Optimal. Leaf size=144 \[ -\frac {i \sqrt {1-x^2} \text {Li}_2\left (-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \text {Li}_2\left (\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}-\frac {2 \sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \tanh ^{-1}(x)}{\sqrt {a-a x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5954, 5950} \[ -\frac {i \sqrt {1-x^2} \text {PolyLog}\left (2,-\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \text {PolyLog}\left (2,\frac {i \sqrt {1-x}}{\sqrt {x+1}}\right )}{\sqrt {a-a x^2}}-\frac {2 \sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {x+1}}\right ) \tanh ^{-1}(x)}{\sqrt {a-a x^2}} \]
Antiderivative was successfully verified.
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Rule 5950
Rule 5954
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(x)}{\sqrt {a-a x^2}} \, dx &=\frac {\sqrt {1-x^2} \int \frac {\tanh ^{-1}(x)}{\sqrt {1-x^2}} \, dx}{\sqrt {a-a x^2}}\\ &=-\frac {2 \sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \tanh ^{-1}(x)}{\sqrt {a-a x^2}}-\frac {i \sqrt {1-x^2} \text {Li}_2\left (-\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}+\frac {i \sqrt {1-x^2} \text {Li}_2\left (\frac {i \sqrt {1-x}}{\sqrt {1+x}}\right )}{\sqrt {a-a x^2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 90, normalized size = 0.62 \[ -\frac {i \sqrt {a \left (1-x^2\right )} \left (\text {Li}_2\left (-i e^{-\tanh ^{-1}(x)}\right )-\text {Li}_2\left (i e^{-\tanh ^{-1}(x)}\right )+\tanh ^{-1}(x) \left (\log \left (1-i e^{-\tanh ^{-1}(x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(x)}\right )\right )\right )}{a \sqrt {1-x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a x^{2} + a} \operatorname {artanh}\relax (x)}{a x^{2} - a}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\relax (x)}{\sqrt {-a x^{2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 210, normalized size = 1.46 \[ \frac {i \ln \left (1+\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right ) \arctanh \relax (x ) \sqrt {-x^{2}+1}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{a \left (x^{2}-1\right )}-\frac {i \ln \left (1-\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right ) \arctanh \relax (x ) \sqrt {-x^{2}+1}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{a \left (x^{2}-1\right )}+\frac {i \dilog \left (1+\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right ) \sqrt {-x^{2}+1}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{a \left (x^{2}-1\right )}-\frac {i \dilog \left (1-\frac {i \left (1+x \right )}{\sqrt {-x^{2}+1}}\right ) \sqrt {-x^{2}+1}\, \sqrt {-\left (-1+x \right ) \left (1+x \right ) a}}{a \left (x^{2}-1\right )} \]
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 \[ \int \frac {\operatorname {artanh}\relax (x)}{\sqrt {-a x^{2} + a}}\,{d x} \]
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 \[ \int \frac {\mathrm {atanh}\relax (x)}{\sqrt {a-a\,x^2}} \,d x \]
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 \[ \int \frac {\operatorname {atanh}{\relax (x )}}{\sqrt {- a \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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