Optimal. Leaf size=89 \[ -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{c}+\frac {b \text {Li}_2\left (-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{2 c}-\frac {b \text {Li}_2\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{2 c} \]
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Rubi [A] time = 0.05, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {206, 6681, 5912} \[ \frac {b \text {PolyLog}\left (2,-\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{2 c}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{2 c}-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 5912
Rule 6681
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(x)}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {b \text {Li}_2\left (-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c}-\frac {b \text {Li}_2\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 43, normalized size = 0.48 \[ \frac {a \tanh ^{-1}(c x)}{c}+\frac {b \left (\text {Li}_2\left (-e^{-\tanh ^{-1}(c x)}\right )-\text {Li}_2\left (e^{-\tanh ^{-1}(c x)}\right )\right )}{2 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1}, x\right ) \]
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 \[ \int -\frac {b \operatorname {artanh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 118, normalized size = 1.33 \[ \frac {a \ln \left (c x +1\right )}{2 c}-\frac {a \ln \left (c x -1\right )}{2 c}-\frac {b \dilog \left (\frac {-\frac {-c x +1}{c x +1}+1}{\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}\right )}{c}+\frac {b \dilog \left (\frac {\left (-\frac {-c x +1}{c x +1}+1\right )^{2}}{\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{4}}\right )}{4 c} \]
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 \[ \frac {1}{4} \, b {\left (\frac {{\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \log \left (\sqrt {c x + 1} + \sqrt {-c x + 1}\right ) - {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \log \left (\sqrt {c x + 1} - \sqrt {-c x + 1}\right )}{c} - 2 \, \int -\frac {\sqrt {c x + 1} {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{2 \, {\left ({\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} + {\left (c^{2} x^{2} - 1\right )} \sqrt {-c x + 1}\right )}}\,{d x} - 2 \, \int \frac {\sqrt {c x + 1} {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{2 \, {\left ({\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} - {\left (c^{2} x^{2} - 1\right )} \sqrt {-c x + 1}\right )}}\,{d x}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} \]
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 \[ \int -\frac {a+b\,\mathrm {atanh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \]
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 \[ - \int \frac {a}{c^{2} x^{2} - 1}\, dx - \int \frac {b \operatorname {atanh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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