Integrand size = 20, antiderivative size = 107 \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}-e^{2 c (a+b x)}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}-e^{2 c (a+b x)}\right )}{2 b c} \]
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Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {2225, 6411, 2320, 12, 1261, 646, 31} \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {\left (1-\sqrt {2}\right ) \log \left (-e^{2 c (a+b x)}+3-2 \sqrt {2}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (-e^{2 c (a+b x)}+3+2 \sqrt {2}\right )}{2 b c}+\frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c} \]
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Rule 12
Rule 31
Rule 646
Rule 1261
Rule 2225
Rule 2320
Rule 6411
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^x \coth ^{-1}(\sinh (x)) \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int \frac {e^x \cosh (x)}{1-\sinh ^2(x)} \, dx,x,a c+b c x\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int \frac {2 x \left (-1-x^2\right )}{1-6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}-\frac {2 \text {Subst}\left (\int \frac {x \left (-1-x^2\right )}{1-6 x^2+x^4} \, dx,x,e^{a c+b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}-\frac {\text {Subst}\left (\int \frac {-1-x}{1-6 x+x^2} \, dx,x,e^{2 a c+2 b c x}\right )}{b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-3+2 \sqrt {2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-3-2 \sqrt {2}+x} \, dx,x,e^{2 a c+2 b c x}\right )}{2 b c} \\ & = \frac {e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c}+\frac {\left (1-\sqrt {2}\right ) \log \left (3-2 \sqrt {2}-e^{2 a c+2 b c x}\right )}{2 b c}+\frac {\left (1+\sqrt {2}\right ) \log \left (3+2 \sqrt {2}-e^{2 a c+2 b c x}\right )}{2 b c} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.43 \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {-2 e^{c (a+b x)} \coth ^{-1}\left (\frac {1}{2} e^{-c (a+b x)}-\frac {1}{2} e^{c (a+b x)}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {-1+e^{c (a+b x)}}{\sqrt {2}}\right )+2 \sqrt {2} \text {arctanh}\left (\frac {1+e^{c (a+b x)}}{\sqrt {2}}\right )+\log \left (1-2 e^{c (a+b x)}-e^{2 c (a+b x)}\right )+\log \left (1+2 e^{c (a+b x)}-e^{2 c (a+b x)}\right )}{2 b c} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.45 (sec) , antiderivative size = 794, normalized size of antiderivative = 7.42
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (90) = 180\).
Time = 0.27 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.18 \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {{\left (\cosh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )\right )} \log \left (\frac {\sinh \left (b c x + a c\right ) + 1}{\sinh \left (b c x + a c\right ) - 1}\right ) + \sqrt {2} \log \left (\frac {3 \, {\left (2 \, \sqrt {2} + 3\right )} \cosh \left (b c x + a c\right )^{2} - 4 \, {\left (3 \, \sqrt {2} + 4\right )} \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 3 \, {\left (2 \, \sqrt {2} + 3\right )} \sinh \left (b c x + a c\right )^{2} - 2 \, \sqrt {2} - 3}{\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} - 3}\right ) + \log \left (\frac {2 \, {\left (\cosh \left (b c x + a c\right )^{2} + \sinh \left (b c x + a c\right )^{2} - 3\right )}}{\cosh \left (b c x + a c\right )^{2} - 2 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + \sinh \left (b c x + a c\right )^{2}}\right )}{2 \, b c} \]
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\[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=e^{a c} \int e^{b c x} \operatorname {acoth}{\left (\sinh {\left (a c + b c x \right )} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (90) = 180\).
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.72 \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {\operatorname {arcoth}\left (\sinh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}}{b c} + \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (b c x + a c\right )} + 1}{\sqrt {2} + e^{\left (b c x + a c\right )} - 1}\right )}{2 \, b c} - \frac {\sqrt {2} \log \left (-\frac {\sqrt {2} - e^{\left (b c x + a c\right )} - 1}{\sqrt {2} + e^{\left (b c x + a c\right )} + 1}\right )}{2 \, b c} + \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} + 2 \, e^{\left (b c x + a c\right )} - 1\right )}{2 \, b c} + \frac {\log \left (e^{\left (2 \, b c x + 2 \, a c\right )} - 2 \, e^{\left (b c x + a c\right )} - 1\right )}{2 \, b c} \]
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Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.56 \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {e^{\left ({\left (b x + a\right )} c\right )} \log \left (-\frac {\frac {2}{e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}} + 1}{\frac {2}{e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}} - 1}\right )}{2 \, b c} + \frac {\sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} - 6 \right |}}{{\left | 4 \, \sqrt {2} + 2 \, e^{\left (2 \, b c x + 2 \, a c\right )} - 6 \right |}}\right ) + \log \left ({\left | e^{\left (4 \, b c x + 4 \, a c\right )} - 6 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1 \right |}\right )}{2 \, b c} \]
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Time = 4.57 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.75 \[ \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx=\frac {\ln \left (6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-2\,\sqrt {2}-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}+1\right )}{2\,b\,c}-\frac {{\mathrm {e}}^{a\,c+b\,c\,x}\,\ln \left (1-\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}\right )}{2\,b\,c}-\frac {\ln \left (2\,\sqrt {2}-8\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}-6\,\sqrt {2}\,{\mathrm {e}}^{2\,c\,\left (a+b\,x\right )}\right )\,\left (\sqrt {2}-1\right )}{2\,b\,c}+\frac {\ln \left (\frac {1}{\frac {{\mathrm {e}}^{b\,c\,x}\,{\mathrm {e}}^{a\,c}}{2}-\frac {{\mathrm {e}}^{-b\,c\,x}\,{\mathrm {e}}^{-a\,c}}{2}}+1\right )\,{\mathrm {e}}^{a\,c+b\,c\,x}}{2\,b\,c} \]
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