Optimal. Leaf size=189 \[ -i a^3 \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )+i a^3 \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-2 a^3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac {7}{6} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right ) \]
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Rubi [A] time = 0.31, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6014, 6008, 266, 47, 63, 208, 5950} \[ -i a^3 \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )+i a^3 \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {7}{6} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 208
Rule 266
Rule 5950
Rule 6008
Rule 6014
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^4} \, dx &=-\left (a^2 \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^2} \, dx\right )+\int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^4} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}+\frac {1}{3} a \int \frac {\sqrt {1-a^2 x^2}}{x^3} \, dx-a^2 \int \frac {\tanh ^{-1}(a x)}{x^2 \sqrt {1-a^2 x^2}} \, dx+a^4 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)-i a^3 \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+i a^3 \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x^2} \, dx,x,x^2\right )-a^3 \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)-i a^3 \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+i a^3 \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{12} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {1}{2} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)-i a^3 \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+i a^3 \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )+a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{3 x^3}-2 a^3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)+\frac {7}{6} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-i a^3 \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )+i a^3 \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 1.27, size = 199, normalized size = 1.05 \[ -\frac {\left (1-a^2 x^2\right )^{3/2} \left (\log \left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right ) \left (\sinh \left (3 \tanh ^{-1}(a x)\right )-\frac {3 a x}{\sqrt {1-a^2 x^2}}\right )+8 \tanh ^{-1}(a x)+2 \sinh \left (2 \tanh ^{-1}(a x)\right )\right )}{24 x^3}-\left (a^3 \left (-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a x}+i \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-i \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+\log \left (\tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.47, size = 220, normalized size = 1.16 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (8 a^{2} x^{2} \arctanh \left (a x \right )-a x -2 \arctanh \left (a x \right )\right )}{6 x^{3}}-\frac {7 a^{3} \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-1\right )}{6}+\frac {7 a^{3} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{6}+i a^{3} \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )+i a^{3} \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )-i a^{3} \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )-i a^{3} \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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