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 = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 = {5613, 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 2637
Rule 5613
Rubi steps
\begin {align*} \int \sinh (a+b x) \sinh (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\\ &=-\left (\frac {1}{2} \int \cosh (a-c+(b-d) x) \, dx\right )+\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*}
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Mathematica [A] time = 0.23, size = 43, normalized size = 1.00 \[ \frac {\sinh (a+x (b+d)+c)}{2 (b+d)}-\frac {\sinh (a+x (b-d)-c)}{2 (b-d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 72, normalized size = 1.67 \[ -\frac {d \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) - b \cosh \left (b x + a\right ) \sinh \left (d x + c\right )}{{\left (b^{2} - d^{2}\right )} \cosh \left (b x + a\right )^{2} - {\left (b^{2} - d^{2}\right )} \sinh \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.12, size = 85, normalized size = 1.98 \[ \frac {e^{\left (b x + d x + a + c\right )}}{4 \, {\left (b + d\right )}} - \frac {e^{\left (b x - d x + a - c\right )}}{4 \, {\left (b - d\right )}} + \frac {e^{\left (-b x + d x - a + c\right )}}{4 \, {\left (b - d\right )}} - \frac {e^{\left (-b x - d x - a - c\right )}}{4 \, {\left (b + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 40, normalized size = 0.93 \[ -\frac {\sinh \left (a -c +\left (b -d \right ) x \right )}{2 \left (b -d \right )}+\frac {\sinh \left (a +c +\left (b +d \right ) x \right )}{2 b +2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 42, normalized size = 0.98 \[ \frac {b\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )-d\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{b^2-d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.47, size = 153, normalized size = 3.56 \[ \begin {cases} x \sinh {\relax (a )} \sinh {\relax (c )} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh {\left (a - d x \right )} \sinh {\left (c + d x \right )}}{2} + \frac {x \cosh {\left (a - d x \right )} \cosh {\left (c + d x \right )}}{2} - \frac {\sinh {\left (c + d x \right )} \cosh {\left (a - d x \right )}}{2 d} & \text {for}\: b = - d \\\frac {x \sinh {\left (a + d x \right )} \sinh {\left (c + d x \right )}}{2} - \frac {x \cosh {\left (a + d x \right )} \cosh {\left (c + d x \right )}}{2} + \frac {\sinh {\left (c + d x \right )} \cosh {\left (a + d x \right )}}{2 d} & \text {for}\: b = d \\\frac {b \sinh {\left (c + d x \right )} \cosh {\left (a + b x \right )}}{b^{2} - d^{2}} - \frac {d \sinh {\left (a + b x \right )} \cosh {\left (c + d x \right )}}{b^{2} - d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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