Optimal. Leaf size=139 \[ -\frac {5 x \sqrt {1-a^2 x^2}}{24 a^5}-\frac {x^3 \sqrt {1-a^2 x^2}}{20 a^3}+\frac {89 \text {ArcSin}(a x)}{120 a^6}-\frac {8 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac {4 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2} \]
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Rubi [A]
time = 0.14, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6163, 327, 222,
6141} \begin {gather*} \frac {89 \text {ArcSin}(a x)}{120 a^6}-\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}-\frac {8 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac {5 x \sqrt {1-a^2 x^2}}{24 a^5}-\frac {4 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{20 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 6141
Rule 6163
Rubi steps
\begin {align*} \int \frac {x^5 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=-\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}+\frac {4 \int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{5 a^2}+\frac {\int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx}{5 a}\\ &=-\frac {x^3 \sqrt {1-a^2 x^2}}{20 a^3}-\frac {4 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}+\frac {8 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a^4}+\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{20 a^3}+\frac {4 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{15 a^3}\\ &=-\frac {5 x \sqrt {1-a^2 x^2}}{24 a^5}-\frac {x^3 \sqrt {1-a^2 x^2}}{20 a^3}-\frac {8 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac {4 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{40 a^5}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{15 a^5}+\frac {8 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{15 a^5}\\ &=-\frac {5 x \sqrt {1-a^2 x^2}}{24 a^5}-\frac {x^3 \sqrt {1-a^2 x^2}}{20 a^3}+\frac {89 \sin ^{-1}(a x)}{120 a^6}-\frac {8 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^6}-\frac {4 x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{5 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 79, normalized size = 0.57 \begin {gather*} -\frac {a x \sqrt {1-a^2 x^2} \left (25+6 a^2 x^2\right )-89 \text {ArcSin}(a x)+8 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \tanh ^{-1}(a x)}{120 a^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 2.71, size = 120, normalized size = 0.86
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (24 a^{4} x^{4} \arctanh \left (a x \right )+6 a^{3} x^{3}+32 a^{2} x^{2} \arctanh \left (a x \right )+25 a x +64 \arctanh \left (a x \right )\right )}{120 a^{6}}+\frac {89 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{120 a^{6}}-\frac {89 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{120 a^{6}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 163, normalized size = 1.17 \begin {gather*} -\frac {1}{120} \, a {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )}}{a^{2}} + \frac {16 \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )}}{a^{4}} - \frac {64 \, \arcsin \left (a x\right )}{a^{7}}\right )} - \frac {1}{15} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 91, normalized size = 0.65 \begin {gather*} -\frac {{\left (6 \, a^{3} x^{3} + 25 \, a x + 4 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} + 178 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{6}} \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 {x^{5} \operatorname {atanh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, 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 = -0.01 \begin {gather*} \int \frac {x^5\,\mathrm {atanh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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