Optimal. Leaf size=27 \[ \frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac {\text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \]
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Rubi [A]
time = 0.07, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6115, 3393,
3382} \begin {gather*} \frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac {\text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 6115
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cosh ^3(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 x}+\frac {\cosh (3 x)}{4 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {\text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}\\ &=\frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac {\text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 22, normalized size = 0.81 \begin {gather*} \frac {3 \text {Chi}\left (\tanh ^{-1}(a x)\right )+\text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.88, size = 21, normalized size = 0.78
method | result | size |
default | \(\frac {3 \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right )+\hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right )}{4 a}\) | \(21\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \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 = -0.04 \begin {gather*} \int \frac {1}{\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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