Optimal. Leaf size=43 \[ \frac{\sinh (a+x (b-d)-c)}{2 (b-d)}+\frac{\sinh (a+x (b+d)+c)}{2 (b+d)} \]
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Rubi [A] time = 0.0398964, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5614, 2637} \[ \frac{\sinh (a+x (b-d)-c)}{2 (b-d)}+\frac{\sinh (a+x (b+d)+c)}{2 (b+d)} \]
Antiderivative was successfully verified.
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Rule 5614
Rule 2637
Rubi steps
\begin{align*} \int \cosh (a+b x) \cosh (c+d x) \, dx &=\int \left (\frac{1}{2} \cosh (a-c+(b-d) x)+\frac{1}{2} \cosh (a+c+(b+d) x)\right ) \, dx\\ &=\frac{1}{2} \int \cosh (a-c+(b-d) x) \, dx+\frac{1}{2} \int \cosh (a+c+(b+d) x) \, dx\\ &=\frac{\sinh (a-c+(b-d) x)}{2 (b-d)}+\frac{\sinh (a+c+(b+d) x)}{2 (b+d)}\\ \end{align*}
Mathematica [A] time = 0.173012, size = 43, normalized size = 1. \[ \frac{\sinh (a+x (b-d)-c)}{2 (b-d)}+\frac{\sinh (a+x (b+d)+c)}{2 (b+d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 40, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( a-c+ \left ( b-d \right ) x \right ) }{2\,b-2\,d}}+{\frac{\sinh \left ( a+c+ \left ( b+d \right ) x \right ) }{2\,b+2\,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 [A] time = 1.92083, size = 169, normalized size = 3.93 \begin{align*} \frac{b \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - d \cosh \left (b x + a\right ) \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} -{\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.04283, size = 153, normalized size = 3.56 \begin{align*} \begin{cases} x \cosh{\left (a \right )} \cosh{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x \sinh{\left (a - d x \right )} \sinh{\left (c + d x \right )}}{2} + \frac{x \cosh{\left (a - d x \right )} \cosh{\left (c + d x \right )}}{2} - \frac{\sinh{\left (a - d x \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = - d \\- \frac{x \sinh{\left (a + d x \right )} \sinh{\left (c + d x \right )}}{2} + \frac{x \cosh{\left (a + d x \right )} \cosh{\left (c + d x \right )}}{2} + \frac{\sinh{\left (a + d x \right )} \cosh{\left (c + d x \right )}}{2 d} & \text{for}\: b = d \\\frac{b \sinh{\left (a + b x \right )} \cosh{\left (c + d x \right )}}{b^{2} - d^{2}} - \frac{d \sinh{\left (c + d x \right )} \cosh{\left (a + b x \right )}}{b^{2} - d^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17167, size = 115, normalized size = 2.67 \begin{align*} \frac{e^{\left (b x + d x + a + c\right )}}{4 \,{\left (b + d\right )}} + \frac{e^{\left (b x - d x + a - c\right )}}{4 \,{\left (b - d\right )}} - \frac{e^{\left (-b x + d x - a + c\right )}}{4 \,{\left (b - d\right )}} - \frac{e^{\left (-b x - d x - a - c\right )}}{4 \,{\left (b + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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