Optimal. Leaf size=243 \[ -\frac {a \sqrt {1-a^2 x^2}}{30 x^5}+\frac {19 a^3 \sqrt {1-a^2 x^2}}{360 x^3}+\frac {31 a^5 \sqrt {1-a^2 x^2}}{720 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {7 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}-\frac {a^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{16 x^2}-\frac {1}{8} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{16} a^6 \text {PolyLog}\left (2,-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{16} a^6 \text {PolyLog}\left (2,\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \]
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Rubi [A]
time = 0.55, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6161, 6157,
6173, 277, 270, 6165} \begin {gather*} \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 {7 a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{24 x^4}+\frac {31 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 {19 a^3 \sqrt {1-a^2 x^2}}{360 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rule 6157
Rule 6161
Rule 6165
Rule 6173
Rubi steps
\begin {align*} \int \frac {\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}{x^7} \, dx &=-\left (a^2 \int \frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{x^5} \, dx\right )+\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 {a^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 x^4}-\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 {1}{3} a^2 \int \frac {\tanh ^{-1}(a x)}{x^5 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{3} a^3 \int \frac {1}{x^4 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {a \sqrt {1-a^2 x^2}}{25 x^5}+\frac {a^3 \sqrt {1-a^2 x^2}}{9 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)}{4 x^4}-\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}{12} a^3 \int \frac {1}{x^4 \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 {1}{4} a^4 \int \frac {\tanh ^{-1}(a x)}{x^3 \sqrt {1-a^2 x^2}} \, dx-\frac {1}{9} \left (2 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 {3 a^3 \sqrt {1-a^2 x^2}}{100 x^3}+\frac {2 a^5 \sqrt {1-a^2 x^2}}{9 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {7 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)}{8 x^2}-\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}{18} a^5 \int \frac {1}{x^2 \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 {1}{8} a^5 \int \frac {1}{x^2 \sqrt {1-a^2 x^2}} \, dx+\frac {1}{8} 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 {19 a^3 \sqrt {1-a^2 x^2}}{360 x^3}-\frac {13 a^5 \sqrt {1-a^2 x^2}}{200 x}-\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{6 x^6}+\frac {7 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}{4} a^6 \tanh ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )+\frac {1}{8} a^6 \text {Li}_2\left (-\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\frac {1}{8} a^6 \text {Li}_2\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right )-\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 {19 a^3 \sqrt {1-a^2 x^2}}{360 x^3}+\frac {31 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 {7 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 = 4.46, size = 474, normalized size = 1.95 \begin {gather*} \frac {82 a^7 x^4 \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+90 a^6 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \text {csch}^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+4 a^7 x^4 \text {csch}^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-3 a^7 x^4 \text {csch}^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-15 a^6 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \text {csch}^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+360 a^6 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \log \left (1-e^{-\tanh ^{-1}(a x)}\right )-360 a^6 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \log \left (1+e^{-\tanh ^{-1}(a x)}\right )+360 a^6 x^3 \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{-\tanh ^{-1}(a x)}\right )-360 a^6 x^3 \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,e^{-\tanh ^{-1}(a x)}\right )+328 a^5 x^2 \left (-1+a^2 x^2\right ) \sinh ^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+360 a^4 x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \sinh ^2\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+64 a^3 \sinh ^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-128 a^5 x^2 \sinh ^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )+64 a^7 x^4 \sinh ^4\left (\frac {1}{2} \tanh ^{-1}(a x)\right )-\frac {192 a \left (-1+a^2 x^2\right )^3 \sinh ^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{x^2}-\frac {960 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x) \sinh ^6\left (\frac {1}{2} \tanh ^{-1}(a x)\right )}{x^3}}{5760 x^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.57, size = 184, normalized size = 0.76
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (-31 a^{5} x^{5}+45 a^{4} x^{4} \arctanh \left (a x \right )-38 a^{3} x^{3}-210 a^{2} x^{2} \arctanh \left (a x \right )+24 a x +120 \arctanh \left (a x \right )\right )}{720 x^{6}}+\frac {a^{6} \arctanh \left (a x \right ) \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}+\frac {a^{6} \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \arctanh \left (a x \right ) \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}-\frac {a^{6} \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )}{16}\) | \(184\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}}{x^{7}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2}}{x^7} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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