Optimal. Leaf size=33 \[ \frac{a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.237485, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {2967, 2854} \[ \frac{a^3 B c^3 \cos ^7(e+f x) (a \sin (e+f x)+a)^{m-3}}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2967
Rule 2854
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^m (c-c \sin (e+f x))^3 (B (-3+m)-B (4+m) \sin (e+f x)) \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (a+a \sin (e+f x))^{-3+m} (B (-3+m)-B (4+m) \sin (e+f x)) \, dx\\ &=\frac{a^3 B c^3 \cos ^7(e+f x) (a+a \sin (e+f x))^{-3+m}}{f}\\ \end{align*}
Mathematica [A] time = 1.12281, size = 66, normalized size = 2. \[ \frac{B c^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) (a (\sin (e+f x)+1))^m}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 2.378, 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 ) ^{3} \left ( B \left ( m-3 \right ) -B \left ( 4+m \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 + 4\right )} \sin \left (f x + e\right ) - B{\left (m - 3\right )}\right )}{\left (c \sin \left (f x + e\right ) - c\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.09549, size = 184, normalized size = 5.58 \begin{align*} -\frac{{\left (3 \, B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right ) -{\left (B c^{3} \cos \left (f x + e\right )^{3} - 4 \, B c^{3} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B{\left (m + 4\right )} \sin \left (f x + e\right ) - B{\left (m - 3\right )}\right )}{\left (c \sin \left (f x + e\right ) - c\right )}^{3}{\left (a \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]