Optimal. Leaf size=74 \[ -\frac {x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\text {ArcSin}(a x)}{a^4}+\frac {\tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^4} \]
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Rubi [A]
time = 0.12, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6175, 6141,
222, 197} \begin {gather*} -\frac {\text {ArcSin}(a x)}{a^4}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^4}+\frac {\tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}-\frac {x}{a^3 \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 222
Rule 6141
Rule 6175
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=\frac {\int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^2}-\frac {\int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{a^2}\\ &=\frac {\tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^4}-\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3}\\ &=-\frac {x}{a^3 \sqrt {1-a^2 x^2}}-\frac {\sin ^{-1}(a x)}{a^4}+\frac {\tanh ^{-1}(a x)}{a^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{a^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 76, normalized size = 1.03 \begin {gather*} \frac {a x \sqrt {1-a^2 x^2}+\left (1-a^2 x^2\right ) \text {ArcSin}(a x)+\sqrt {1-a^2 x^2} \left (-2+a^2 x^2\right ) \tanh ^{-1}(a x)}{a^4 \left (-1+a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.59, size = 144, normalized size = 1.95
method | result | size |
default | \(-\frac {\left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{4} \left (a x -1\right )}+\frac {\left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{4} \left (a x +1\right )}+\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \arctanh \left (a x \right )}{a^{4}}+\frac {i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{a^{4}}-\frac {i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{a^{4}}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 96, normalized size = 1.30 \begin {gather*} a {\left (\frac {\frac {x}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}}{a^{2}} - \frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} - {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\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 = 0.39, size = 94, normalized size = 1.27 \begin {gather*} \frac {4 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x + {\left (a^{2} x^{2} - 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )}}{2 \, {\left (a^{6} x^{2} - a^{4}\right )}} \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^{3} \operatorname {atanh}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, 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^3\,\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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