Optimal. Leaf size=162 \[ \frac{e^{-2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{16 b c}-\frac{3 e^{2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{16 b c}+\frac{e^{4 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{32 b c}+\frac{3}{8} x \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x) \]
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Rubi [A] time = 0.139462, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 2282, 12, 266, 43} \[ \frac{e^{-2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{16 b c}-\frac{3 e^{2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{16 b c}+\frac{e^{4 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{32 b c}+\frac{3}{8} x \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x) \]
Antiderivative was successfully verified.
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Rule 6720
Rule 2282
Rule 12
Rule 266
Rule 43
Rubi steps
\begin{align*} \int e^{c (a+b x)} \sinh ^2(a c+b c x)^{3/2} \, dx &=\left (\text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\right ) \int e^{c (a+b x)} \sinh ^3(a c+b c x) \, dx\\ &=\frac{\left (\text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3}{8 x^3} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (\text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^3}{x^3} \, dx,x,e^{c (a+b x)}\right )}{8 b c}\\ &=\frac{\left (\text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \frac{(-1+x)^3}{x^2} \, dx,x,e^{2 c (a+b x)}\right )}{16 b c}\\ &=\frac{\left (\text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\right ) \operatorname{Subst}\left (\int \left (-3-\frac{1}{x^2}+\frac{3}{x}+x\right ) \, dx,x,e^{2 c (a+b x)}\right )}{16 b c}\\ &=\frac{e^{-2 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{16 b c}-\frac{3 e^{2 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{16 b c}+\frac{e^{4 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{32 b c}+\frac{3}{8} x \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\\ \end{align*}
Mathematica [A] time = 0.0582139, size = 76, normalized size = 0.47 \[ \frac{\left (e^{-2 c (a+b x)}-3 e^{2 c (a+b x)}+\frac{1}{2} e^{4 c (a+b x)}+6 b c x\right ) \sinh ^2(c (a+b x))^{3/2} \text{csch}^3(c (a+b x))}{16 b c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 152, normalized size = 0.9 \begin{align*}{\frac{1}{8\,\sinh \left ( c \left ( bx+a \right ) \right ) cb} \left ( 2\,\sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}} \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{3}\cosh \left ( c \left ( bx+a \right ) \right ) +2\,\sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}} \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{4}-3\,\cosh \left ( c \left ( bx+a \right ) \right ) \sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}}\sinh \left ( c \left ( bx+a \right ) \right ) +3\,\ln \left ( \cosh \left ( c \left ( bx+a \right ) \right ) +\sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}} \right ) \sinh \left ( c \left ( bx+a \right ) \right ) -2\,\sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.66537, size = 84, normalized size = 0.52 \begin{align*} \frac{{\left (e^{\left (6 \, b c x + 6 \, a c\right )} - 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 2\right )} e^{\left (-2 \, b c x - 2 \, a c\right )}}{32 \, b c} + \frac{3 \,{\left (b c x + a c\right )}}{8 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88162, size = 319, normalized size = 1.97 \begin{align*} \frac{3 \, \cosh \left (b c x + a c\right )^{3} + 9 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} - \sinh \left (b c x + a c\right )^{3} + 6 \,{\left (2 \, b c x - 1\right )} \cosh \left (b c x + a c\right ) - 3 \,{\left (4 \, b c x + \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right )}{32 \,{\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \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 [A] time = 1.15974, size = 263, normalized size = 1.62 \begin{align*} \frac{12 \, b c x \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 2 \,{\left (3 \, e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-2 \, b c x - 2 \, a c\right )} +{\left (e^{\left (4 \, b c x + 8 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 6 \, e^{\left (2 \, b c x + 6 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-4 \, a c\right )}}{32 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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