Optimal. Leaf size=197 \[ -\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+2 i a \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )-2 i a \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )-2 i a \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )+2 i a \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )-4 a \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)-2 a \tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.38, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6014, 6008, 6018, 5952, 4180, 2531, 2282, 6589} \[ 2 a \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 a \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+2 i a \tanh ^{-1}(a x) \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )-2 i a \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )-2 i a \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )+2 i a \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-4 a \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)-2 a \tanh ^{-1}(a x)^2 \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 2531
Rule 4180
Rule 5952
Rule 6008
Rule 6014
Rule 6018
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x^2} \, dx &=-\left (a^2 \int \frac {\tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {\tanh ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-a \operatorname {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )+(2 a) \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-2 a \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+(2 i a) \operatorname {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )-(2 i a) \operatorname {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-2 a \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 i a \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )-2 i a \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-(2 i a) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )+(2 i a) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-2 a \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 i a \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )-2 i a \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-(2 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )+(2 i a) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x}-2 a \tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^2-4 a \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+2 i a \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )-2 i a \tanh ^{-1}(a x) \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )+2 a \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 a \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-2 i a \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )+2 i a \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.78, size = 223, normalized size = 1.13 \[ a \left (-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{a x}+2 i \tanh ^{-1}(a x) \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-2 i \tanh ^{-1}(a x) \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+2 \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-2 \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )+2 i \text {Li}_3\left (-i e^{-\tanh ^{-1}(a x)}\right )-2 i \text {Li}_3\left (i e^{-\tanh ^{-1}(a x)}\right )+i \tanh ^{-1}(a x)^2 \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-i \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+2 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-2 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {\arctanh \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2}}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________