Optimal. Leaf size=88 \[ \frac{\sinh (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac{\sinh (2 (a+c)+2 x (b+d))}{16 (b+d)}-\frac{\sinh (2 a+2 b x)}{8 b}-\frac{\sinh (2 c+2 d x)}{8 d}+\frac{x}{4} \]
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Rubi [A] time = 0.0685166, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {5613, 2637} \[ \frac{\sinh (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac{\sinh (2 (a+c)+2 x (b+d))}{16 (b+d)}-\frac{\sinh (2 a+2 b x)}{8 b}-\frac{\sinh (2 c+2 d x)}{8 d}+\frac{x}{4} \]
Antiderivative was successfully verified.
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Rule 5613
Rule 2637
Rubi steps
\begin{align*} \int \sinh ^2(a+b x) \sinh ^2(c+d x) \, dx &=\int \left (\frac{1}{4}-\frac{1}{4} \cosh (2 a+2 b x)+\frac{1}{8} \cosh (2 (a-c)+2 (b-d) x)-\frac{1}{4} \cosh (2 c+2 d x)+\frac{1}{8} \cosh (2 (a+c)+2 (b+d) x)\right ) \, dx\\ &=\frac{x}{4}+\frac{1}{8} \int \cosh (2 (a-c)+2 (b-d) x) \, dx+\frac{1}{8} \int \cosh (2 (a+c)+2 (b+d) x) \, dx-\frac{1}{4} \int \cosh (2 a+2 b x) \, dx-\frac{1}{4} \int \cosh (2 c+2 d x) \, dx\\ &=\frac{x}{4}-\frac{\sinh (2 a+2 b x)}{8 b}+\frac{\sinh (2 (a-c)+2 (b-d) x)}{16 (b-d)}-\frac{\sinh (2 c+2 d x)}{8 d}+\frac{\sinh (2 (a+c)+2 (b+d) x)}{16 (b+d)}\\ \end{align*}
Mathematica [A] time = 0.723742, size = 106, normalized size = 1.2 \[ \frac{\left (2 d^3-2 b^2 d\right ) \sinh (2 (a+b x))+b d (b+d) \sinh (2 (a+x (b-d)-c))+b (b-d) (d (\sinh (2 (a+x (b+d)+c))+4 x (b+d))-2 (b+d) \sinh (2 (c+d x)))}{16 b d (b-d) (b+d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 83, normalized size = 0.9 \begin{align*}{\frac{x}{4}}-{\frac{\sinh \left ( 2\,bx+2\,a \right ) }{8\,b}}-{\frac{\sinh \left ( 2\,dx+2\,c \right ) }{8\,d}}+{\frac{\sinh \left ( \left ( 2\,b-2\,d \right ) x+2\,a-2\,c \right ) }{16\,b-16\,d}}+{\frac{\sinh \left ( \left ( 2\,b+2\,d \right ) x+2\,a+2\,c \right ) }{16\,b+16\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [B] time = 1.88068, size = 455, normalized size = 5.17 \begin{align*} \frac{b^{2} d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} +{\left (b^{3} d - b d^{3}\right )} x +{\left (b^{2} d \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} -{\left (b^{2} d - d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) -{\left (b d^{2} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} +{\left (b d^{2} \cosh \left (b x + a\right )^{2} + b^{3} - b d^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{4 \,{\left ({\left (b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{3} d - b d^{3}\right )} \sinh \left (b x + a\right )^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 37.4244, size = 1027, normalized size = 11.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2311, size = 211, normalized size = 2.4 \begin{align*} \frac{1}{4} \, x + \frac{e^{\left (2 \, b x + 2 \, d x + 2 \, a + 2 \, c\right )}}{32 \,{\left (b + d\right )}} + \frac{e^{\left (2 \, b x - 2 \, d x + 2 \, a - 2 \, c\right )}}{32 \,{\left (b - d\right )}} - \frac{e^{\left (2 \, b x + 2 \, a\right )}}{16 \, b} - \frac{e^{\left (-2 \, b x + 2 \, d x - 2 \, a + 2 \, c\right )}}{32 \,{\left (b - d\right )}} - \frac{e^{\left (-2 \, b x - 2 \, d x - 2 \, a - 2 \, c\right )}}{32 \,{\left (b + d\right )}} + \frac{e^{\left (-2 \, b x - 2 \, a\right )}}{16 \, b} - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{16 \, d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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