Optimal. Leaf size=136 \[ \frac {x \sqrt {1-a^2 x^2}}{24 a^3}+\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}+\frac {11 \text {ArcSin}(a x)}{120 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \]
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Rubi [A]
time = 0.15, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6157, 6163,
327, 222, 6141} \begin {gather*} \frac {11 \text {ArcSin}(a x)}{120 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}+\frac {x \sqrt {1-a^2 x^2}}{24 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 6141
Rule 6157
Rule 6163
Rubi steps
\begin {align*} \int x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \, dx &=\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {1}{5} \int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx-\frac {1}{5} a \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {2 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a^2}+\frac {\int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{15 a}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{20 a}\\ &=\frac {x \sqrt {1-a^2 x^2}}{24 a^3}+\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{30 a^3}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{40 a^3}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{15 a^3}\\ &=\frac {x \sqrt {1-a^2 x^2}}{24 a^3}+\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}+\frac {11 \sin ^{-1}(a x)}{120 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 79, normalized size = 0.58 \begin {gather*} \frac {a x \sqrt {1-a^2 x^2} \left (5+6 a^2 x^2\right )+11 \text {ArcSin}(a x)+8 \sqrt {1-a^2 x^2} \left (-2-a^2 x^2+3 a^4 x^4\right ) \tanh ^{-1}(a x)}{120 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.37, size = 120, normalized size = 0.88
method | result | size |
default | \(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (24 a^{4} x^{4} \arctanh \left (a x \right )+6 a^{3} x^{3}-8 a^{2} x^{2} \arctanh \left (a x \right )+5 a x -16 \arctanh \left (a x \right )\right )}{120 a^{4}}+\frac {11 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{120 a^{4}}-\frac {11 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{120 a^{4}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 128, normalized size = 0.94 \begin {gather*} -\frac {1}{120} \, a {\left (\frac {3 \, {\left (\frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} x + \frac {\arcsin \left (a x\right )}{a}\right )}}{a^{4}}\right )} - \frac {1}{15} \, {\left (\frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\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 = 0.41, size = 91, normalized size = 0.67 \begin {gather*} \frac {{\left (6 \, a^{3} x^{3} + 5 \, a x + 4 \, {\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} - 22 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{4}} \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^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \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^3\,\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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