Optimal. Leaf size=191 \[ \frac {3}{8} a^4 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{8} a^4 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{4} a^4 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {5 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}-\frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {11 a^3 \sqrt {1-a^2 x^2}}{24 x} \]
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Rubi [A] time = 0.57, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6014, 6010, 6026, 271, 264, 6018} \[ \frac {3}{8} a^4 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {3}{8} a^4 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {11 a^3 \sqrt {1-a^2 x^2}}{24 x}-\frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {5 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}-\frac {3}{4} a^4 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 6010
Rule 6014
Rule 6018
Rule 6026
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^5} \, dx &=-\left (a^2 \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^3} \, dx\right )+\int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^5} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^4}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^2}-\frac {1}{3} \int \frac {\tanh ^{-1}(a x)}{x^5 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{3} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx-a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{9 x^3}+\frac {a^3 \sqrt {1-a^2 x^2}}{x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{2 x^2}-\frac {1}{12} a \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{4} a^2 \int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{9} \left (2 a^3\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{2} a^4 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {5 a^3 \sqrt {1-a^2 x^2}}{18 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}+\frac {5 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-a^4 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{2} a^4 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{2} a^4 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{18} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{8} a^3 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{8} a^4 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{12 x^3}+\frac {11 a^3 \sqrt {1-a^2 x^2}}{24 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{4 x^4}+\frac {5 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 x^2}-\frac {3}{4} a^4 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {3}{8} a^4 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {3}{8} a^4 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 3.83, size = 282, normalized size = 1.48 \[ \frac {1}{192} a \left (72 a^3 \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-72 a^3 \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )-40 a^3 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )+72 a^3 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-72 a^3 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )+40 a^3 \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-3 a^3 \tanh ^{-1}(a x) \text {csch}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+18 a^3 \tanh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+3 a^3 \tanh ^{-1}(a x) \text {sech}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+18 a^3 \tanh ^{-1}(a x) \text {sech}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+\frac {16 a^2 \sqrt {1-a^2 x^2} \sinh ^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{x}-\frac {16 \sqrt {1-a^2 x^2} \sinh ^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{x^3}-\frac {a^4 x \text {csch}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{5}}, x\right ) \]
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 = 0.44, size = 164, normalized size = 0.86 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (11 x^{3} a^{3}+15 a^{2} x^{2} \arctanh \left (a x \right )-2 a x -6 \arctanh \left (a x \right )\right )}{24 x^{4}}-\frac {3 a^{4} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}-\frac {3 a^{4} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}+\frac {3 a^{4} \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8}+\frac {3 a^{4} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{8} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )}{x^{5}}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^5} \,d x \]
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 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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