Optimal. Leaf size=65 \[ -\frac {2^{m+\frac {1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2651} \[ -\frac {2^{m+\frac {1}{2}} 3^{-m-1} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {\sin (e+f x)+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2651
Rubi steps
\begin {align*} \int 3^{-1-m} (1+\sin (e+f x))^m \, dx &=3^{-1-m} \int (1+\sin (e+f x))^m \, dx\\ &=-\frac {2^{\frac {1}{2}+m} 3^{-1-m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};\frac {1}{2} (1-\sin (e+f x))\right )}{f \sqrt {1+\sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 95, normalized size = 1.46 \[ \frac {\sqrt {2} 3^{-m-1} \cos (e+f x) (\sin (e+f x)+1)^m \, _2F_1\left (\frac {1}{2},m+\frac {1}{2};m+\frac {3}{2};\frac {1}{4} \cos ^2(e+f x) \csc ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )}{(2 f m+f) \sqrt {1-\sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (3^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int 3^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.53, size = 0, normalized size = 0.00 \[ \int 3^{-1-m} \left (1+\sin \left (f x +e \right )\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 3^{-m - 1} \int {\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{3^{m+1}}\,{\left (\sin \left (e+f\,x\right )+1\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ 3^{- m - 1} \int \left (\sin {\left (e + f x \right )} + 1\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________