Optimal. Leaf size=76 \[ \frac {\text {Li}_2\left (-e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}+1\right )}{b} \]
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Rubi [A] time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2282, 5660, 3718, 2190, 2279, 2391} \[ \frac {\text {PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}-\frac {\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}+1\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2282
Rule 2391
Rule 3718
Rule 5660
Rubi steps
\begin {align*} \int \cosh ^{-1}\left (c e^{a+b x}\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cosh ^{-1}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {2 \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (c e^{a+b x}\right )\right )}{b}\\ &=-\frac {\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{b}-\frac {\operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}\\ &=-\frac {\cosh ^{-1}\left (c e^{a+b x}\right )^2}{2 b}+\frac {\cosh ^{-1}\left (c e^{a+b x}\right ) \log \left (1+e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{b}+\frac {\text {Li}_2\left (-e^{2 \cosh ^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}\\ \end {align*}
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Mathematica [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \cosh ^{-1}\left (c e^{a+b x}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcosh}\left (c e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 115, normalized size = 1.51 \[ -\frac {\mathrm {arccosh}\left (c \,{\mathrm e}^{b x +a}\right )^{2}}{2 b}+\frac {\mathrm {arccosh}\left (c \,{\mathrm e}^{b x +a}\right ) \ln \left (1+\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{b}+\frac {\polylog \left (2, -\left (c \,{\mathrm e}^{b x +a}+\sqrt {c \,{\mathrm e}^{b x +a}-1}\, \sqrt {c \,{\mathrm e}^{b x +a}+1}\right )^{2}\right )}{2 b} \]
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 \[ b c \int \frac {x e^{\left (b x + a\right )}}{c^{3} e^{\left (3 \, b x + 3 \, a\right )} - c e^{\left (b x + a\right )} + {\left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (c e^{\left (b x + a\right )} + 1\right ) + \frac {1}{2} \, \log \left (c e^{\left (b x + a\right )} - 1\right )\right )}}\,{d x} + x \log \left (c e^{\left (b x + a\right )} + \sqrt {c e^{\left (b x + a\right )} + 1} \sqrt {c e^{\left (b x + a\right )} - 1}\right ) - \frac {b x \log \left (c e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-c e^{\left (b x + a\right )}\right )}{2 \, b} - \frac {b x \log \left (-c e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (c e^{\left (b x + a\right )}\right )}{2 \, b} \]
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 \[ \int \mathrm {acosh}\left (c\,{\mathrm {e}}^{a+b\,x}\right ) \,d x \]
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 \[ \int \operatorname {acosh}{\left (c e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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