Optimal. Leaf size=35 \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a}-\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]
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Rubi [A] time = 0.13, 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 = {5966, 6034, 3298} \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )}{a}-\frac {1}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5966
Rule 6034
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 {\operatorname {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 = 0.09, size = 32, normalized size = 0.91 \[ \frac {\text {Shi}\left (\tanh ^{-1}(a x)\right )-\frac {1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 62, normalized size = 1.77 \[ \frac {\arctanh \left (a x \right ) \Shi \left (\arctanh \left (a x \right )\right ) x^{2} a^{2}-\Shi \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+\sqrt {-a^{2} x^{2}+1}}{a \arctanh \left (a x \right ) \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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