Optimal. Leaf size=186 \[ \frac {17 \sin ^{-1}(a x)}{560 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^2}-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^4}+\frac {3 x \sqrt {1-a^2 x^2}}{112 a^3} \]
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Rubi [A] time = 0.57, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6014, 6010, 6016, 321, 216, 5994} \[ -\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}+\frac {3 x \sqrt {1-a^2 x^2}}{112 a^3}-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^2}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^4}+\frac {17 \sin ^{-1}(a x)}{560 a^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 321
Rule 5994
Rule 6010
Rule 6014
Rule 6016
Rubi steps
\begin {align*} \int x^3 \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=-\left (a^2 \int x^5 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \, dx\right )+\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}{7} a^2 x^6 \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 {1}{7} a^2 \int \frac {x^5 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\frac {1}{7} a^3 \int \frac {x^6}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x^3 \sqrt {1-a^2 x^2}}{20 a}-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{15 a^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {4}{35} \int \frac {x^3 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\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 {1}{35} a \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx+\frac {1}{42} (5 a) \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x \sqrt {1-a^2 x^2}}{24 a^3}+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}-\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)}{35 a^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^6 \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 {8 \int \frac {x \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{105 a^2}-\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{140 a}-\frac {4 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{105 a}+\frac {5 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{56 a}\\ &=\frac {3 x \sqrt {1-a^2 x^2}}{112 a^3}+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}+\frac {11 \sin ^{-1}(a x)}{120 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{280 a^3}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{105 a^3}+\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{112 a^3}-\frac {8 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{105 a^3}\\ &=\frac {3 x \sqrt {1-a^2 x^2}}{112 a^3}+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}+\frac {17 \sin ^{-1}(a x)}{560 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 79, normalized size = 0.42 \[ \frac {-48 \left (5 a^2 x^2+2\right ) \left (1-a^2 x^2\right )^{5/2} \tanh ^{-1}(a x)+a x \left (-40 a^4 x^4+46 a^2 x^2+45\right ) \sqrt {1-a^2 x^2}+51 \sin ^{-1}(a x)}{1680 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 106, normalized size = 0.57 \[ -\frac {{\left (40 \, a^{5} x^{5} - 46 \, a^{3} x^{3} - 45 \, a x + 24 \, {\left (5 \, a^{6} x^{6} - 8 \, a^{4} x^{4} + a^{2} x^{2} + 2\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1} + 102 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{1680 \, a^{4}} \]
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 = 0.38, size = 140, normalized size = 0.75 \[ -\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (240 \arctanh \left (a x \right ) x^{6} a^{6}+40 x^{5} a^{5}-384 a^{4} x^{4} \arctanh \left (a x \right )-46 x^{3} a^{3}+48 a^{2} x^{2} \arctanh \left (a x \right )-45 a x +96 \arctanh \left (a x \right )\right )}{1680 a^{4}}+\frac {17 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{560 a^{4}}-\frac {17 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{560 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 163, normalized size = 0.88 \[ -\frac {1}{1680} \, a {\left (\frac {5 \, {\left (\frac {8 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{a^{2}} - \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{2}} - \frac {3 \, \arcsin \left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {12 \, {\left (2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x + 3 \, \sqrt {-a^{2} x^{2} + 1} x + \frac {3 \, \arcsin \left (a x\right )}{a}\right )}}{a^{4}}\right )} - \frac {1}{35} \, {\left (\frac {5 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{a^{2}} + \frac {2 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{a^{4}}\right )} \operatorname {artanh}\left (a x\right ) \]
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 \[ \int x^3\,\mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2} \,d x \]
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 \[ \int x^{3} \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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