Optimal. Leaf size=66 \[ \frac{b e^{c+d x} \cosh (2 a+2 b x)}{4 b^2-d^2}-\frac{d e^{c+d x} \sinh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0471871, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {5509, 12, 5474} \[ \frac{b e^{c+d x} \cosh (2 a+2 b x)}{4 b^2-d^2}-\frac{d e^{c+d x} \sinh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5509
Rule 12
Rule 5474
Rubi steps
\begin{align*} \int e^{c+d x} \cosh (a+b x) \sinh (a+b x) \, dx &=\int \frac{1}{2} e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{2} \int e^{c+d x} \sinh (2 a+2 b x) \, dx\\ &=\frac{b e^{c+d x} \cosh (2 a+2 b x)}{4 b^2-d^2}-\frac{d e^{c+d x} \sinh (2 a+2 b x)}{2 \left (4 b^2-d^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0526181, size = 47, normalized size = 0.71 \[ \frac{e^{c+d x} (2 b \cosh (2 (a+b x))-d \sinh (2 (a+b x)))}{2 \left (4 b^2-d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 102, normalized size = 1.6 \begin{align*} -{\frac{\sinh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{8\,b-4\,d}}+{\frac{\sinh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{8\,b+4\,d}}+{\frac{\cosh \left ( 2\,a-c+ \left ( 2\,b-d \right ) x \right ) }{8\,b-4\,d}}+{\frac{\cosh \left ( 2\,a+c+ \left ( 2\,b+d \right ) x \right ) }{8\,b+4\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.76207, size = 360, normalized size = 5.45 \begin{align*} \frac{b \cosh \left (b x + a\right )^{2} \cosh \left (d x + c\right ) - d \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) + b \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} +{\left (b \cosh \left (b x + a\right )^{2} - d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )}{{\left (4 \, b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} -{\left (4 \, b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 16.925, size = 347, normalized size = 5.26 \begin{align*} \begin{cases} x e^{c} \sinh{\left (a \right )} \cosh{\left (a \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x e^{c} e^{d x} \sinh ^{2}{\left (a - \frac{d x}{2} \right )}}{4} + \frac{x e^{c} e^{d x} \sinh{\left (a - \frac{d x}{2} \right )} \cosh{\left (a - \frac{d x}{2} \right )}}{2} + \frac{x e^{c} e^{d x} \cosh ^{2}{\left (a - \frac{d x}{2} \right )}}{4} + \frac{e^{c} e^{d x} \sinh{\left (a - \frac{d x}{2} \right )} \cosh{\left (a - \frac{d x}{2} \right )}}{2 d} & \text{for}\: b = - \frac{d}{2} \\- \frac{x e^{c} e^{d x} \sinh ^{2}{\left (a + \frac{d x}{2} \right )}}{4} + \frac{x e^{c} e^{d x} \sinh{\left (a + \frac{d x}{2} \right )} \cosh{\left (a + \frac{d x}{2} \right )}}{2} - \frac{x e^{c} e^{d x} \cosh ^{2}{\left (a + \frac{d x}{2} \right )}}{4} + \frac{e^{c} e^{d x} \sinh ^{2}{\left (a + \frac{d x}{2} \right )}}{d} - \frac{3 e^{c} e^{d x} \sinh{\left (a + \frac{d x}{2} \right )} \cosh{\left (a + \frac{d x}{2} \right )}}{2 d} + \frac{e^{c} e^{d x} \cosh ^{2}{\left (a + \frac{d x}{2} \right )}}{d} & \text{for}\: b = \frac{d}{2} \\\frac{b e^{c} e^{d x} \sinh ^{2}{\left (a + b x \right )}}{4 b^{2} - d^{2}} + \frac{b e^{c} e^{d x} \cosh ^{2}{\left (a + b x \right )}}{4 b^{2} - d^{2}} - \frac{d e^{c} e^{d x} \sinh{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{4 b^{2} - d^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18206, size = 63, normalized size = 0.95 \begin{align*} \frac{e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{4 \,{\left (2 \, b + d\right )}} + \frac{e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{4 \,{\left (2 \, b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]