Optimal. Leaf size=243 \[ -\frac {1}{16} a^6 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {1}{16} a^6 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac {a^5 \sqrt {1-a^2 x^2}}{720 x}+\frac {a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac {11 a^3 \sqrt {1-a^2 x^2}}{360 x^3} \]
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Rubi [A] time = 0.41, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6010, 6026, 271, 264, 6018} \[ -\frac {1}{16} a^6 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {1}{16} a^6 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right )+\frac {a^5 \sqrt {1-a^2 x^2}}{720 x}-\frac {11 a^3 \sqrt {1-a^2 x^2}}{360 x^3}-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}+\frac {a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 6010
Rule 6018
Rule 6026
Rubi steps
\begin {align*} \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^7} \, dx &=-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 x^6}-\frac {1}{5} \int \frac {\tanh ^{-1}(a x)}{x^7 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{5} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{25 x^5}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}-\frac {1}{30} a \int \frac {1}{x^6 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{6} a^2 \int \frac {\tanh ^{-1}(a x)}{x^5 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{25} \left (4 a^3\right ) \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}-\frac {4 a^3 \sqrt {1-a^2 x^2}}{75 x^3}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac {1}{75} \left (2 a^3\right ) \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{24} a^3 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{8} a^4 \int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{75} \left (8 a^5\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}-\frac {11 a^3 \sqrt {1-a^2 x^2}}{360 x^3}-\frac {8 a^5 \sqrt {1-a^2 x^2}}{75 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac {a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac {1}{225} \left (4 a^5\right ) \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{36} a^5 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{16} a^5 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{16} a^6 \int \frac {\tanh ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{30 x^5}-\frac {11 a^3 \sqrt {1-a^2 x^2}}{360 x^3}+\frac {a^5 \sqrt {1-a^2 x^2}}{720 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac {a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}+\frac {1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{16} a^6 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{16} a^6 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )\\ \end {align*}
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Mathematica [A] time = 3.70, size = 307, normalized size = 1.26 \[ \frac {a^6 \left (-\frac {3 a x \text {csch}^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {26 a x \text {csch}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}-\frac {416 \left (1-a^2 x^2\right )^{3/2} \sinh ^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{a^3 x^3}-360 \text {Li}_2\left (-e^{-\tanh ^{-1}(a x)}\right )+360 \text {Li}_2\left (e^{-\tanh ^{-1}(a x)}\right )+76 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-360 \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )+360 \tanh ^{-1}(a x) \log \left (e^{-\tanh ^{-1}(a x)}+1\right )-76 \coth \left (\frac {1}{2} \tanh ^{-1}(a x)\right )-15 \tanh ^{-1}(a x) \text {csch}^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-90 \tanh ^{-1}(a x) \text {csch}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-90 \tanh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-15 \tanh ^{-1}(a x) \text {sech}^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+90 \tanh ^{-1}(a x) \text {sech}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+6 \tanh \left (\frac {1}{2} \tanh ^{-1}(a x)\right ) \text {sech}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-90 \tanh ^{-1}(a x) \text {sech}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )\right )}{5760} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{7}}, x\right ) \]
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 = 183, normalized size = 0.75 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (x^{5} a^{5}+45 a^{4} x^{4} \arctanh \left (a x \right )-22 x^{3} a^{3}+30 a^{2} x^{2} \arctanh \left (a x \right )-24 a x -120 \arctanh \left (a x \right )\right )}{720 x^{6}}+\frac {a^{6} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}+\frac {a^{6} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right )}{x^{7}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}}{x^7} \,d x \]
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 \[ \int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}}{x^{7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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