Optimal. Leaf size=36 \[ \frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.131828, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 47, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {2970} \[ \frac{B \cos (e+f x) (a \sin (e+f x)+a)^m (c-c \sin (e+f x))^n}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2970
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n (B (m-n)-B (1+m+n) \sin (e+f x)) \, dx &=\frac{B \cos (e+f x) (a+a \sin (e+f x))^m (c-c \sin (e+f x))^n}{f}\\ \end{align*}
Mathematica [A] time = 0.463766, size = 36, normalized size = 1. \[ \frac{B \cos (e+f x) (a (\sin (e+f x)+1))^m (c-c \sin (e+f x))^n}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.552, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{m} \left ( c-c\sin \left ( fx+e \right ) \right ) ^{n} \left ( B \left ( m-n \right ) -B \left ( m+n+1 \right ) \sin \left ( fx+e \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (B{\left (m + n + 1\right )} \sin \left (f x + e\right ) - B{\left (m - n\right )}\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.05146, size = 88, normalized size = 2.44 \begin{align*} \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{m}{\left (-c \sin \left (f x + e\right ) + c\right )}^{n} B \cos \left (f x + e\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]