Optimal. Leaf size=168 \[ \frac {g 2^{\frac {p+1}{2}} (1-\sin (e+f x))^{\frac {1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (\frac {1}{2} (2 m+p+1);\frac {1-p}{2},-n;\frac {1}{2} (2 m+p+3);\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (2 m+p+1)} \]
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Rubi [A] time = 0.28, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2921, 140, 139, 138} \[ \frac {g 2^{\frac {p+1}{2}} (1-\sin (e+f x))^{\frac {1-p}{2}} (a \sin (e+f x)+a)^{m+1} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (\frac {1}{2} (2 m+p+1);\frac {1-p}{2},-n;\frac {1}{2} (2 m+p+3);\frac {1}{2} (\sin (e+f x)+1),-\frac {d (\sin (e+f x)+1)}{c-d}\right )}{a f (2 m+p+1)} \]
Antiderivative was successfully verified.
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Rule 138
Rule 139
Rule 140
Rule 2921
Rubi steps
\begin {align*} \int (g \cos (e+f x))^p (a+a \sin (e+f x))^m (c+d \sin (e+f x))^n \, dx &=\frac {\left (g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{\frac {1-p}{2}} (a+a \sin (e+f x))^{\frac {1-p}{2}}\right ) \operatorname {Subst}\left (\int (a-a x)^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (2^{-\frac {1}{2}+\frac {p}{2}} g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{-\frac {1}{2}+\frac {1-p}{2}+\frac {p}{2}} \left (\frac {a-a \sin (e+f x)}{a}\right )^{\frac {1}{2}-\frac {p}{2}} (a+a \sin (e+f x))^{\frac {1-p}{2}}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {\left (2^{-\frac {1}{2}+\frac {p}{2}} g (g \cos (e+f x))^{-1+p} (a-a \sin (e+f x))^{-\frac {1}{2}+\frac {1-p}{2}+\frac {p}{2}} \left (\frac {a-a \sin (e+f x)}{a}\right )^{\frac {1}{2}-\frac {p}{2}} (a+a \sin (e+f x))^{\frac {1-p}{2}} (c+d \sin (e+f x))^n \left (\frac {a (c+d \sin (e+f x))}{a c-a d}\right )^{-n}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}-\frac {x}{2}\right )^{\frac {1}{2} (-1+p)} (a+a x)^{m+\frac {1}{2} (-1+p)} \left (\frac {a c}{a c-a d}+\frac {a d x}{a c-a d}\right )^n \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac {2^{\frac {1+p}{2}} g F_1\left (\frac {1}{2} (1+2 m+p);\frac {1-p}{2},-n;\frac {1}{2} (3+2 m+p);\frac {1}{2} (1+\sin (e+f x)),-\frac {d (1+\sin (e+f x))}{c-d}\right ) (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac {1-p}{2}} (a+a \sin (e+f x))^{1+m} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n}}{a f (1+2 m+p)}\\ \end {align*}
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Mathematica [B] time = 10.40, size = 798, normalized size = 4.75 \[ -\frac {2 F_1\left (\frac {p+1}{2};m+n+p+1,-n;\frac {p+3}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) (g \cos (e+f x))^p \cos \left (\frac {1}{4} (2 e+2 f x+\pi )\right ) (a (\sin (e+f x)+1))^m (c+d \sin (e+f x))^n \sin \left (\frac {1}{4} (2 e+2 f x+\pi )\right )}{f \left (-\frac {d n F_1\left (\frac {p+1}{2};m+n+p+1,-n;\frac {p+3}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \cos ^2(e+f x)}{c+d \sin (e+f x)}+\frac {2 (p+1) \left ((c-d) n F_1\left (\frac {p+3}{2};m+n+p+1,1-n;\frac {p+5}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right )-(c+d) (m+n+p+1) F_1\left (\frac {p+3}{2};m+n+p+2,-n;\frac {p+5}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right )\right ) \cot ^2\left (\frac {1}{4} (2 e+2 f x+\pi )\right )}{(c+d) (p+3)}+2 (n+p) F_1\left (\frac {p+1}{2};m+n+p+1,-n;\frac {p+3}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \sin ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )-\frac {2 (c-d) n F_1\left (\frac {p+1}{2};m+n+p+1,-n;\frac {p+3}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \sin ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d \sin (e+f x)}+F_1\left (\frac {p+1}{2};m+n+p+1,-n;\frac {p+3}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right )+p F_1\left (\frac {p+1}{2};m+n+p+1,-n;\frac {p+3}{2};-\tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right ),-\frac {(c-d) \tan ^2\left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{c+d}\right ) \sin (e+f x)\right )} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{p} \left (a +a \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x + e\right )\right )^{p} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (g\,\cos \left (e+f\,x\right )\right )}^p\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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