Optimal. Leaf size=235 \[ -\frac {i c \sqrt {1-a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {i c \sqrt {1-a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)-\frac {c \sqrt {1-a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.10, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5942, 5954, 5950} \[ -\frac {i c \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {i c \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1-a x}}{\sqrt {a x+1}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)-\frac {c \sqrt {1-a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {a x+1}}\right ) \tanh ^{-1}(a x)}{a \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 5942
Rule 5950
Rule 5954
Rubi steps
\begin {align*} \int \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x) \, dx &=\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {1}{2} c \int \frac {\tanh ^{-1}(a x)}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)+\frac {\left (c \sqrt {1-a^2 x^2}\right ) \int \frac {\tanh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2}}{2 a}+\frac {1}{2} x \sqrt {c-a^2 c x^2} \tanh ^{-1}(a x)-\frac {c \sqrt {1-a^2 x^2} \tan ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {1+a x}}\right ) \tanh ^{-1}(a x)}{a \sqrt {c-a^2 c x^2}}-\frac {i c \sqrt {1-a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a \sqrt {c-a^2 c x^2}}+\frac {i c \sqrt {1-a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1-a x}}{\sqrt {1+a x}}\right )}{2 a \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 119, normalized size = 0.51 \[ \frac {\sqrt {c \left (1-a^2 x^2\right )} \left (-\frac {i \left (\text {Li}_2\left (-i e^{-\tanh ^{-1}(a x)}\right )-\text {Li}_2\left (i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )}{\sqrt {1-a^2 x^2}}+a x \tanh ^{-1}(a x)+1\right )}{2 a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-a^{2} c x^{2} + c} \operatorname {artanh}\left (a x\right ), x\right ) \]
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 = 0.55, size = 319, normalized size = 1.36 \[ \frac {\left (a x \arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{2 a}+\frac {i \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \arctanh \left (a x \right ) \ln \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}-\frac {i \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \arctanh \left (a x \right ) \ln \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}+\frac {i \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \dilog \left (1+\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )}-\frac {i \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \dilog \left (1-\frac {i \left (a x +1\right )}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a \left (a x +1\right ) \left (a x -1\right )} \]
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 \[ \int \sqrt {-a^{2} c x^{2} + c} \operatorname {artanh}\left (a x\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \mathrm {atanh}\left (a\,x\right )\,\sqrt {c-a^2\,c\,x^2} \,d x \]
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 \[ \int \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {atanh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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