Optimal. Leaf size=62 \[ -\frac {4^{-m-1} \cos (e+f x) \, _2F_1\left (\frac {1}{2},m+1;\frac {3}{2};\frac {1-\sin (e+f x)}{4 (\sin (e+f x)+1)}\right )}{f (\sin (e+f x)+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.79, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2788, 132} \[ -\frac {2^{-2 m-1} \cos (e+f x) (\sin (e+f x)+1)^{m-1} \left (\frac {\sin (e+f x)+1}{5 \sin (e+f x)+3}\right )^{\frac {1}{2}-m} (5 \sin (e+f x)+3)^{-m} \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};-\frac {1-\sin (e+f x)}{5 \sin (e+f x)+3}\right )}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 132
Rule 2788
Rubi steps
\begin {align*} \int (1+\sin (e+f x))^m (3+5 \sin (e+f x))^{-1-m} \, dx &=\frac {\cos (e+f x) \operatorname {Subst}\left (\int \frac {(1+x)^{-\frac {1}{2}+m} (3+5 x)^{-1-m}}{\sqrt {1-x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {2^{-1-2 m} \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1}{2}-m;\frac {3}{2};-\frac {1-\sin (e+f x)}{3+5 \sin (e+f x)}\right ) (1+\sin (e+f x))^{-1+m} \left (\frac {1+\sin (e+f x)}{3+5 \sin (e+f x)}\right )^{\frac {1}{2}-m} (3+5 \sin (e+f x))^{-m}}{f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 1.40, size = 238, normalized size = 3.84 \[ \frac {4^m (\cosh (m \log (4))-\sinh (m \log (4))) (\sin (e+f x)+1)^m (5 \sin (e+f x)+3)^{-m} (\sin (e+f x)+i \cos (e+f x)+1) \left (-\frac {2 \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )+\cos \left (\frac {1}{4} (2 e+2 f x+\pi )\right )}{\sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right )+2 \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}\right )^m \, _2F_1\left (m+1,2 m+1;2 (m+1);\frac {4 \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{2 \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )+\sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}\right )}{f (2 m+1) ((2+i) \sin (e+f x)+(-1+2 i) \cos (e+f x)+(2-i))} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (5 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (5 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (1+\sin \left (f x +e \right )\right )^{m} \left (3+5 \sin \left (f x +e \right )\right )^{-1-m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (5 \, \sin \left (f x + e\right ) + 3\right )}^{-m - 1} {\left (\sin \left (f x + e\right ) + 1\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (\sin \left (e+f\,x\right )+1\right )}^m}{{\left (5\,\sin \left (e+f\,x\right )+3\right )}^{m+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________