Optimal. Leaf size=29 \[ -\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^4} \]
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Rubi [A]
time = 0.09, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6181, 5556,
3379} \begin {gather*} \frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5556
Rule 6181
Rubi steps
\begin {align*} \int \frac {x^3}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 x}+\frac {\sinh (4 x)}{8 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a^4}\\ &=\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{8 a^4}-\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a^4}\\ &=-\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{4 a^4}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 24, normalized size = 0.83 \begin {gather*} \frac {-2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{8 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.96, size = 24, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {\frac {\hyperbolicSineIntegral \left (4 \arctanh \left (a x \right )\right )}{8}-\frac {\hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{4}}{a^{4}}\) | \(24\) |
default | \(\frac {\frac {\hyperbolicSineIntegral \left (4 \arctanh \left (a x \right )\right )}{8}-\frac {\hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{4}}{a^{4}}\) | \(24\) |
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. 102 vs.
\(2 (25) = 50\).
time = 0.41, size = 102, normalized size = 3.52 \begin {gather*} \frac {\operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - \operatorname {log\_integral}\left (\frac {a^{2} x^{2} - 2 \, a x + 1}{a^{2} x^{2} + 2 \, a x + 1}\right ) - 2 \, \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + 2 \, \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )}{16 \, a^{4}} \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}}{a^{6} x^{6} \operatorname {atanh}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}{\left (a x \right )} - \operatorname {atanh}{\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 = -0.03 \begin {gather*} -\int \frac {x^3}{\mathrm {atanh}\left (a\,x\right )\,{\left (a^2\,x^2-1\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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