Optimal. Leaf size=89 \[ -\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{3 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6107, 6105}
\begin {gather*} -\frac {2}{3 a \sqrt {1-a^2 x^2}}-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6105
Rule 6107
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{3} \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {1}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2}{3 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x \tanh ^{-1}(a x)}{3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 49, normalized size = 0.55 \begin {gather*} -\frac {7-6 a^2 x^2+\left (-9 a x+6 a^3 x^3\right ) \tanh ^{-1}(a x)}{9 a \left (1-a^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.95, size = 59, normalized size = 0.66
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (6 a^{3} x^{3} \arctanh \left (a x \right )-6 a^{2} x^{2}-9 a x \arctanh \left (a x \right )+7\right )}{9 a \left (a^{2} x^{2}-1\right )^{2}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 74, normalized size = 0.83 \begin {gather*} -\frac {1}{9} \, a {\left (\frac {6}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}}\right )} + \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\right )} \operatorname {artanh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 73, normalized size = 0.82 \begin {gather*} \frac {{\left (12 \, a^{2} x^{2} - 3 \, {\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right ) - 14\right )} \sqrt {-a^{2} x^{2} + 1}}{18 \, {\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\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 (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 90, normalized size = 1.01 \begin {gather*} -\frac {{\left (2 \, a^{2} x^{2} - 3\right )} \sqrt {-a^{2} x^{2} + 1} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{6 \, {\left (a^{2} x^{2} - 1\right )}^{2}} - \frac {6 \, a^{2} x^{2} - 7}{9 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} 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 = -0.01 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (1-a^2\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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