Optimal. Leaf size=52 \[ -\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \text {Shi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \]
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Rubi [A] time = 0.16, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5966, 6034, 5448, 3298} \[ -\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \text {Shi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )}{4 a} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 5448
Rule 5966
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)^2} \, dx &=-\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+(3 a) \int \frac {x}{\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 x}+\frac {\sinh (3 x)}{4 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{4 a}\\ &=-\frac {1}{a \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}+\frac {3 \text {Shi}\left (\tanh ^{-1}(a x)\right )}{4 a}+\frac {3 \text {Shi}\left (3 \tanh ^{-1}(a x)\right )}{4 a}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 45, normalized size = 0.87 \[ \frac {3 \left (\text {Shi}\left (\tanh ^{-1}(a x)\right )+\text {Shi}\left (3 \tanh ^{-1}(a x)\right )\right )-\frac {4}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)}}{4 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \operatorname {artanh}\left (a x\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 120, normalized size = 2.31 \[ \frac {3 \arctanh \left (a x \right ) \Shi \left (\arctanh \left (a x \right )\right ) x^{2} a^{2}+3 \arctanh \left (a x \right ) \Shi \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}-\cosh \left (3 \arctanh \left (a x \right )\right ) x^{2} a^{2}-3 \Shi \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )-3 \Shi \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}+\cosh \left (3 \arctanh \left (a x \right )\right )}{4 a \arctanh \left (a x \right ) \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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