Optimal. Leaf size=89 \[ -\frac {a \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}+\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{2 c} \]
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Rubi [A]
time = 0.04, 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 = {212, 6813,
6031} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 6031
Rule 6813
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 {\text {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 = 0.19, size = 43, normalized size = 0.48 \begin {gather*} \frac {a \tanh ^{-1}(c x)}{c}+\frac {b \left (\text {PolyLog}\left (2,-e^{-\tanh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,e^{-\tanh ^{-1}(c x)}\right )\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 117, normalized size = 1.31
method | result | size |
default | \(\frac {a \ln \left (c x +1\right )}{2 c}-\frac {a \ln \left (c x -1\right )}{2 c}-\frac {b \left (4 \dilog \left (\frac {-\frac {-c x +1}{c x +1}+1}{\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{2}}\right )-\dilog \left (\frac {\left (-\frac {-c x +1}{c x +1}+1\right )^{2}}{\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1\right )^{4}}\right )\right )}{4 c}\) | \(117\) |
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 \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 \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 {atanh}\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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