Optimal. Leaf size=107 \[ \frac{\left (1-\sqrt{2}\right ) \log \left (-e^{2 c (a+b x)}+3-2 \sqrt{2}\right )}{2 b c}+\frac{\left (1+\sqrt{2}\right ) \log \left (-e^{2 c (a+b x)}+3+2 \sqrt{2}\right )}{2 b c}+\frac{e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c} \]
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Rubi [A] time = 0.1546, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2194, 6276, 2282, 12, 1247, 632, 31} \[ \frac{\left (1-\sqrt{2}\right ) \log \left (-e^{2 c (a+b x)}+3-2 \sqrt{2}\right )}{2 b c}+\frac{\left (1+\sqrt{2}\right ) \log \left (-e^{2 c (a+b x)}+3+2 \sqrt{2}\right )}{2 b c}+\frac{e^{a c+b c x} \coth ^{-1}(\sinh (c (a+b x)))}{b c} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 6276
Rule 2282
Rule 12
Rule 1247
Rule 632
Rule 31
Rubi steps
\begin{align*} \int e^{c (a+b x)} \coth ^{-1}(\sinh (a c+b c x)) \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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{\operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [A] time = 0.164186, size = 153, normalized size = 1.43 \[ \frac{\log \left (-2 e^{c (a+b x)}-e^{2 c (a+b x)}+1\right )+\log \left (2 e^{c (a+b x)}-e^{2 c (a+b x)}+1\right )-2 \sqrt{2} \tanh ^{-1}\left (\frac{e^{c (a+b x)}-1}{\sqrt{2}}\right )+2 \sqrt{2} \tanh ^{-1}\left (\frac{e^{c (a+b x)}+1}{\sqrt{2}}\right )-2 e^{c (a+b x)} \coth ^{-1}\left (\frac{1}{2} e^{-c (a+b x)}-\frac{1}{2} e^{c (a+b x)}\right )}{2 b c} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.329, size = 794, normalized size = 7.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.61425, size = 248, normalized size = 2.32 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.74446, size = 624, normalized size = 5.83 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{arcoth}\left (\sinh \left (b c x + a c\right )\right ) e^{\left ({\left (b x + a\right )} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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