(x*c^(1/2))*(2*a-b*ln(-c*x^2+1))/c^(5/2)+2/15*b^2*x^3*ln(c*x^2+1)/c+1/5*a*b*x^5*ln(c*x^2
+1)+1/5*b^2*arctan(x*c^(1/2))*ln(c*x^2+1)/c^(5/2)-1/5*b^2*arctanh(x*c^(1/2))*ln(c*x^2+1)/c^(5/2)-1/10*b^2*x^5*
ln(-c*x^2+1)*ln(c*x^2+1)+2/5*b^2*arctanh(x*c^(1/2))*ln(2/(1-x*c^(1/2)))/c^(5/2)-2/5*b^2*arctan(x*c^(1/2))*ln(2
/(1-I*x*c^(1/2)))/c^(5/2)+1/5*b^2*arctan(x*c^(1/2))*ln((1+I)*(1-x*c^(1/2))/(1-I*x*c^(1/2)))/c^(5/2)+2/5*b^2*ar
ctan(x*c^(1/2))*ln(2/(1+I*x*c^(1/2)))/c^(5/2)-2/5*b^2*arctanh(x*c^(1/2))*ln(2/(1+x*c^(1/2)))/c^(5/2)+1/5*b^2*a
rctanh(x*c^(1/2))*ln(-2*(1-x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(1+x*c^(1/2)))/c^(5/2)+1/5*b^2*arctanh(x
*c^(1/2))*ln(2*(1+x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+x*c^(1/2)))/c^(5/2)+1/5*b^2*arctan(x*c^(1/2))*
ln((1-I)*(1+x*c^(1/2))/(1-I*x*c^(1/2)))/c^(5/2)-1/10*I*b^2*polylog(2,1-(1+I)*(1-x*c^(1/2))/(1-I*x*c^(1/2)))/c^
(5/2)-1/10*I*b^2*polylog(2,1+(-1+I)*(1+x*c^(1/2))/(1-I*x*c^(1/2)))/c^(5/2)-4/15*b^2*arctan(x*c^(1/2))/c^(5/2)-
4/15*b^2*arctanh(x*c^(1/2))/c^(5/2)-1/5*b^2*arctanh(x*c^(1/2))^2/c^(5/2)+1/25*b^2*x^5*ln(-c*x^2+1)+1/25*b*x^5*
(2*a-b*ln(-c*x^2+1))+1/20*b^2*x^5*ln(c*x^2+1)^2+1/5*b^2*polylog(2,1-2/(1-x*c^(1/2)))/c^(5/2)+1/5*b^2*polylog(2
,1-2/(1+x*c^(1/2)))/c^(5/2)-1/10*b^2*polylog(2,1+2*(1-x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)-c^(1/2))/(1+x*c^(1/2))
)/c^(5/2)-1/10*b^2*polylog(2,1-2*(1+x*(-c)^(1/2))*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+x*c^(1/2)))/c^(5/2)+2/15*a*b
*x^3/c

________________________________________________________________________________________

Rubi [A]
time = 1.64, antiderivative size = 1173, normalized size of antiderivative = 1.00, number of steps used = 102, number of rules used = 26, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.625, Rules used = {6041, 2507, 2526, 2498, 327, 212, 2505, 308, 2520, 12, 6131, 6055, 2449, 2352, 6874, 209, 30, 2637, 213, 6139, 6057, 2497, 5048, 4966, 5040, 4964} \begin {gather*} \frac {1}{20} \left (2 a-b \log \left (1-c x^2\right )\right )^2 x^5+\frac {1}{20} b^2 \log ^2\left (c x^2+1\right ) x^5-\frac {2}{25} a b x^5+\frac {1}{25} b^2 \log \left (1-c x^2\right ) x^5+\frac {1}{25} b \left (2 a-b \log \left (1-c x^2\right )\right ) x^5+\frac {1}{5} a b \log \left (c x^2+1\right ) x^5-\frac {1}{10} b^2 \log \left (1-c x^2\right ) \log \left (c x^2+1\right ) x^5-\frac {b^2 \log \left (1-c x^2\right ) x^3}{15 c}+\frac {b \left (2 a-b \log \left (1-c x^2\right )\right ) x^3}{15 c}+\frac {2 b^2 \log \left (c x^2+1\right ) x^3}{15 c}+\frac {2 a b x^3}{15 c}+\frac {8 b^2 x}{15 c^2}+\frac {i b^2 \text {ArcTan}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \text {ArcTan}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {2 a b \text {ArcTan}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {b^2 \text {ArcTan}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}+1\right )}{10 c^{5/2}}-\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*ArcTanh[c*x^2])^2,x]

[Out]

(8*b^2*x)/(15*c^2) + (2*a*b*x^3)/(15*c) - (2*a*b*x^5)/25 + (2*a*b*ArcTan[Sqrt[c]*x])/(5*c^(5/2)) - (4*b^2*ArcT
an[Sqrt[c]*x])/(15*c^(5/2)) + ((I/5)*b^2*ArcTan[Sqrt[c]*x]^2)/c^(5/2) - (4*b^2*ArcTanh[Sqrt[c]*x])/(15*c^(5/2)
) - (b^2*ArcTanh[Sqrt[c]*x]^2)/(5*c^(5/2)) + (2*b^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) - (
2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 + I)*(1 - Sqrt[
c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) + (2*b^2*ArcTan[Sqrt[c]*x]*Log[2/(1 + I*Sqrt[c]*x)])/(5*c^(5/2)) - (2*b
^2*ArcTanh[Sqrt[c]*x]*Log[2/(1 + Sqrt[c]*x)])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(-2*Sqrt[c]*(1 - Sqrt[
-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTanh[Sqrt[c]*x]*Log[(2*Sqrt[c]*(1 + Sqr
t[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(5*c^(5/2)) + (b^2*ArcTan[Sqrt[c]*x]*Log[((1 - I)*(1 + Sqrt
[c]*x))/(1 - I*Sqrt[c]*x)])/(5*c^(5/2)) - (b^2*x^3*Log[1 - c*x^2])/(15*c) + (b^2*x^5*Log[1 - c*x^2])/25 - (b^2
*ArcTan[Sqrt[c]*x]*Log[1 - c*x^2])/(5*c^(5/2)) + (b*x^3*(2*a - b*Log[1 - c*x^2]))/(15*c) + (b*x^5*(2*a - b*Log
[1 - c*x^2]))/25 - (b*ArcTanh[Sqrt[c]*x]*(2*a - b*Log[1 - c*x^2]))/(5*c^(5/2)) + (x^5*(2*a - b*Log[1 - c*x^2])
^2)/20 + (2*b^2*x^3*Log[1 + c*x^2])/(15*c) + (a*b*x^5*Log[1 + c*x^2])/5 + (b^2*ArcTan[Sqrt[c]*x]*Log[1 + c*x^2
])/(5*c^(5/2)) - (b^2*ArcTanh[Sqrt[c]*x]*Log[1 + c*x^2])/(5*c^(5/2)) - (b^2*x^5*Log[1 - c*x^2]*Log[1 + c*x^2])
/10 + (b^2*x^5*Log[1 + c*x^2]^2)/20 + (b^2*PolyLog[2, 1 - 2/(1 - Sqrt[c]*x)])/(5*c^(5/2)) + ((I/5)*b^2*PolyLog
[2, 1 - 2/(1 - I*Sqrt[c]*x)])/c^(5/2) - ((I/10)*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]*x))/(1 - I*Sqrt[c]*x)
])/c^(5/2) + ((I/5)*b^2*PolyLog[2, 1 - 2/(1 + I*Sqrt[c]*x)])/c^(5/2) + (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]*x)])
/(5*c^(5/2)) - (b^2*PolyLog[2, 1 + (2*Sqrt[c]*(1 - Sqrt[-c]*x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c
^(5/2)) - (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1 + Sqrt[-c]*x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]*x))])/(10*c^(5/2
)) - ((I/10)*b^2*PolyLog[2, 1 - ((1 - I)*(1 + Sqrt[c]*x))/(1 - I*Sqrt[c]*x)])/c^(5/2)

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 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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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 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 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 4964

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTan[c*x])^p)*(
Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[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 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 5040

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*e*(p + 1))), x] - Dist[1/(c*d), Int[(a + b*ArcTan[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b
, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[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 6041

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

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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 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 6131

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[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 6874

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

Rubi steps

\begin {align*} \int x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {1}{2} b x^4 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )+\frac {1}{4} b^2 x^4 \log ^2\left (1+c x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int x^4 \left (2 a-b \log \left (1-c x^2\right )\right )^2 \, dx-\frac {1}{2} b \int x^4 \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right ) \, dx+\frac {1}{4} b^2 \int x^4 \log ^2\left (1+c x^2\right ) \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{2} b \int \left (-2 a x^4 \log \left (1+c x^2\right )+b x^4 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )\right ) \, dx-\frac {1}{5} (b c) \int \frac {x^6 \left (2 a-b \log \left (1-c x^2\right )\right )}{1-c x^2} \, dx-\frac {1}{5} \left (b^2 c\right ) \int \frac {x^6 \log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+(a b) \int x^4 \log \left (1+c x^2\right ) \, dx-\frac {1}{2} b^2 \int x^4 \log \left (1-c x^2\right ) \log \left (1+c x^2\right ) \, dx-\frac {1}{5} (b c) \int \left (-\frac {2 a-b \log \left (1-c x^2\right )}{c^3}-\frac {x^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{c^2}-\frac {x^4 \left (2 a-b \log \left (1-c x^2\right )\right )}{c}+\frac {2 a-b \log \left (1-c x^2\right )}{c^3 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{c^3}-\frac {x^2 \log \left (1+c x^2\right )}{c^2}+\frac {x^4 \log \left (1+c x^2\right )}{c}-\frac {\log \left (1+c x^2\right )}{c^3 \left (1+c x^2\right )}\right ) \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {1}{5} b \int x^4 \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx-\frac {1}{5} b^2 \int x^4 \log \left (1+c x^2\right ) \, dx+\frac {1}{2} b^2 \int \frac {2 c x^6 \log \left (1-c x^2\right )}{5+5 c x^2} \, dx+\frac {1}{2} b^2 \int \frac {2 c x^6 \log \left (1+c x^2\right )}{-5+5 c x^2} \, dx+\frac {b \int \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx}{5 c^2}-\frac {b \int \frac {2 a-b \log \left (1-c x^2\right )}{1-c x^2} \, dx}{5 c^2}-\frac {b^2 \int \log \left (1+c x^2\right ) \, dx}{5 c^2}+\frac {b^2 \int \frac {\log \left (1+c x^2\right )}{1+c x^2} \, dx}{5 c^2}+\frac {b \int x^2 \left (2 a-b \log \left (1-c x^2\right )\right ) \, dx}{5 c}+\frac {b^2 \int x^2 \log \left (1+c x^2\right ) \, dx}{5 c}-\frac {1}{5} (2 a b c) \int \frac {x^6}{1+c x^2} \, dx\\ &=\frac {2 a b x}{5 c^2}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac {b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1-c x^2} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1+c x^2} \, dx-\frac {b^2 \int \log \left (1-c x^2\right ) \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {x^2}{1+c x^2} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx}{5 c}+\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx}{5 c}-\frac {1}{5} (2 a b c) \int \left (\frac {1}{c^3}-\frac {x^2}{c^2}+\frac {x^4}{c}-\frac {1}{c^3 \left (1+c x^2\right )}\right ) \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1-c x^2} \, dx+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1+c x^2} \, dx+\left (b^2 c\right ) \int \frac {x^6 \log \left (1-c x^2\right )}{5+5 c x^2} \, dx+\left (b^2 c\right ) \int \frac {x^6 \log \left (1+c x^2\right )}{-5+5 c x^2} \, dx\\ &=\frac {2 b^2 x}{5 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5-\frac {b^2 x \log \left (1-c x^2\right )}{5 c^2}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac {b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}-\frac {x^2}{c}+\frac {1}{c^2 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}+\frac {x^2}{c}+\frac {1}{c^2 \left (1+c x^2\right )}\right ) \, dx+\frac {(2 a b) \int \frac {1}{1+c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx}{5 c^{3/2}}-\frac {\left (2 b^2\right ) \int \frac {x^2}{1-c x^2} \, dx}{5 c}-\frac {1}{25} \left (2 b^2 c\right ) \int \left (-\frac {1}{c^3}-\frac {x^2}{c^2}-\frac {x^4}{c}+\frac {1}{c^3 \left (1-c x^2\right )}\right ) \, dx+\frac {1}{25} \left (2 b^2 c\right ) \int \left (\frac {1}{c^3}-\frac {x^2}{c^2}+\frac {x^4}{c}-\frac {1}{c^3 \left (1+c x^2\right )}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac {\log \left (1-c x^2\right )}{5 c^3}-\frac {x^2 \log \left (1-c x^2\right )}{5 c^2}+\frac {x^4 \log \left (1-c x^2\right )}{5 c}-\frac {\log \left (1-c x^2\right )}{c^3 \left (5+5 c x^2\right )}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{5 c^3}+\frac {x^2 \log \left (1+c x^2\right )}{5 c^2}+\frac {x^4 \log \left (1+c x^2\right )}{5 c}+\frac {\log \left (1+c x^2\right )}{c^3 \left (-5+5 c x^2\right )}\right ) \, dx\\ &=\frac {92 b^2 x}{75 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {4 b^2 x^5}{125}+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {b^2 x \log \left (1-c x^2\right )}{5 c^2}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2-\frac {b^2 x \log \left (1+c x^2\right )}{5 c^2}+\frac {b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )-\frac {1}{25} b^2 x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {1}{5} b^2 \int x^4 \log \left (1-c x^2\right ) \, dx+\frac {1}{5} b^2 \int x^4 \log \left (1+c x^2\right ) \, dx-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{15 c^2}+\frac {b^2 \int \log \left (1-c x^2\right ) \, dx}{5 c^2}+\frac {b^2 \int \log \left (1+c x^2\right ) \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{i-\sqrt {c} x} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (1-c x^2\right )}{5+5 c x^2} \, dx}{c^2}+\frac {b^2 \int \frac {\log \left (1+c x^2\right )}{-5+5 c x^2} \, dx}{c^2}-\frac {b^2 \int x^2 \log \left (1-c x^2\right ) \, dx}{5 c}+\frac {b^2 \int x^2 \log \left (1+c x^2\right ) \, dx}{5 c}\\ &=\frac {92 b^2 x}{75 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {4 b^2 x^5}{125}+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {46 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {46 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1-c x^2} \, dx-\frac {1}{15} \left (2 b^2\right ) \int \frac {x^4}{1+c x^2} \, dx-\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1-\sqrt {c} x}\right )}{1-c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1+i \sqrt {c} x}\right )}{1+c x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {x^2}{1-c x^2} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {x^2}{1+c x^2} \, dx}{5 c}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{5 \sqrt {c} \left (1-c x^2\right )} \, dx}{c}-\frac {\left (2 b^2\right ) \int -\frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{5 \sqrt {c} \left (1+c x^2\right )} \, dx}{c}+\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1-c x^2} \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \frac {x^6}{1+c x^2} \, dx\\ &=\frac {32 b^2 x}{75 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {4 b^2 x^5}{125}+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {46 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {46 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}-\frac {x^2}{c}+\frac {1}{c^2 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{15} \left (2 b^2\right ) \int \left (-\frac {1}{c^2}+\frac {x^2}{c}+\frac {1}{c^2 \left (1+c x^2\right )}\right ) \, dx+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{5 c^2}+\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{5 c^2}-\frac {\left (2 b^2\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx}{5 c^{3/2}}+\frac {1}{25} \left (2 b^2 c\right ) \int \left (-\frac {1}{c^3}-\frac {x^2}{c^2}-\frac {x^4}{c}+\frac {1}{c^3 \left (1-c x^2\right )}\right ) \, dx-\frac {1}{25} \left (2 b^2 c\right ) \int \left (\frac {1}{c^3}-\frac {x^2}{c^2}+\frac {x^4}{c}-\frac {1}{c^3 \left (1+c x^2\right )}\right ) \, dx\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {16 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {16 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{75 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{25 c^2}+\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{25 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1-c x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \frac {1}{1+c x^2} \, dx}{15 c^2}-\frac {\left (2 b^2\right ) \int \left (\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1-\sqrt {c} x\right )}-\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1+\sqrt {c} x\right )}\right ) \, dx}{5 c^{3/2}}+\frac {\left (2 b^2\right ) \int \left (-\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1-\sqrt {-c} x\right )}+\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1+\sqrt {-c} x\right )}\right ) \, dx}{5 c^{3/2}}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx}{5 c^2}+\frac {b^2 \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {c} x} \, dx}{5 c^2}+\frac {b^2 \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {-c} x} \, dx}{5 \sqrt {-c} c^{3/2}}-\frac {b^2 \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {-c} x} \, dx}{5 \sqrt {-c} c^{3/2}}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+2 \frac {b^2 \int \frac {\log \left (\frac {2}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx}{5 c^2}+2 \frac {b^2 \int \frac {\log \left (\frac {2}{1+\sqrt {c} x}\right )}{1-c x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (-\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {\log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx}{5 c^2}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{10 c^{5/2}}-\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+2 \frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+2 \frac {b^2 \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt {c} x}\right )}{5 c^{5/2}}\\ &=\frac {8 b^2 x}{15 c^2}+\frac {2 a b x^3}{15 c}-\frac {2}{25} a b x^5+\frac {2 a b \tan ^{-1}\left (\sqrt {c} x\right )}{5 c^{5/2}}-\frac {4 b^2 \tan ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}+\frac {i b^2 \tan ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}-\frac {4 b^2 \tanh ^{-1}\left (\sqrt {c} x\right )}{15 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right )^2}{5 c^{5/2}}+\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {2 b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {2 b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{5 c^{5/2}}+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 x^3 \log \left (1-c x^2\right )}{15 c}+\frac {1}{25} b^2 x^5 \log \left (1-c x^2\right )-\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )}{5 c^{5/2}}+\frac {b x^3 \left (2 a-b \log \left (1-c x^2\right )\right )}{15 c}+\frac {1}{25} b x^5 \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {b \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )}{5 c^{5/2}}+\frac {1}{20} x^5 \left (2 a-b \log \left (1-c x^2\right )\right )^2+\frac {2 b^2 x^3 \log \left (1+c x^2\right )}{15 c}+\frac {1}{5} a b x^5 \log \left (1+c x^2\right )+\frac {b^2 \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {b^2 \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )}{5 c^{5/2}}-\frac {1}{10} b^2 x^5 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+c x^2\right )+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )}{5 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )}{5 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )}{5 c^{5/2}}+\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+\sqrt {c} x}\right )}{5 c^{5/2}}-\frac {b^2 \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{10 c^{5/2}}-\frac {b^2 \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{10 c^{5/2}}-\frac {i b^2 \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{10 c^{5/2}}\\ \end {align*}

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Mathematica [F]
time = 8.45, size = 0, normalized size = 0.00 \begin {gather*} \int x^4 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^4*(a + b*ArcTanh[c*x^2])^2,x]

[Out]

Integrate[x^4*(a + b*ArcTanh[c*x^2])^2, x]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{4} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*arctanh(c*x^2))^2,x)

[Out]

int(x^4*(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^4*(a+b*arctanh(c*x^2))^2,x, algorithm="maxima")

[Out]

1/5*a^2*x^5 + 1/15*(6*x^5*arctanh(c*x^2) + c*(4*x^3/c^2 + 6*arctan(sqrt(c)*x)/c^(7/2) + 3*log((c*x - sqrt(c))/
(c*x + sqrt(c)))/c^(7/2)))*a*b + 1/20*(x^5*log(-c*x^2 + 1)^2 - 5*integrate(-1/5*(5*(c*x^6 - x^4)*log(c*x^2 + 1
)^2 - 2*(2*c*x^6 + 5*(c*x^6 - x^4)*log(c*x^2 + 1))*log(-c*x^2 + 1))/(c*x^2 - 1), 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^4*(a+b*arctanh(c*x^2))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^4*arctanh(c*x^2)^2 + 2*a*b*x^4*arctanh(c*x^2) + a^2*x^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{4} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*atanh(c*x**2))**2,x)

[Out]

Integral(x**4*(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^4*(a+b*arctanh(c*x^2))^2,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^2) + a)^2*x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*atanh(c*x^2))^2,x)

[Out]

int(x^4*(a + b*atanh(c*x^2))^2, x)

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