Optimal. Leaf size=194 \[ \frac {\sqrt {1-a^2 x^2}}{8 a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{4 a^3}-\frac {i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a^3}+\frac {i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a^3} \]
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Rubi [A]
time = 0.13, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6157, 6163,
267, 6097, 272, 45} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{4 a^3}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a^3}+\frac {i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a^3}+\frac {\sqrt {1-a^2 x^2}}{8 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 267
Rule 272
Rule 6097
Rule 6157
Rule 6163
Rubi steps
\begin {align*} \int x^2 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x) \, dx &=\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {1}{4} \int \frac {x^2 \tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx-\frac {1}{4} a \int \frac {x^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)+\frac {\int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a^2}+\frac {\int \frac {x}{\sqrt {1-a^2 x^2}} \, dx}{8 a}-\frac {1}{8} a \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1-a^2 x^2}}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{4 a^3}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a^3}+\frac {i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a^3}-\frac {1}{8} a \text {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 {\sqrt {1-a^2 x^2}}{8 a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{12 a^3}-\frac {x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)}{8 a^2}+\frac {1}{4} x^3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)-\frac {\tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{4 a^3}-\frac {i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a^3}+\frac {i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 160, normalized size = 0.82 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (1+2 a^2 x^2+3 a x \tanh ^{-1}(a x)+6 a x \left (-1+a^2 x^2\right ) \tanh ^{-1}(a x)-\frac {3 i \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 )}{\sqrt {1-a^2 x^2}}-\frac {3 i \left (\text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}}\right )}{24 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.38, size = 175, normalized size = 0.90
method | result | size |
default | \(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \left (6 a^{3} x^{3} \arctanh \left (a x \right )+2 a^{2} x^{2}-3 a x \arctanh \left (a x \right )+1\right )}{24 a^{3}}-\frac {i \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{8 a^{3}}+\frac {i \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right ) \arctanh \left (a x \right )}{8 a^{3}}-\frac {i \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{3}}+\frac {i \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{3}}\) | \(175\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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^2\,\mathrm {atanh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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