Optimal. Leaf size=302 \[ \frac {6 \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {\text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a}-\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {3 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \tanh ^{-1}(a x) \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \text {PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {3 i \text {PolyLog}\left (4,i e^{\tanh ^{-1}(a x)}\right )}{a} \]
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Rubi [A]
time = 0.14, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {6091, 6099,
4265, 2611, 6744, 2320, 6724, 6097} \begin {gather*} \frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {\tanh ^{-1}(a x)^3 \text {ArcTan}\left (e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {6 \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{a}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {3 i \text {Li}_4\left (i e^{\tanh ^{-1}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 4265
Rule 6091
Rule 6097
Rule 6099
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3 \, dx &=\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {1}{2} \int \frac {\tanh ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx-3 \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {\text {Subst}\left (\int x^3 \text {sech}(x) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a}+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {(3 i) \text {Subst}\left (\int x^2 \log \left (1-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {(3 i) \text {Subst}\left (\int x^2 \log \left (1+i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}\\ &=\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {(3 i) \text {Subst}\left (\int x \text {Li}_2\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}-\frac {(3 i) \text {Subst}\left (\int x \text {Li}_2\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {(3 i) \text {Subst}\left (\int \text {Li}_3\left (-i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}+\frac {(3 i) \text {Subst}\left (\int \text {Li}_3\left (i e^x\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {(3 i) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{\tanh ^{-1}(a x)}\right )}{a}\\ &=\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a}+\frac {3 \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2}{2 a}+\frac {1}{2} x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3+\frac {\tan ^{-1}\left (e^{\tanh ^{-1}(a x)}\right ) \tanh ^{-1}(a x)^3}{a}-\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (-i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \tanh ^{-1}(a x)^2 \text {Li}_2\left (i e^{\tanh ^{-1}(a x)}\right )}{2 a}+\frac {3 i \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}-\frac {3 i \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{a}+\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \tanh ^{-1}(a x) \text {Li}_3\left (i e^{\tanh ^{-1}(a x)}\right )}{a}-\frac {3 i \text {Li}_4\left (-i e^{\tanh ^{-1}(a x)}\right )}{a}+\frac {3 i \text {Li}_4\left (i e^{\tanh ^{-1}(a x)}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 2.46, size = 569, normalized size = 1.88 \begin {gather*} -\frac {i \left (7 \pi ^4+8 i \pi ^3 \tanh ^{-1}(a x)+24 \pi ^2 \tanh ^{-1}(a x)^2+192 i \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^2-32 i \pi \tanh ^{-1}(a x)^3+64 i a x \sqrt {1-a^2 x^2} \tanh ^{-1}(a x)^3-16 \tanh ^{-1}(a x)^4-384 \tanh ^{-1}(a x) \log \left (1-i e^{-\tanh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+384 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )+48 \pi ^2 \tanh ^{-1}(a x) \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-96 i \pi \tanh ^{-1}(a x)^2 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-64 \tanh ^{-1}(a x)^3 \log \left (1+i e^{-\tanh ^{-1}(a x)}\right )-48 \pi ^2 \tanh ^{-1}(a x) \log \left (1-i e^{\tanh ^{-1}(a x)}\right )+96 i \pi \tanh ^{-1}(a x)^2 \log \left (1-i e^{\tanh ^{-1}(a x)}\right )-8 i \pi ^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )+64 \tanh ^{-1}(a x)^3 \log \left (1+i e^{\tanh ^{-1}(a x)}\right )+8 i \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (\pi +2 i \tanh ^{-1}(a x)\right )\right )\right )-48 \left (8+\pi ^2-4 i \pi \tanh ^{-1}(a x)-4 \tanh ^{-1}(a x)^2\right ) \text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+192 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{\tanh ^{-1}(a x)}\right )-48 \pi ^2 \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )+192 i \pi \tanh ^{-1}(a x) \text {PolyLog}\left (2,i e^{\tanh ^{-1}(a x)}\right )+192 i \pi \text {PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )+384 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{-\tanh ^{-1}(a x)}\right )-384 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-i e^{\tanh ^{-1}(a x)}\right )-192 i \pi \text {PolyLog}\left (3,i e^{\tanh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (4,-i e^{-\tanh ^{-1}(a x)}\right )+384 \text {PolyLog}\left (4,-i e^{\tanh ^{-1}(a x)}\right )\right )}{128 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.08, size = 0, normalized size = 0.00 \[\int \sqrt {-a^{2} x^{2}+1}\, \arctanh \left (a x \right )^{3}\, dx\]
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 \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}^{3}{\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 = -0.00 \begin {gather*} \int {\mathrm {atanh}\left (a\,x\right )}^3\,\sqrt {1-a^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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