Optimal. Leaf size=49 \[ \frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}+\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.08, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2194, 6276, 2282, 12, 260} \[ \frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}+\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rule 2194
Rule 2282
Rule 6276
Rubi steps
\begin {align*} \int e^{c (a+b x)} \coth ^{-1}(\cosh (a c+b c x)) \, dx &=\frac {\operatorname {Subst}\left (\int e^x \coth ^{-1}(\cosh (x)) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\operatorname {Subst}\left (\int e^x \text {csch}(x) \, dx,x,a c+b c x\right )}{b c}\\ &=\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\operatorname {Subst}\left (\int \frac {2 x}{-1+x^2} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {2 \operatorname {Subst}\left (\int \frac {x}{-1+x^2} \, dx,x,e^{a c+b c x}\right )}{b c}\\ &=\frac {e^{a c+b c x} \coth ^{-1}(\cosh (c (a+b x)))}{b c}+\frac {\log \left (1-e^{2 c (a+b x)}\right )}{b c}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 60, normalized size = 1.22 \[ \frac {\log \left (1-e^{2 c (a+b x)}\right )+e^{c (a+b x)} \coth ^{-1}\left (\frac {1}{2} e^{-c (a+b x)} \left (e^{2 c (a+b x)}+1\right )\right )}{b c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.50, size = 92, normalized size = 1.88 \[ \frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (\frac {\cosh \left (b c x + a c\right ) + 1}{\cosh \left (b c x + a c\right ) - 1}\right ) + 2 \, \log \left (\frac {2 \, \sinh \left (b c x + a c\right )}{\cosh \left (b c x + a c\right ) - \sinh \left (b c x + a c\right )}\right )}{2 \, b c} \]
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 \[ \int \operatorname {arcoth}\left (\cosh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.43, size = 824, normalized size = 16.82 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 64, normalized size = 1.31 \[ \frac {\operatorname {arcoth}\left (\cosh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} + 1\right )}{b c} + \frac {\log \left (e^{\left (b c x + a c\right )} - 1\right )}{b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.83, size = 119, normalized size = 2.43 \[ \frac {\ln \left ({\mathrm {e}}^{2\,b\,c\,x}\,{\mathrm {e}}^{2\,a\,c}-1\right )}{b\,c}-\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\ln \left (1-\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{2\,b\,c}+\frac {\ln \left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}+\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}+1\right )\,{\mathrm {e}}^{a\,c+b\,c\,x}}{2\,b\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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