Optimal. Leaf size=32 \[ -\frac {F^{a+b x} \left (e^{-c-d x}\right )^n}{d n-b \log (F)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5648, 2281, 2287, 2194} \[ -\frac {F^{a+b x} \left (e^{-c-d x}\right )^n}{d n-b \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 37, normalized size = 1.16 \[ -\frac {F^{a+b x} (\cosh (c+d x)-\sinh (c+d x))^n}{d n-b \log (F)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 76, normalized size = 2.38 \[ -\frac {{\left (\cosh \left (d n x + c n\right ) - \sinh \left (d n x + c n\right )\right )} \cosh \left ({\left (b x + a\right )} \log \relax (F)\right ) + {\left (\cosh \left (d n x + c n\right ) - \sinh \left (d n x + c n\right )\right )} \sinh \left ({\left (b x + a\right )} \log \relax (F)\right )}{d n - b \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [C] time = 0.17, size = 282, normalized size = 8.81 \[ -2 \, {\left (\frac {2 \, {\left (d n - b \log \left ({\left | F \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi a\right )}{{\left (\pi b \mathrm {sgn}\relax (F) - \pi b\right )}^{2} + 4 \, {\left (d n - b \log \left ({\left | F \right |}\right )\right )}^{2}} + \frac {{\left (\pi b \mathrm {sgn}\relax (F) - \pi b\right )} \sin \left (-\frac {1}{2} \, \pi b x \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi b x - \frac {1}{2} \, \pi a \mathrm {sgn}\relax (F) + \frac {1}{2} \, \pi a\right )}{{\left (\pi b \mathrm {sgn}\relax (F) - \pi b\right )}^{2} + 4 \, {\left (d n - b \log \left ({\left | F \right |}\right )\right )}^{2}}\right )} e^{\left (-c n - {\left (d n - b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} - \frac {1}{2} i \, {\left (-\frac {2 i \, e^{\left (\frac {1}{2} i \, \pi b x \mathrm {sgn}\relax (F) - \frac {1}{2} i \, \pi b x + \frac {1}{2} i \, \pi a \mathrm {sgn}\relax (F) - \frac {1}{2} i \, \pi a\right )}}{i \, \pi b \mathrm {sgn}\relax (F) - i \, \pi b - 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )} + \frac {2 i \, e^{\left (-\frac {1}{2} i \, \pi b x \mathrm {sgn}\relax (F) + \frac {1}{2} i \, \pi b x - \frac {1}{2} i \, \pi a \mathrm {sgn}\relax (F) + \frac {1}{2} i \, \pi a\right )}}{-i \, \pi b \mathrm {sgn}\relax (F) + i \, \pi b - 2 \, d n + 2 \, b \log \left ({\left | F \right |}\right )}\right )} e^{\left (-c n - {\left (d n - b \log \left ({\left | F \right |}\right )\right )} x + a \log \left ({\left | F \right |}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 37, normalized size = 1.16 \[ \frac {F^{b x +a} \left (\cosh \left (d x +c \right )-\sinh \left (d x +c \right )\right )^{n}}{b \ln \relax (F )-d n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 36, normalized size = 1.12 \[ -\frac {F^{a} e^{\left (-d n x + b x \log \relax (F)\right )}}{d n e^{\left (c n\right )} - b e^{\left (c n\right )} \log \relax (F)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int F^{a+b\,x}\,{\left (\mathrm {cosh}\left (c+d\,x\right )-\mathrm {sinh}\left (c+d\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.00, size = 92, normalized size = 2.88 \[ \begin {cases} \frac {F^{a} F^{b x} \left (- \sinh {\left (c + d x \right )} + \cosh {\left (c + d x \right )}\right )^{n}}{b \log {\relax (F )} - d n} & \text {for}\: b \neq \frac {d n}{\log {\relax (F )}} \\F^{a} x \left (- \sinh {\left (c + d x \right )} + \cosh {\left (c + d x \right )}\right )^{n} e^{d n x} - \frac {F^{a} \left (- \sinh {\left (c + d x \right )} + \cosh {\left (c + d x \right )}\right )^{n} e^{d n x}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________