Optimal. Leaf size=65 \[ -\frac {x}{2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a} \]
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Rubi [A] time = 0.16, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5966, 6006, 5968, 3301} \[ -\frac {x}{2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}-\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5966
Rule 5968
Rule 6006
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}+\frac {1}{2} a \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {1}{2} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=-\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 44, normalized size = 0.68 \[ \frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )-\frac {a x \tanh ^{-1}(a x)+1}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}}{2 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \operatorname {artanh}\left (a x\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 86, normalized size = 1.32 \[ \frac {\arctanh \left (a x \right )^{2} \Chi \left (\arctanh \left (a x \right )\right ) x^{2} a^{2}+\sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-\Chi \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+\sqrt {-a^{2} x^{2}+1}}{2 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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