Optimal. Leaf size=64 \[ -\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6153, 6179,
6181, 3393, 3382, 6115} \begin {gather*} \frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^3}-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 6115
Rule 6153
Rule 6179
Rule 6181
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+\frac {\int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx}{a}\\ &=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx}{a^2}+\int \frac {x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {\text {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}+\frac {\text {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^3}\\ &=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a^3}\\ &=-\frac {x^2}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{a^2 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 47, normalized size = 0.73 \begin {gather*} \frac {\frac {a x \left (a x+2 \tanh ^{-1}(a x)\right )}{\left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^2}+2 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.86, size = 51, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )}{a^{3}}\) | \(51\) |
default | \(\frac {\frac {1}{4 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicCosineIntegral \left (2 \arctanh \left (a x \right )\right )}{a^{3}}\) | \(51\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (59) = 118\).
time = 0.44, size = 131, normalized size = 2.05 \begin {gather*} \frac {4 \, a^{2} x^{2} + 4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}}{2 \, {\left (a^{5} x^{2} - a^{3}\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \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^{2}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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