Optimal. Leaf size=49 \[ -\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a} \]
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Rubi [A]
time = 0.09, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6113, 6181,
5556, 3379} \begin {gather*} -\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 5556
Rule 6113
Rule 6181
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)^2} \, dx &=-\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+(4 a) \int \frac {x}{\left (1-a^2 x^2\right )^3 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {4 \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {4 \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}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (4 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {\text {Subst}\left (\int \frac {\sinh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (2 \tanh ^{-1}(a x)\right )}{a}+\frac {\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 43, normalized size = 0.88 \begin {gather*} \frac {-\frac {2}{\left (-1+a^2 x^2\right )^2 \tanh ^{-1}(a x)}+2 \text {Shi}\left (2 \tanh ^{-1}(a x)\right )+\text {Shi}\left (4 \tanh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.70, size = 60, normalized size = 1.22
method | result | size |
derivativedivides | \(\frac {-\frac {3}{8 \arctanh \left (a x \right )}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{8 \arctanh \left (a x \right )}+\frac {\hyperbolicSineIntegral \left (4 \arctanh \left (a x \right )\right )}{2}}{a}\) | \(60\) |
default | \(\frac {-\frac {3}{8 \arctanh \left (a x \right )}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\hyperbolicSineIntegral \left (2 \arctanh \left (a x \right )\right )-\frac {\cosh \left (4 \arctanh \left (a x \right )\right )}{8 \arctanh \left (a x \right )}+\frac {\hyperbolicSineIntegral \left (4 \arctanh \left (a x \right )\right )}{2}}{a}\) | \(60\) |
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. 222 vs.
\(2 (46) = 92\).
time = 0.37, size = 222, normalized size = 4.53 \begin {gather*} \frac {{\left ({\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (\frac {a^{2} x^{2} + 2 \, a x + 1}{a^{2} x^{2} - 2 \, a x + 1}\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 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 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) - 2 \, {\left (a^{4} x^{4} - 2 \, 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 ) - 8}{4 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\right )} \log \left (-\frac {a x + 1}{a x - 1}\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 {1}{a^{6} x^{6} \operatorname {atanh}^{2}{\left (a x \right )} - 3 a^{4} x^{4} \operatorname {atanh}^{2}{\left (a x \right )} + 3 a^{2} x^{2} \operatorname {atanh}^{2}{\left (a x \right )} - \operatorname {atanh}^{2}{\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 {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\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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