Optimal. Leaf size=35 \[ -\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a} \]
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Rubi [A]
time = 0.09, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6113, 6181,
3379} \begin {gather*} \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a}-\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 6113
Rule 6181
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx &=-\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+a \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 32, normalized size = 0.91 \begin {gather*} \frac {-\frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 62, normalized size = 1.77
method | result | size |
default | \(\frac {\arctanh \left (a x \right ) \hyperbolicSineIntegral \left (\arctanh \left (a x \right )\right ) a^{2} x^{2}-\hyperbolicSineIntegral \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+\sqrt {-a^{2} x^{2}+1}}{a \arctanh \left (a x \right ) \left (a^{2} x^{2}-1\right )}\) | \(62\) |
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 {3}{2}} \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 = -0.03 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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