Optimal. Leaf size=58 \[ -\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a} \]
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Rubi [A] time = 0.23, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5966, 6032, 6034, 3312, 3301, 5968} \[ -\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 3312
Rule 5966
Rule 5968
Rule 6032
Rule 6034
Rubi steps
\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}+a \int \frac {x}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+a^2 \int \frac {x^2}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx+\int \frac {1}{\left (1-a^2 x^2\right )^2 \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {\operatorname {Subst}\left (\int \frac {\sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {\operatorname {Subst}\left (\int \left (\frac {1}{2 x}+\frac {\cosh (2 x)}{2 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+2 \frac {\operatorname {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right ) \tanh ^{-1}(a x)^2}-\frac {x}{\left (1-a^2 x^2\right ) \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \tanh ^{-1}(a x)\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 58, normalized size = 1.00 \[ \frac {2 \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2 \text {Chi}\left (2 \tanh ^{-1}(a x)\right )+2 a x \tanh ^{-1}(a x)+1}{2 a \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 122, normalized size = 2.10 \[ \frac {4 \, a x \log \left (-\frac {a x + 1}{a x - 1}\right ) + {\left ({\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x + 1}{a x - 1}\right ) + {\left (a^{2} x^{2} - 1\right )} \operatorname {log\_integral}\left (-\frac {a x - 1}{a x + 1}\right )\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 4}{2 \, {\left (a^{3} x^{2} - a\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2}} \]
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 \[ \int \frac {1}{{\left (a^{2} x^{2} - 1\right )}^{2} \operatorname {artanh}\left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 51, normalized size = 0.88 \[ \frac {-\frac {1}{4 \arctanh \left (a x \right )^{2}}-\frac {\cosh \left (2 \arctanh \left (a x \right )\right )}{4 \arctanh \left (a x \right )^{2}}-\frac {\sinh \left (2 \arctanh \left (a x \right )\right )}{2 \arctanh \left (a x \right )}+\Chi \left (2 \arctanh \left (a x \right )\right )}{a} \]
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 \[ \frac {2 \, {\left (a x \log \left (a x + 1\right ) - a x \log \left (-a x + 1\right ) + 1\right )}}{{\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right )^{2} - 2 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + 1\right ) \log \left (-a x + 1\right ) + {\left (a^{3} x^{2} - a\right )} \log \left (-a x + 1\right )^{2}} - \int -\frac {2 \, {\left (a^{2} x^{2} + 1\right )}}{{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )}\,{d x} \]
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 \[ \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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