Optimal. Leaf size=189 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {3 \sqrt {1-a^2 x^2}}{8 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {3}{8} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{4 a} \]
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Rubi [A] time = 0.09, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5942, 5950} \[ -\frac {3 i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a}+\frac {3 i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a}+\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {3 \sqrt {1-a^2 x^2}}{8 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {3}{8} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{4 a} \]
Antiderivative was successfully verified.
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Rule 5942
Rule 5950
Rubi steps
\begin {align*} \int \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {3}{4} \int \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2}}{8 a}+\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {3}{8} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {3}{8} \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {3 \sqrt {1-a^2 x^2}}{8 a}+\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a}+\frac {3}{8} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (1-a^2 x^2\right )^{3/2} \tanh ^{-1}(a x)-\frac {3 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{4 a}-\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a}+\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 176, normalized size = 0.93 \[ \frac {-2 a^2 x^2 \sqrt {1-a^2 x^2}+11 \sqrt {1-a^2 x^2}+15 a x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-6 a^3 x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-9 i \text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )+9 i \text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )-9 i \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+9 i \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )}{24 a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1} \operatorname {artanh}\left (a x\right ), x\right ) \]
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 = 0.91, size = 173, normalized size = 0.92 \[ -\frac {\left (6 a^{3} x^{3} \arctanh \left (a x \right )+2 a^{2} x^{2}-15 a x \arctanh \left (a x \right )-11\right ) \sqrt {-a^{2} x^{2}+1}}{24 a}-\frac {3 i \arctanh \left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a}+\frac {3 i \arctanh \left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a}-\frac {3 i \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a}+\frac {3 i \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a} \]
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 \[ \int {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \operatorname {artanh}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {atanh}\left (a\,x\right )\,{\left (1-a^2\,x^2\right )}^{3/2} \,d x \]
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 \[ \int \left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {atanh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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