Optimal. Leaf size=281 \[ -\frac {11 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{60 a^4}+\frac {11 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{60 a^4}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{30 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac {\left (1-a^2 x^2\right )^{3/2}}{30 a^4}+\frac {11 \sqrt {1-a^2 x^2}}{60 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^4}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3} \]
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Rubi [A] time = 1.07, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6014, 6016, 261, 5950, 5994, 266, 43} \[ -\frac {11 i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{60 a^4}+\frac {11 i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{60 a^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{30 a^4}+\frac {11 \sqrt {1-a^2 x^2}}{60 a^4}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2+\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^4}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{30 a^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 5950
Rule 5994
Rule 6014
Rule 6016
Rubi steps
\begin {align*} \int x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2 \, dx &=-\left (a^2 \int \frac {x^5 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\right )+\int \frac {x^3 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {4}{5} \int \frac {x^3 \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx+\frac {2 \int \frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}+\frac {2 \int \frac {x^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a}-\frac {1}{5} (2 a) \int \frac {x^4 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{3 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{3 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {1}{10} \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx+\frac {\int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^3}+\frac {4 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{3 a^3}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{3 a^2}-\frac {8 \int \frac {x \tanh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx}{15 a^2}-\frac {3 \int \frac {x^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{10 a}-\frac {8 \int \frac {x^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a}\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 a^4}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac {10 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{3 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {5 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^4}+\frac {5 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{3 a^4}-\frac {1}{20} \operatorname {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )-\frac {3 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{20 a^3}-\frac {4 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a^3}-\frac {16 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{15 a^3}-\frac {3 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{20 a^2}-\frac {4 \int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{15 a^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{12 a^4}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{30 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {11 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{60 a^4}+\frac {11 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{60 a^4}-\frac {1}{20} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 \sqrt {1-a^2 x}}-\frac {\sqrt {1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )\\ &=\frac {11 \sqrt {1-a^2 x^2}}{60 a^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{30 a^4}+\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{12 a^3}+\frac {x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{10 a}-\frac {11 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{30 a^4}-\frac {2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^4}-\frac {x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-\frac {11 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{60 a^4}+\frac {11 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{60 a^4}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 175, normalized size = 0.62 \[ \frac {\sqrt {1-a^2 x^2} \left (-\frac {11 i \left (\text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-\text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )}{\sqrt {1-a^2 x^2}}+12 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2+6 a x \left (a^2 x^2-1\right ) \tanh ^{-1}(a x)+2 \left (a^2 x^2-1\right ) \left (10 \tanh ^{-1}(a x)^2+1\right )+11 a x \tanh ^{-1}(a x)+11\right )}{60 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} x^{2} + 1} x^{3} \operatorname {artanh}\left (a x\right )^{2}, x\right ) \]
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 [A] time = 0.42, size = 211, normalized size = 0.75 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (12 a^{4} x^{4} \arctanh \left (a x \right )^{2}+6 a^{3} x^{3} \arctanh \left (a x \right )-4 a^{2} x^{2} \arctanh \left (a x \right )^{2}+2 a^{2} x^{2}+5 a x \arctanh \left (a x \right )-8 \arctanh \left (a x \right )^{2}+9\right )}{60 a^{4}}-\frac {11 i \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{60 a^{4}}+\frac {11 i \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{60 a^{4}}-\frac {11 i \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{60 a^{4}}+\frac {11 i \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{60 a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-a^{2} x^{2} + 1} x^{3} \operatorname {artanh}\left (a x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {atanh}\left (a\,x\right )}^2\,\sqrt {1-a^2\,x^2} \,d x \]
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 \[ \int x^{3} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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