Optimal. Leaf size=27 \[ \frac {F^{a+b x} \left (e^{c+d x}\right )^n}{b \log (F)+d n} \]
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Rubi [A] time = 0.09, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5648, 2281, 2287, 2194} \[ \frac {F^{a+b x} \left (e^{c+d x}\right )^n}{b \log (F)+d n} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2281
Rule 2287
Rule 5648
Rubi steps
\begin {align*} \int F^{a+b x} (\cosh (c+d x)+\sinh (c+d x))^n \, dx &=\int \left (e^{c+d x}\right )^n F^{a+b x} \, dx\\ &=\left (e^{-n (c+d x)} \left (e^{c+d x}\right )^n\right ) \int e^{n (c+d x)} F^{a+b x} \, dx\\ &=\left (e^{-n (c+d x)} \left (e^{c+d x}\right )^n\right ) \int e^{c n+a \log (F)+x (d n+b \log (F))} \, dx\\ &=\frac {\left (e^{c+d x}\right )^n F^{a+b x}}{d n+b \log (F)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 33, normalized size = 1.22 \[ \frac {F^{a+b x} (\sinh (c+d x)+\cosh (c+d x))^n}{b \log (F)+d n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 70, normalized size = 2.59 \[ \frac {{\left (\cosh \left (d n x + c n\right ) + \sinh \left (d n x + c n\right )\right )} \cosh \left ({\left (b x + a\right )} \log \relax (F)\right ) + {\left (\cosh \left (d n x + c n\right ) + \sinh \left (d n x + c n\right )\right )} \sinh \left ({\left (b x + a\right )} \log \relax (F)\right )}{d n + b \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.17, size = 274, normalized size = 10.15 \[ 2 \, {\left (\frac {2 \, {\left (d n + b \log \left ({\left | F \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi a\right )}{{\left (\pi b \mathrm {sgn}\relax (F) - \pi b\right )}^{2} + 4 \, {\left (d n + b \log \left ({\left | F \right |}\right )\right )}^{2}} - \frac {{\left (\pi b \mathrm {sgn}\relax (F) - \pi b\right )} \sin \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi a\right )}{{\left (\pi b \mathrm {sgn}\relax (F) - \pi b\right )}^{2} + 4 \, {\left (d n + b \log \left ({\left | F \right |}\right )\right )}^{2}}\right )} e^{\left (c n + {\left (d n + b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} - \frac {1}{2} i \, {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi b x \mathrm {sgn}\relax (F) - \frac {1}{2} i \, \pi b x + \frac {1}{2} i \, \pi a \mathrm {sgn}\relax (F) - \frac {1}{2} i \, \pi a\right )}}{i \, \pi b \mathrm {sgn}\relax (F) - i \, \pi b + 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi b x \mathrm {sgn}\relax (F) + \frac {1}{2} i \, \pi b x - \frac {1}{2} i \, \pi a \mathrm {sgn}\relax (F) + \frac {1}{2} i \, \pi a\right )}}{-i \, \pi b \mathrm {sgn}\relax (F) + i \, \pi b + 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )}\right )} e^{\left (c n + {\left (d n + b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 34, normalized size = 1.26 \[ \frac {F^{b x +a} \left (\cosh \left (d x +c \right )+\sinh \left (d x +c \right )\right )^{n}}{d n +b \ln \relax (F )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 28, normalized size = 1.04 \[ \frac {F^{a} e^{\left (d n x + b x \log \relax (F) + c n\right )}}{d n + b \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int F^{a+b\,x}\,{\left (\mathrm {cosh}\left (c+d\,x\right )+\mathrm {sinh}\left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.80, size = 94, normalized size = 3.48 \[ \begin {cases} \frac {F^{a} F^{b x} \left (\sinh {\left (c + d x \right )} + \cosh {\left (c + d x \right )}\right )^{n}}{b \log {\relax (F )} + d n} & \text {for}\: b \neq - \frac {d n}{\log {\relax (F )}} \\F^{a} x \left (\sinh {\left (c + d x \right )} + \cosh {\left (c + d x \right )}\right )^{n} e^{- d n x} + \frac {F^{a} \left (\sinh {\left (c + d x \right )} + \cosh {\left (c + d x \right )}\right )^{n} e^{- d n x}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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