Optimal. Leaf size=81 \[ \frac {3 x \sqrt {1-a^2 x^2}}{40 a}+\frac {x \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {3 \text {ArcSin}(a x)}{40 a^2}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 a^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6141, 201, 222}
\begin {gather*} \frac {3 \text {ArcSin}(a x)}{40 a^2}+\frac {x \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {3 x \sqrt {1-a^2 x^2}}{40 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 6141
Rubi steps
\begin {align*} \int x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 a^2}+\frac {\int \left (1-a^2 x^2\right )^{3/2} \, dx}{5 a}\\ &=\frac {x \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 a^2}+\frac {3 \int \sqrt {1-a^2 x^2} \, dx}{20 a}\\ &=\frac {3 x \sqrt {1-a^2 x^2}}{40 a}+\frac {x \left (1-a^2 x^2\right )^{3/2}}{20 a}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 a^2}+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{40 a}\\ &=\frac {3 x \sqrt {1-a^2 x^2}}{40 a}+\frac {x \left (1-a^2 x^2\right )^{3/2}}{20 a}+\frac {3 \sin ^{-1}(a x)}{40 a^2}-\frac {\left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{5 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 61, normalized size = 0.75 \begin {gather*} \frac {a x \left (5-2 a^2 x^2\right ) \sqrt {1-a^2 x^2}+3 \text {ArcSin}(a x)-8 \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)}{40 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.35, size = 120, normalized size = 1.48
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (8 a^{4} x^{4} \arctanh \left (a x \right )+2 a^{3} x^{3}-16 a^{2} x^{2} \arctanh \left (a x \right )-5 a x +8 \arctanh \left (a x \right )\right )}{40 a^{2}}+\frac {3 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{40 a^{2}}-\frac {3 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{40 a^{2}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 67, normalized size = 0.83 \begin {gather*} -\frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} \operatorname {artanh}\left (a x\right )}{5 \, a^{2}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{a}}{40 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 90, normalized size = 1.11 \begin {gather*} -\frac {{\left (2 \, a^{3} x^{3} - 5 \, a x + 4 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} + 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{40 \, a^{2}} \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 x \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\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 = -0.01 \begin {gather*} \int x\,\mathrm {atanh}\left (a\,x\right )\,{\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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