Optimal. Leaf size=186 \[ \frac {3 x \sqrt {1-a^2 x^2}}{112 a^3}+\frac {23 x^3 \sqrt {1-a^2 x^2}}{840 a}-\frac {1}{42} a x^5 \sqrt {1-a^2 x^2}+\frac {17 \text {ArcSin}(a x)}{560 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{35 a^2}+\frac {8}{35} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {1}{7} a^2 x^6 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \]
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Rubi [A]
time = 0.41, 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 = {6161, 6157,
6163, 327, 222, 6141} \begin {gather*} \frac {17 \text {ArcSin}(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} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 327
Rule 6141
Rule 6157
Rule 6161
Rule 6163
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 = 0.06, size = 79, normalized size = 0.42 \begin {gather*} \frac {a x \sqrt {1-a^2 x^2} \left (45+46 a^2 x^2-40 a^4 x^4\right )+51 \text {ArcSin}(a x)-48 \left (1-a^2 x^2\right )^{5/2} \left (2+5 a^2 x^2\right ) \tanh ^{-1}(a x)}{1680 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 = 140, normalized size = 0.75
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (240 \arctanh \left (a x \right ) a^{6} x^{6}+40 a^{5} x^{5}-384 a^{4} x^{4} \arctanh \left (a x \right )-46 a^{3} x^{3}+48 a^{2} x^{2} \arctanh \left (a x \right )-45 a x +96 \arctanh \left (a x \right )\right )}{1680 a^{4}}+\frac {17 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}+i\right )}{560 a^{4}}-\frac {17 i \ln \left (\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}-i\right )}{560 a^{4}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 163, normalized size = 0.88 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 106, normalized size = 0.57 \begin {gather*} -\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}} \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} \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^3\,\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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