Optimal. Leaf size=136 \[ \frac {11 \sin ^{-1}(a x)}{120 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}+\frac {x \sqrt {1-a^2 x^2}}{24 a^3} \]
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Rubi [A] time = 0.20, 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 = {6010, 6016, 321, 216, 5994} \[ \frac {x^3 \sqrt {1-a^2 x^2}}{20 a}+\frac {x \sqrt {1-a^2 x^2}}{24 a^3}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^4}+\frac {11 \sin ^{-1}(a x)}{120 a^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 5994
Rule 6010
Rule 6016
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 = 0.07, size = 79, normalized size = 0.58 \[ \frac {a x \sqrt {1-a^2 x^2} \left (6 a^2 x^2+5\right )+8 \sqrt {1-a^2 x^2} \left (3 a^4 x^4-a^2 x^2-2\right ) \tanh ^{-1}(a x)+11 \sin ^{-1}(a x)}{120 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 91, normalized size = 0.67 \[ \frac {{\left (6 \, a^{3} x^{3} + 5 \, a x + 4 \, {\left (3 \, a^{4} x^{4} - a^{2} x^{2} - 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} - 22 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{120 \, a^{4}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.39, size = 120, normalized size = 0.88 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (24 a^{4} x^{4} \arctanh \left (a x \right )+6 x^{3} a^{3}-8 a^{2} x^{2} \arctanh \left (a x \right )+5 a x -16 \arctanh \left (a x \right )\right )}{120 a^{4}}+\frac {11 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{120 a^{4}}-\frac {11 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{120 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 128, normalized size = 0.94 \[ -\frac {1}{120} \, a {\left (\frac {3 \, {\left (\frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {\arcsin \left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {-a^{2} x^{2} + 1} x + \frac {\arcsin \left (a x\right )}{a}\right )}}{a^{4}}\right )} - \frac {1}{15} \, {\left (\frac {3 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int x^3\,\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2} \,d x \]
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 \[ \int x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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