Optimal. Leaf size=169 \[ -\frac {1}{3} a^3 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {1}{3} a^3 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {2}{3} a^3 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 x^3} \]
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Rubi [A] time = 0.30, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6008, 6010, 6026, 264, 6018} \[ -\frac {1}{3} a^3 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {1}{3} a^3 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 264
Rule 6008
Rule 6010
Rule 6018
Rule 6026
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{x^4} \, dx &=-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} (2 a) \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^3} \, dx\\ &=-\frac {2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} (2 a) \int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^2 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{3} a^3 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a^2 \sqrt {1-a^2 x^2}}{3 x}-\frac {a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^2}-\frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2}{3 x^3}+\frac {2}{3} a^3 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{3} a^3 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{3} a^3 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 2.06, size = 177, normalized size = 1.05 \[ -\frac {1}{3} a^3 \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )-\frac {\left (1-a^2 x^2\right )^{3/2} \left (\tanh ^{-1}(a x) \left (\frac {\left (\log \left (1-e^{-\tanh ^{-1}(a x)}\right )-\log \left (e^{-\tanh ^{-1}(a x)}+1\right )\right ) \left (\sqrt {1-a^2 x^2} \sinh \left (3 \tanh ^{-1}(a x)\right )-3 a x\right )}{\sqrt {1-a^2 x^2}}+2 \sinh \left (2 \tanh ^{-1}(a x)\right )\right )-\frac {4 a^3 x^3 \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )}{\left (1-a^2 x^2\right )^{3/2}}+4 \tanh ^{-1}(a x)^2+2 \left (\cosh \left (2 \tanh ^{-1}(a x)\right )-1\right )\right )}{12 x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x^{4}}, x\right ) \]
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 = 0.44, size = 171, normalized size = 1.01 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (a^{2} x^{2} \arctanh \left (a x \right )^{2}-a^{2} x^{2}-a x \arctanh \left (a x \right )-\arctanh \left (a x \right )^{2}\right )}{3 x^{3}}+\frac {a^{3} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}+\frac {a^{3} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {a^{3} \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3}-\frac {a^{3} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{3} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )^{2}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2}}{x^4} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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