Integrand size = 16, antiderivative size = 75 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\frac {e F^{c (a+b x)} \cosh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)} \]
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Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5582} \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\frac {e \cosh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c \log (F) \sinh (d+e x) F^{c (a+b x)}}{e^2-b^2 c^2 \log ^2(F)} \]
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Rule 5582
Rubi steps \begin{align*} \text {integral}& = \frac {e F^{c (a+b x)} \cosh (d+e x)}{e^2-b^2 c^2 \log ^2(F)}-\frac {b c F^{c (a+b x)} \log (F) \sinh (d+e x)}{e^2-b^2 c^2 \log ^2(F)} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.67 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\frac {F^{c (a+b x)} (e \cosh (d+e x)-b c \log (F) \sinh (d+e x))}{(e-b c \log (F)) (e+b c \log (F))} \]
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Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68
method | result | size |
parallelrisch | \(\frac {F^{c \left (b x +a \right )} \left (\sinh \left (e x +d \right ) \ln \left (F \right ) b c -e \cosh \left (e x +d \right )\right )}{b^{2} c^{2} \ln \left (F \right )^{2}-e^{2}}\) | \(51\) |
risch | \(\frac {\left (\ln \left (F \right ) b c \,{\mathrm e}^{2 e x +2 d}-b c \ln \left (F \right )-e \,{\mathrm e}^{2 e x +2 d}-e \right ) {\mathrm e}^{-e x -d} F^{c \left (b x +a \right )}}{2 \left (b c \ln \left (F \right )-e \right ) \left (e +b c \ln \left (F \right )\right )}\) | \(77\) |
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (77) = 154\).
Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 3.25 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=-\frac {{\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} - b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) + e\right )} \cosh \left ({\left (b c x + a c\right )} \log \left (F\right )\right ) + {\left (e \cosh \left (e x + d\right )^{2} - {\left (b c \log \left (F\right ) - e\right )} \sinh \left (e x + d\right )^{2} - {\left (b c \cosh \left (e x + d\right )^{2} - b c\right )} \log \left (F\right ) - 2 \, {\left (b c \cosh \left (e x + d\right ) \log \left (F\right ) - e \cosh \left (e x + d\right )\right )} \sinh \left (e x + d\right ) + e\right )} \sinh \left ({\left (b c x + a c\right )} \log \left (F\right )\right )}{2 \, {\left (b^{2} c^{2} \cosh \left (e x + d\right ) \log \left (F\right )^{2} - e^{2} \cosh \left (e x + d\right ) + {\left (b^{2} c^{2} \log \left (F\right )^{2} - e^{2}\right )} \sinh \left (e x + d\right )\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (68) = 136\).
Time = 0.58 (sec) , antiderivative size = 323, normalized size of antiderivative = 4.31 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\begin {cases} x \sinh {\left (d \right )} & \text {for}\: F = 1 \wedge e = 0 \\F^{a c} x \sinh {\left (d \right )} & \text {for}\: b = 0 \wedge e = 0 \\x \sinh {\left (d \right )} & \text {for}\: c = 0 \wedge e = 0 \\- \frac {F^{a c + b c x} x \sinh {\left (b c x \log {\left (F \right )} - d \right )}}{2} + \frac {F^{a c + b c x} x \cosh {\left (b c x \log {\left (F \right )} - d \right )}}{2} + \frac {F^{a c + b c x} \sinh {\left (b c x \log {\left (F \right )} - d \right )}}{2 b c \log {\left (F \right )}} - \frac {F^{a c + b c x} \cosh {\left (b c x \log {\left (F \right )} - d \right )}}{b c \log {\left (F \right )}} & \text {for}\: e = - b c \log {\left (F \right )} \\\frac {F^{a c + b c x} x \sinh {\left (b c x \log {\left (F \right )} + d \right )}}{2} - \frac {F^{a c + b c x} x \cosh {\left (b c x \log {\left (F \right )} + d \right )}}{2} - \frac {F^{a c + b c x} \sinh {\left (b c x \log {\left (F \right )} + d \right )}}{2 b c \log {\left (F \right )}} + \frac {F^{a c + b c x} \cosh {\left (b c x \log {\left (F \right )} + d \right )}}{b c \log {\left (F \right )}} & \text {for}\: e = b c \log {\left (F \right )} \\\frac {F^{a c + b c x} b c \log {\left (F \right )} \sinh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - e^{2}} - \frac {F^{a c + b c x} e \cosh {\left (d + e x \right )}}{b^{2} c^{2} \log {\left (F \right )}^{2} - e^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\frac {F^{a c} e^{\left (b c x \log \left (F\right ) + e x + d\right )}}{2 \, {\left (b c \log \left (F\right ) + e\right )}} - \frac {F^{a c} e^{\left (b c x \log \left (F\right ) - e x\right )}}{2 \, {\left (b c e^{d} \log \left (F\right ) - e e^{d}\right )}} \]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 598, normalized size of antiderivative = 7.97 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\text {Too large to display} \]
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Time = 1.38 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int F^{c (a+b x)} \sinh (d+e x) \, dx=\frac {F^{a\,c+b\,c\,x}\,{\mathrm {e}}^{-d-e\,x}\,\left (e+e\,{\mathrm {e}}^{2\,d+2\,e\,x}+b\,c\,\ln \left (F\right )-b\,c\,{\mathrm {e}}^{2\,d+2\,e\,x}\,\ln \left (F\right )\right )}{2\,\left (e^2-b^2\,c^2\,{\ln \left (F\right )}^2\right )} \]
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