Optimal. Leaf size=250 \[ \frac{e^{-4 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{128 b c}-\frac{5 e^{-2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{64 b c}+\frac{5 e^{2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{32 b c}-\frac{5 e^{4 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{128 b c}+\frac{e^{6 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{192 b c}-\frac{5}{16} x \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.247979, antiderivative size = 250, 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} \[ \frac{e^{-4 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{128 b c}-\frac{5 e^{-2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{64 b c}+\frac{5 e^{2 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{32 b c}-\frac{5 e^{4 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{128 b c}+\frac{e^{6 c (a+b x)} \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x)}{192 b c}-\frac{5}{16} x \sqrt{\sinh ^2(a c+b c x)} \text{csch}(a c+b c x) \]
Antiderivative was successfully verified.
[In]
[Out]
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)^{5/2} \, 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 ^5(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 )^5}{32 x^5} \, 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 )^5}{x^5} \, dx,x,e^{c (a+b x)}\right )}{32 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)^5}{x^3} \, dx,x,e^{2 c (a+b x)}\right )}{64 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 (10-\frac{1}{x^3}+\frac{5}{x^2}-\frac{10}{x}-5 x+x^2\right ) \, dx,x,e^{2 c (a+b x)}\right )}{64 b c}\\ &=\frac{e^{-4 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{128 b c}-\frac{5 e^{-2 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{64 b c}+\frac{5 e^{2 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{32 b c}-\frac{5 e^{4 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{128 b c}+\frac{e^{6 c (a+b x)} \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}}{192 b c}-\frac{5}{16} x \text{csch}(a c+b c x) \sqrt{\sinh ^2(a c+b c x)}\\ \end{align*}
Mathematica [A] time = 0.109684, size = 106, normalized size = 0.42 \[ \frac{\left (\frac{1}{2} e^{-4 c (a+b x)}-5 e^{-2 c (a+b x)}+10 e^{2 c (a+b x)}-\frac{5}{2} e^{4 c (a+b x)}+\frac{1}{3} e^{6 c (a+b x)}-20 b c x\right ) \sinh ^2(c (a+b x))^{5/2} \text{csch}^5(c (a+b x))}{64 b c} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.163, size = 184, normalized size = 0.7 \begin{align*}{\frac{1}{48\,\sinh \left ( c \left ( bx+a \right ) \right ) cb} \left ( 8\,\sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}}\cosh \left ( c \left ( bx+a \right ) \right ) \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{5}+8\,\sqrt{ \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{2}} \left ( \sinh \left ( c \left ( bx+a \right ) \right ) \right ) ^{6}-10\,\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 ) +15\,\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 ) -15\,\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 ) +8\,\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.59572, size = 122, normalized size = 0.49 \begin{align*} \frac{{\left (2 \, e^{\left (10 \, b c x + 10 \, a c\right )} - 15 \, e^{\left (8 \, b c x + 8 \, a c\right )} + 60 \, e^{\left (6 \, b c x + 6 \, a c\right )} - 30 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 3\right )} e^{\left (-4 \, b c x - 4 \, a c\right )}}{384 \, b c} - \frac{5 \,{\left (b c x + a c\right )}}{16 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.85523, size = 559, normalized size = 2.24 \begin{align*} \frac{5 \, \cosh \left (b c x + a c\right )^{5} + 25 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{4} - \sinh \left (b c x + a c\right )^{5} - 5 \,{\left (2 \, \cosh \left (b c x + a c\right )^{2} - 3\right )} \sinh \left (b c x + a c\right )^{3} - 45 \, \cosh \left (b c x + a c\right )^{3} + 5 \,{\left (10 \, \cosh \left (b c x + a c\right )^{3} - 27 \, \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{2} - 60 \,{\left (2 \, b c x - 1\right )} \cosh \left (b c x + a c\right ) - 5 \,{\left (\cosh \left (b c x + a c\right )^{4} - 24 \, b c x - 9 \, \cosh \left (b c x + a c\right )^{2} - 12\right )} \sinh \left (b c x + a c\right )}{384 \,{\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23678, size = 363, normalized size = 1.45 \begin{align*} -\frac{120 \, b c x \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 3 \,{\left (30 \, e^{\left (4 \, b c x + 4 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 10 \, 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 (-4 \, b c x - 4 \, a c\right )} -{\left (2 \, e^{\left (6 \, b c x + 18 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 15 \, e^{\left (4 \, b c x + 16 \, a c\right )} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) + 60 \, e^{\left (2 \, b c x + 14 \, 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 (-12 \, a c\right )}}{384 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]