Optimal. Leaf size=98 \[ -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {i b \text {PolyLog}\left (2,-\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c}-\frac {i b \text {PolyLog}\left (2,\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c} \]
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Rubi [A]
time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {212, 6813,
4941, 2438} \begin {gather*} -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}{c}+\frac {i b \text {Li}_2\left (-\frac {i \sqrt {c x+1}}{\sqrt {1-c x}}\right )}{2 c}-\frac {i b \text {Li}_2\left (\frac {i \sqrt {c x+1}}{\sqrt {1-c x}}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2438
Rule 4941
Rule 6813
Rubi steps
\begin {align*} \int \frac {a+b \cot ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {a+b \cot ^{-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 {(i b) \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c}+\frac {(i b) \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{x}\right )}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c}\\ &=-\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {i b \text {Li}_2\left (-\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c}-\frac {i b \text {Li}_2\left (\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 73, normalized size = 0.74 \begin {gather*} \frac {a \tanh ^{-1}(c x)}{c}+\frac {i b \left (\text {PolyLog}\left (2,-\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )-\text {PolyLog}\left (2,\frac {i \sqrt {1+c x}}{\sqrt {1-c x}}\right )\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 258 vs. \(2 (78 ) = 156\).
time = 0.26, size = 259, normalized size = 2.64
method | result | size |
default | \(-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {a \ln \left (c x +1\right )}{2 c}+\frac {b \,\mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1-\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{c}-\frac {b \,\mathrm {arccot}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}+1\right )}{c}+\frac {i b \dilog \left (\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}+1\right )}{2 c}-\frac {i b \dilog \left (1-\frac {\left (i+\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{\frac {-c x +1}{c x +1}+1}\right )}{2 c}\) | \(259\) |
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int -\frac {a+b\,\mathrm {acot}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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