Optimal. Leaf size=37 \[ -\frac {1}{a \sqrt {a-a x^2}}+\frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6105}
\begin {gather*} \frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}}-\frac {1}{a \sqrt {a-a x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6105
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(x)}{\left (a-a x^2\right )^{3/2}} \, dx &=-\frac {1}{a \sqrt {a-a x^2}}+\frac {x \tanh ^{-1}(x)}{a \sqrt {a-a x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.81 \begin {gather*} \frac {\sqrt {a-a x^2} \left (1-x \tanh ^{-1}(x)\right )}{a^2 \left (-1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.82, size = 52, normalized size = 1.41
method | result | size |
risch | \(\frac {x \ln \left (1+x \right )}{2 a \sqrt {-a \left (x^{2}-1\right )}}-\frac {x \ln \left (1-x \right )+2}{2 a \sqrt {-a \left (x^{2}-1\right )}}\) | \(47\) |
default | \(-\frac {\left (\arctanh \left (x \right )-1\right ) \sqrt {-\left (x -1\right ) \left (1+x \right ) a}}{2 a^{2} \left (x -1\right )}-\frac {\left (1+\arctanh \left (x \right )\right ) \sqrt {-\left (x -1\right ) \left (1+x \right ) a}}{2 a^{2} \left (1+x \right )}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 63, normalized size = 1.70 \begin {gather*} \frac {x \operatorname {artanh}\left (x\right )}{\sqrt {-a x^{2} + a} a} - \frac {\frac {\sqrt {-a x^{2} + a}}{a x + a} - \frac {\sqrt {-a x^{2} + a}}{a x - a}}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 42, normalized size = 1.14 \begin {gather*} -\frac {\sqrt {-a x^{2} + a} {\left (x \log \left (-\frac {x + 1}{x - 1}\right ) - 2\right )}}{2 \, {\left (a^{2} x^{2} - a^{2}\right )}} \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 {\operatorname {atanh}{\left (x \right )}}{\left (- a \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 54, normalized size = 1.46 \begin {gather*} -\frac {\sqrt {-a x^{2} + a} x \log \left (-\frac {x + 1}{x - 1}\right )}{2 \, {\left (a x^{2} - a\right )} a} - \frac {1}{\sqrt {-a x^{2} + a} a} \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 {\mathrm {atanh}\left (x\right )}{{\left (a-a\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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