Optimal. Leaf size=133 \[ -\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 b c}-\frac {\log \left (e^{-2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c}+\frac {b \text {Li}_2\left (-e^{-2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 c} \]
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Rubi [A] time = 0.12, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {206, 6681, 5660, 3718, 2190, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}\right )}{2 c}+\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{2 b c}-\frac {\log \left (e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )}+1\right ) \left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )}{c} \]
Warning: Unable to verify antiderivative.
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Rule 206
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 6681
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-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 \cosh ^{-1}(x)}{x} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{c}\\ &=-\frac {\operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c}-\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c}-\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c}\\ &=\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c}-\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}\\ &=\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{2 b c}-\frac {\left (a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right ) \log \left (1+e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{c}-\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}\right )}{2 c}\\ \end {align*}
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Mathematica [F] time = 3.16, size = 0, normalized size = 0.00 \[ \int \frac {a+b \cosh ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )}{1-c^2 x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {arcosh}\left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a}{c^{2} x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 = 0.01, size = 207, normalized size = 1.56 \[ \frac {a \ln \left (c x +1\right )}{2 c}-\frac {a \ln \left (c x -1\right )}{2 c}+\frac {b \mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right )^{2}}{2 c}-\frac {b \,\mathrm {arccosh}\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}\right ) \ln \left (1+\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{c}-\frac {b \polylog \left (2, -\left (\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+\sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}-1}\, \sqrt {\frac {\sqrt {-c x +1}}{\sqrt {c x +1}}+1}\right )^{2}\right )}{2 c} \]
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 \[ -\frac {1}{8} \, b {\left (\frac {2 \, {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) - \log \left (c x + 1\right )^{2} + 2 \, \log \left (c x + 1\right ) \log \left (-c x + 1\right ) - \log \left (-c x + 1\right )^{2} - 4 \, {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \log \left (\sqrt {\sqrt {c x + 1} + \sqrt {-c x + 1}} \sqrt {-\sqrt {c x + 1} + \sqrt {-c x + 1}} + \sqrt {-c x + 1}\right )}{c} + 8 \, \int \frac {{\left (c x + 1\right )} \sqrt {-c x + 1} {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{2 \, {\left ({\left (c^{2} x^{2} - 1\right )} {\left (c x + 1\right )} \sqrt {-c x + 1} - {\left (c^{2} x^{2} - 1\right )} {\left (-c x + 1\right )}^{\frac {3}{2}} + {\left ({\left (c^{2} x^{2} - 1\right )} {\left (c x + 1\right )} + {\left (c^{2} x^{2} - 1\right )} {\left (c x - 1\right )}\right )} \sqrt {\sqrt {c x + 1} + \sqrt {-c x + 1}} \sqrt {-\sqrt {c x + 1} + \sqrt {-c x + 1}}\right )}}\,{d x} + 8 \, \int -\frac {\sqrt {c x + 1} {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{4 \, {\left ({\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} + {\left (c^{2} x^{2} - 1\right )} \sqrt {-c x + 1}\right )}}\,{d x} - 8 \, \int \frac {\sqrt {c x + 1} {\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )}}{4 \, {\left ({\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} - {\left (c^{2} x^{2} - 1\right )} \sqrt {-c x + 1}\right )}}\,{d x}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} \]
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 \[ \int -\frac {a+b\,\mathrm {acosh}\left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )}{c^2\,x^2-1} \,d x \]
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 \[ - \int \frac {a}{c^{2} x^{2} - 1}\, dx - \int \frac {b \operatorname {acosh}{\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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