Optimal. Leaf size=291 \[ \frac {3 c \sqrt {c-a^2 c x^2}}{8 a}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)-\frac {3 c^2 \sqrt {1-a^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{4 a \sqrt {c-a^2 c x^2}}-\frac {3 i c^2 \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a \sqrt {c-a^2 c x^2}}+\frac {3 i c^2 \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6089, 6101,
6097} \begin {gather*} -\frac {3 c^2 \sqrt {1-a^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{4 a \sqrt {c-a^2 c x^2}}-\frac {3 i c^2 \sqrt {1-a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a \sqrt {c-a^2 c x^2}}+\frac {3 i c^2 \sqrt {1-a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{8 a \sqrt {c-a^2 c x^2}}+\frac {3 c \sqrt {c-a^2 c x^2}}{8 a}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6089
Rule 6097
Rule 6101
Rubi steps
\begin {align*} \int \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x) \, dx &=\frac {\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x) \, dx\\ &=\frac {3 c \sqrt {c-a^2 c x^2}}{8 a}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {1}{8} \left (3 c^2\right ) \int \frac {\tanh ^{-1}(a x)}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {3 c \sqrt {c-a^2 c x^2}}{8 a}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)+\frac {\left (3 c^2 \sqrt {1-a^2 x^2}\right ) \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {c-a^2 c x^2}}\\ &=\frac {3 c \sqrt {c-a^2 c x^2}}{8 a}+\frac {\left (c-a^2 c x^2\right )^{3/2}}{12 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \tanh ^{-1}(a x)-\frac {3 c^2 \sqrt {1-a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{4 a \sqrt {c-a^2 c x^2}}-\frac {3 i c^2 \sqrt {1-a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a \sqrt {c-a^2 c x^2}}+\frac {3 i c^2 \sqrt {1-a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{8 a \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 206, normalized size = 0.71 \begin {gather*} -\frac {c \sqrt {c-a^2 c x^2} \left (-11 \sqrt {1-a^2 x^2}+2 a^2 x^2 \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 \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 )+9 i \text {PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-9 i \text {PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )\right )}{24 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.62, size = 345, normalized size = 1.19
method | result | size |
default | \(-\frac {c \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \left (6 a^{3} x^{3} \arctanh \left (a x \right )+2 a^{2} x^{2}-15 a x \arctanh \left (a x \right )-11\right )}{24 a}+\frac {3 i c \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \arctanh \left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a \left (a x +1\right ) \left (a x -1\right )}-\frac {3 i c \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \arctanh \left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a \left (a x +1\right ) \left (a x -1\right )}+\frac {3 i c \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a \left (a x +1\right ) \left (a x -1\right )}-\frac {3 i c \sqrt {-a^{2} x^{2}+1}\, \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a \left (a x +1\right ) \left (a x -1\right )}\) | \(345\) |
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 \left (- c \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 = -0.00 \begin {gather*} \int \mathrm {atanh}\left (a\,x\right )\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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