Optimal. Leaf size=97 \[ -\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a} \]
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Rubi [A]
time = 0.10, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6113, 6143,
6181, 5556, 12, 3379} \begin {gather*} -\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {a^2 x^2+1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}+\frac {2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 5556
Rule 6113
Rule 6143
Rule 6181
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^4} \, dx &=-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}+\frac {1}{3} (2 a) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx\\ &=-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {1}{3} (4 a) \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {4 \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}\\ &=-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {4 \text {Subst}\left (\int \frac {\sinh (2 x)}{2 x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}\\ &=-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{3 a}\\ &=-\frac {1}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^3}-\frac {x}{3 \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {1+a^2 x^2}{3 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 73, normalized size = 0.75 \begin {gather*} \frac {1+a x \tanh ^{-1}(a x)+\left (1+a^2 x^2\right ) \tanh ^{-1}(a x)^2+2 \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^3 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{3 a \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.91, size = 68, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {-\frac {1}{6 \arctanh \left (a x \right )^{3}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{6 \arctanh \left (a x \right )^{3}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{6 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{3 \arctanh \left (a x \right )}+\frac {2 \hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{3}}{a}\) | \(68\) |
default | \(\frac {-\frac {1}{6 \arctanh \left (a x \right )^{3}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{6 \arctanh \left (a x \right )^{3}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{6 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{3 \arctanh \left (a x \right )}+\frac {2 \hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )}{3}}{a}\) | \(68\) |
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 [A]
time = 0.41, size = 151, normalized size = 1.56 \begin {gather*} \frac {{\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 )^{3} + 4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 8}{3 \, {\left (a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{3}} \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 {1}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{4}{\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 = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^4\,{\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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