Optimal. Leaf size=43 \[ \frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m}{f (1+2 m)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2821}
\begin {gather*} \frac {\cos (e+f x) (3-3 \sin (e+f x))^{-m-1} (\sin (e+f x)+1)^m}{f (2 m+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2821
Rubi steps
\begin {align*} \int (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m \, dx &=\frac {\cos (e+f x) (3-3 \sin (e+f x))^{-1-m} (1+\sin (e+f x))^m}{f (1+2 m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(43)=86\).
time = 0.51, size = 97, normalized size = 2.26 \begin {gather*} \frac {\cos ^{-1-2 m}\left (\frac {1}{4} (2 e+\pi +2 f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{2 m} (6-6 \sin (e+f x))^{-m} (1+\sin (e+f x))^m \sin \left (\frac {1}{4} (2 e+\pi +2 f x)\right )}{3 (f+2 f m)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \left (3-3 \sin \left (f x +e \right )\right )^{-1-m} \left (1+\sin \left (f x +e \right )\right )^{m}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 44, normalized size = 1.02 \begin {gather*} \frac {{\left (\sin \left (f x + e\right ) + 1\right )}^{m} {\left (-3 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} \cos \left (f x + e\right )}{2 \, f m + f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 7.85, size = 43, normalized size = 1.00 \begin {gather*} \frac {\cos \left (e+f\,x\right )\,{\left (\sin \left (e+f\,x\right )+1\right )}^m}{f\,\left (2\,m+1\right )\,{\left (3-3\,\sin \left (e+f\,x\right )\right )}^{m+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________