Optimal. Leaf size=36 \[ -\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6153, 6115,
3382} \begin {gather*} \frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2}-\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 6115
Rule 6153
Rubi steps
\begin {align*} \int \frac {x}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)^2} \, dx &=-\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)} \, dx}{a}\\ &=-\frac {x}{a \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 )}{a^2}\\ &=-\frac {x}{a \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\frac {\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.94 \begin {gather*} \frac {-\frac {a x}{\sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}+\text {Chi}\left (\tanh ^{-1}(a x)\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(89\) vs.
\(2(34)=68\).
time = 1.14, size = 90, normalized size = 2.50
method | result | size |
default | \(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 a^{2} \left (a x -1\right ) \arctanh \left (a x \right )}-\frac {\expIntegral \left (1, -\arctanh \left (a x \right )\right )}{2 a^{2}}+\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}}{2 \arctanh \left (a x \right ) \left (a x +1\right ) a^{2}}-\frac {\expIntegral \left (1, \arctanh \left (a x \right )\right )}{2 a^{2}}\) | \(90\) |
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 {x}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}^{2}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.03 \begin {gather*} \int \frac {x}{{\mathrm {atanh}\left (a\,x\right )}^2\,{\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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