3.2.79 \(\int \frac {(a+b \tanh ^{-1}(\frac {c}{x^2}))^2}{x^2} \, dx\) [179]

Optimal. Leaf size=1117 \[ \frac {2 a b}{x}-\frac {2 a b \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {2 b^2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {2 b^2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {2 b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {i b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2}{\sqrt {c}}+\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2}{\sqrt {c}}+\frac {2 b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{\sqrt {c}}-\frac {b^2 \log \left (1-\frac {c}{x^2}\right )}{x}+\frac {b^2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )}{\sqrt {c}}-\frac {b \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{\sqrt {c}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{4 x}-\frac {a b \log \left (1+\frac {c}{x^2}\right )}{x}+\frac {b^2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{\sqrt {c}}+\frac {b^2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )}{\sqrt {c}}+\frac {b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )}{2 x}-\frac {b^2 \log ^2\left (1+\frac {c}{x^2}\right )}{4 x}+\frac {2 b^2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2}{1-\frac {i \sqrt {c}}{x}}\right )}{\sqrt {c}}-\frac {b^2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (1-\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right )}{\sqrt {c}}+\frac {2 b^2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2}{1+\frac {\sqrt {c}}{x}}\right )}{\sqrt {c}}-\frac {b^2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\frac {\sqrt {c}}{x}\right )}\right )}{\sqrt {c}}-\frac {b^2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (1+\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\frac {\sqrt {c}}{x}\right )}\right )}{\sqrt {c}}-\frac {b^2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (1+\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right )}{\sqrt {c}}-\frac {2 b^2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{\sqrt {c}}-\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1-\frac {i \sqrt {c}}{x}}\right )}{\sqrt {c}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right )}{2 \sqrt {c}}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {c}}{x}}\right )}{\sqrt {c}}+\frac {b^2 \text {PolyLog}\left (2,1+\frac {2 \sqrt {c} \left (1-\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\frac {\sqrt {c}}{x}\right )}\right )}{2 \sqrt {c}}+\frac {b^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (1+\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\frac {\sqrt {c}}{x}\right )}\right )}{2 \sqrt {c}}+\frac {i b^2 \text {PolyLog}\left (2,1-\frac {(1-i) \left (1+\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right )}{2 \sqrt {c}}-\frac {i b^2 \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )}{\sqrt {c}}+\frac {b^2 \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )}{\sqrt {c}} \]

[Out]

-a*b*ln(1+c/x^2)/x+b^2*arccot(x/c^(1/2))*ln(1-c/x^2)/c^(1/2)+1/2*b^2*ln(1-c/x^2)*ln(1+c/x^2)/x-2*a*b*arccot(x/
c^(1/2))/c^(1/2)+2*b^2*arccot(x/c^(1/2))*ln(2/(1-I*c^(1/2)/x))/c^(1/2)+2*b^2*arccoth(x/c^(1/2))*ln(2/(1+1/x*c^
(1/2)))/c^(1/2)+2*b^2*arctan(x/c^(1/2))*ln(2-2*c^(1/2)/(-I*x+c^(1/2)))/c^(1/2)-2*b^2*arctanh(x/c^(1/2))*ln(2-2
*c^(1/2)/(x+c^(1/2)))/c^(1/2)-I*b^2*arctan(x/c^(1/2))^2/c^(1/2)-I*b^2*polylog(2,-1+2*c^(1/2)/(-I*x+c^(1/2)))/c
^(1/2)-I*b^2*polylog(2,1-2/(1-I*c^(1/2)/x))/c^(1/2)+b*arctanh(x/c^(1/2))*(2*a-b*ln(1-c/x^2))/c^(1/2)+b^2*arcco
th(x/c^(1/2))*ln(1+c/x^2)/c^(1/2)+b^2*arctan(x/c^(1/2))*ln(1+c/x^2)/c^(1/2)-b^2*arccot(x/c^(1/2))*ln((1+I)*(1-
1/x*c^(1/2))/(1-I*c^(1/2)/x))/c^(1/2)-b^2*arccoth(x/c^(1/2))*ln(-2*(1-(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/2)-c^(1/2
))/(1+1/x*c^(1/2)))/c^(1/2)-b^2*arccoth(x/c^(1/2))*ln(2*(1+(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+1/x*c
^(1/2)))/c^(1/2)-b^2*arccot(x/c^(1/2))*ln((1-I)*(1+1/x*c^(1/2))/(1-I*c^(1/2)/x))/c^(1/2)+1/2*I*b^2*polylog(2,1
-(1+I)*(1-1/x*c^(1/2))/(1-I*c^(1/2)/x))/c^(1/2)+1/2*I*b^2*polylog(2,1+(-1+I)*(1+1/x*c^(1/2))/(1-I*c^(1/2)/x))/
c^(1/2)+b^2*polylog(2,-1+2*c^(1/2)/(x+c^(1/2)))/c^(1/2)-b^2*polylog(2,1-2/(1+1/x*c^(1/2)))/c^(1/2)+2*a*b/x-1/4
*b^2*ln(1+c/x^2)^2/x-2*b^2*arccot(x/c^(1/2))/c^(1/2)-2*b^2*arccoth(x/c^(1/2))/c^(1/2)-2*b^2*arctan(x/c^(1/2))/
c^(1/2)+2*b^2*arctanh(x/c^(1/2))/c^(1/2)+1/2*b^2*polylog(2,1+2*(1-(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/2)-c^(1/2))/(
1+1/x*c^(1/2)))/c^(1/2)+1/2*b^2*polylog(2,1-2*(1+(-c)^(1/2)/x)*c^(1/2)/((-c)^(1/2)+c^(1/2))/(1+1/x*c^(1/2)))/c
^(1/2)-b^2*ln(1-c/x^2)/x-b*(2*a-b*ln(1-c/x^2))/x-b^2*arctanh(x/c^(1/2))^2/c^(1/2)-1/4*(2*a-b*ln(1-c/x^2))^2/x

________________________________________________________________________________________

Rubi [A]
time = 1.53, antiderivative size = 1117, normalized size of antiderivative = 1.00, number of steps used = 72, number of rules used = 30, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.875, Rules used = {6045, 6042, 2507, 2526, 2505, 269, 331, 213, 212, 2520, 12, 266, 6820, 6135, 6079, 2497, 6847, 2498, 327, 6874, 209, 2636, 6139, 6057, 2449, 2352, 5048, 4966, 5044, 4988} \begin {gather*} -\frac {i \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2 b^2}{\sqrt {c}}-\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2 b^2}{\sqrt {c}}-\frac {\log ^2\left (\frac {c}{x^2}+1\right ) b^2}{4 x}-\frac {2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}-\frac {2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}-\frac {2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}+\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b^2}{\sqrt {c}}+\frac {2 \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right ) b^2}{\sqrt {c}}+\frac {\cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right ) b^2}{\sqrt {c}}-\frac {\log \left (1-\frac {c}{x^2}\right ) b^2}{x}+\frac {\coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{\sqrt {c}}+\frac {\text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{\sqrt {c}}+\frac {\log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right ) b^2}{2 x}+\frac {2 \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}-\frac {\cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (1-\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}+\frac {2 \coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2}{\frac {\sqrt {c}}{x}+1}\right ) b^2}{\sqrt {c}}-\frac {\coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}\right ) b^2}{\sqrt {c}}-\frac {\coth ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (\frac {\sqrt {-c}}{x}+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}\right ) b^2}{\sqrt {c}}-\frac {\cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\frac {\sqrt {c}}{x}+1\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}-\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right ) b^2}{\sqrt {c}}-\frac {i \text {Li}_2\left (1-\frac {2}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{\sqrt {c}}+\frac {i \text {Li}_2\left (1-\frac {(1+i) \left (1-\frac {\sqrt {c}}{x}\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{2 \sqrt {c}}-\frac {\text {Li}_2\left (1-\frac {2}{\frac {\sqrt {c}}{x}+1}\right ) b^2}{\sqrt {c}}+\frac {\text {Li}_2\left (\frac {2 \sqrt {c} \left (1-\frac {\sqrt {-c}}{x}\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}+1\right ) b^2}{2 \sqrt {c}}+\frac {\text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\frac {\sqrt {-c}}{x}+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\frac {\sqrt {c}}{x}+1\right )}\right ) b^2}{2 \sqrt {c}}+\frac {i \text {Li}_2\left (1-\frac {(1-i) \left (\frac {\sqrt {c}}{x}+1\right )}{1-\frac {i \sqrt {c}}{x}}\right ) b^2}{2 \sqrt {c}}-\frac {i \text {Li}_2\left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}-1\right ) b^2}{\sqrt {c}}+\frac {\text {Li}_2\left (\frac {2 \sqrt {c}}{x+\sqrt {c}}-1\right ) b^2}{\sqrt {c}}-\frac {2 a \cot ^{-1}\left (\frac {x}{\sqrt {c}}\right ) b}{\sqrt {c}}+\frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{\sqrt {c}}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) b}{x}-\frac {a \log \left (\frac {c}{x^2}+1\right ) b}{x}+\frac {2 a b}{x}-\frac {\left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])^2/x^2,x]

[Out]

(2*a*b)/x - (2*a*b*ArcCot[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcCot[x/Sqrt[c]])/Sqrt[c] - (2*b^2*ArcCoth[x/Sqrt[c]])
/Sqrt[c] - (2*b^2*ArcTan[x/Sqrt[c]])/Sqrt[c] - (I*b^2*ArcTan[x/Sqrt[c]]^2)/Sqrt[c] + (2*b^2*ArcTanh[x/Sqrt[c]]
)/Sqrt[c] - (b^2*ArcTanh[x/Sqrt[c]]^2)/Sqrt[c] + (2*b^2*ArcTan[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] - I*x)]
)/Sqrt[c] - (b^2*Log[1 - c/x^2])/x + (b^2*ArcCot[x/Sqrt[c]]*Log[1 - c/x^2])/Sqrt[c] - (b*(2*a - b*Log[1 - c/x^
2]))/x + (b*ArcTanh[x/Sqrt[c]]*(2*a - b*Log[1 - c/x^2]))/Sqrt[c] - (2*a - b*Log[1 - c/x^2])^2/(4*x) - (a*b*Log
[1 + c/x^2])/x + (b^2*ArcCoth[x/Sqrt[c]]*Log[1 + c/x^2])/Sqrt[c] + (b^2*ArcTan[x/Sqrt[c]]*Log[1 + c/x^2])/Sqrt
[c] + (b^2*Log[1 - c/x^2]*Log[1 + c/x^2])/(2*x) - (b^2*Log[1 + c/x^2]^2)/(4*x) + (2*b^2*ArcCot[x/Sqrt[c]]*Log[
2/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (b^2*ArcCot[x/Sqrt[c]]*Log[((1 + I)*(1 - Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/S
qrt[c] + (2*b^2*ArcCoth[x/Sqrt[c]]*Log[2/(1 + Sqrt[c]/x)])/Sqrt[c] - (b^2*ArcCoth[x/Sqrt[c]]*Log[(-2*Sqrt[c]*(
1 - Sqrt[-c]/x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]/x))])/Sqrt[c] - (b^2*ArcCoth[x/Sqrt[c]]*Log[(2*Sqrt[c]*(1
+ Sqrt[-c]/x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]/x))])/Sqrt[c] - (b^2*ArcCot[x/Sqrt[c]]*Log[((1 - I)*(1 + Sqr
t[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (2*b^2*ArcTanh[x/Sqrt[c]]*Log[2 - (2*Sqrt[c])/(Sqrt[c] + x)])/Sqrt[c]
- (I*b^2*PolyLog[2, 1 - 2/(1 - (I*Sqrt[c])/x)])/Sqrt[c] + ((I/2)*b^2*PolyLog[2, 1 - ((1 + I)*(1 - Sqrt[c]/x))
/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (b^2*PolyLog[2, 1 - 2/(1 + Sqrt[c]/x)])/Sqrt[c] + (b^2*PolyLog[2, 1 + (2*Sqrt
[c]*(1 - Sqrt[-c]/x))/((Sqrt[-c] - Sqrt[c])*(1 + Sqrt[c]/x))])/(2*Sqrt[c]) + (b^2*PolyLog[2, 1 - (2*Sqrt[c]*(1
+ Sqrt[-c]/x))/((Sqrt[-c] + Sqrt[c])*(1 + Sqrt[c]/x))])/(2*Sqrt[c]) + ((I/2)*b^2*PolyLog[2, 1 - ((1 - I)*(1 +
Sqrt[c]/x))/(1 - (I*Sqrt[c])/x)])/Sqrt[c] - (I*b^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] - I*x)])/Sqrt[c] + (b
^2*PolyLog[2, -1 + (2*Sqrt[c])/(Sqrt[c] + x)])/Sqrt[c]

Rule 12

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

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 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 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 2636

Int[Log[v_]*Log[w_], x_Symbol] :> Simp[x*Log[v]*Log[w], x] + (-Int[SimplifyIntegrand[x*Log[w]*(D[v, x]/v), x],
x] - Int[SimplifyIntegrand[x*Log[v]*(D[w, x]/w), x], x]) /; InverseFunctionFreeQ[v, x] && InverseFunctionFree
Q[w, 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 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 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6874

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

Rubi steps

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

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Mathematica [A]
time = 2.02, size = 568, normalized size = 0.51 \begin {gather*} \frac {-2 a^2-\frac {4 a b \left (\text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )-\tanh ^{-1}\left (\sqrt {\frac {c}{x^2}}\right )\right )}{\sqrt {\frac {c}{x^2}}}-4 a b \tanh ^{-1}\left (\frac {c}{x^2}\right )+\frac {b^2 \left (2 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )^2-4 \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right ) \tanh ^{-1}\left (\frac {c}{x^2}\right )-2 \sqrt {\frac {c}{x^2}} \tanh ^{-1}\left (\frac {c}{x^2}\right )^2-2 \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right ) \log \left (1+e^{4 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )}\right )-2 \tanh ^{-1}\left (\frac {c}{x^2}\right ) \log \left (1-\sqrt {\frac {c}{x^2}}\right )+\log (2) \log \left (1-\sqrt {\frac {c}{x^2}}\right )-\frac {1}{2} \log ^2\left (1-\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i+\sqrt {\frac {c}{x^2}}\right )\right )+2 \tanh ^{-1}\left (\frac {c}{x^2}\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log (2) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\frac {1}{2} \left ((1+i)-(1-i) \sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )-\log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\sqrt {\frac {c}{x^2}}\right )\right ) \log \left (1+\sqrt {\frac {c}{x^2}}\right )+\frac {1}{2} \log ^2\left (1+\sqrt {\frac {c}{x^2}}\right )+\log \left (1-\sqrt {\frac {c}{x^2}}\right ) \log \left (\frac {1}{2} \left ((1+i)+(1-i) \sqrt {\frac {c}{x^2}}\right )\right )+\frac {1}{2} i \text {PolyLog}\left (2,-e^{4 i \text {ArcTan}\left (\sqrt {\frac {c}{x^2}}\right )}\right )-\text {PolyLog}\left (2,\frac {1}{2} \left (1-\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\sqrt {\frac {c}{x^2}}\right )\right )+\text {PolyLog}\left (2,\frac {1}{2} \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )-\text {PolyLog}\left (2,\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\sqrt {\frac {c}{x^2}}\right )\right )\right )}{\sqrt {\frac {c}{x^2}}}}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2 - (4*a*b*(ArcTan[Sqrt[c/x^2]] - ArcTanh[Sqrt[c/x^2]]))/Sqrt[c/x^2] - 4*a*b*ArcTanh[c/x^2] + (b^2*((2*I
)*ArcTan[Sqrt[c/x^2]]^2 - 4*ArcTan[Sqrt[c/x^2]]*ArcTanh[c/x^2] - 2*Sqrt[c/x^2]*ArcTanh[c/x^2]^2 - 2*ArcTan[Sqr
t[c/x^2]]*Log[1 + E^((4*I)*ArcTan[Sqrt[c/x^2]])] - 2*ArcTanh[c/x^2]*Log[1 - Sqrt[c/x^2]] + Log[2]*Log[1 - Sqrt
[c/x^2]] - Log[1 - Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/x^2]]*Log[(1/2 + I/2)*(-I + Sqrt[c/x^2])] + 2*ArcTanh[c/x
^2]*Log[1 + Sqrt[c/x^2]] - Log[2]*Log[1 + Sqrt[c/x^2]] - Log[((1 + I) - (1 - I)*Sqrt[c/x^2])/2]*Log[1 + Sqrt[c
/x^2]] - Log[(-1/2 - I/2)*(I + Sqrt[c/x^2])]*Log[1 + Sqrt[c/x^2]] + Log[1 + Sqrt[c/x^2]]^2/2 + Log[1 - Sqrt[c/
x^2]]*Log[((1 + I) + (1 - I)*Sqrt[c/x^2])/2] + (I/2)*PolyLog[2, -E^((4*I)*ArcTan[Sqrt[c/x^2]])] - PolyLog[2, (
1 - Sqrt[c/x^2])/2] + PolyLog[2, (-1/2 - I/2)*(-1 + Sqrt[c/x^2])] + PolyLog[2, (-1/2 + I/2)*(-1 + Sqrt[c/x^2])
] + PolyLog[2, (1 + Sqrt[c/x^2])/2] - PolyLog[2, (1/2 - I/2)*(1 + Sqrt[c/x^2])] - PolyLog[2, (1/2 + I/2)*(1 +
Sqrt[c/x^2])]))/Sqrt[c/x^2])/(2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

integral((b^2*arctanh(c/x^2)^2 + 2*a*b*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 \frac {\left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )^{2}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c/x**2))**2/x**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((a+b*arctanh(c/x^2))^2/x^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 \frac {{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^2}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))^2/x^2,x)

[Out]

int((a + b*atanh(c/x^2))^2/x^2, x)

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