Optimal. Leaf size=65 \[ -\frac {1}{2 a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}-\frac {x}{2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a} \]
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Rubi [A]
time = 0.11, 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 = {6113, 6153,
6115, 3382} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 6113
Rule 6115
Rule 6153
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 {\text {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 \begin {gather*} \frac {-\frac {1+a x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}+\text {Chi}\left (\tanh ^{-1}(a x)\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.23, size = 86, normalized size = 1.32
method | result | size |
default | \(\frac {\arctanh \left (a x \right )^{2} \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) a^{2} x^{2}+\sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-\hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+\sqrt {-a^{2} x^{2}+1}}{2 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}\) | \(86\) |
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}^{3}{\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 = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\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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