Optimal. Leaf size=149 \[ -\frac {2^{\frac {p}{2}-\frac {3}{2}} (\sin (e+f x)+1)^{\frac {3-p}{2}-2} (g \cos (e+f x))^{p+1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {p+1}{2};\frac {5-p}{2},-n;\frac {p+3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{a^2 f g (p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.23, antiderivative size = 153, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2920, 139, 138} \[ -\frac {g 2^{\frac {p-3}{2}} (1-\sin (e+f x)) (\sin (e+f x)+1)^{\frac {1-p}{2}} (g \cos (e+f x))^{p-1} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {p+1}{2};\frac {5-p}{2},-n;\frac {p+3}{2};\frac {1}{2} (1-\sin (e+f x)),\frac {d (1-\sin (e+f x))}{c+d}\right )}{a^2 f (p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 138
Rule 139
Rule 2920
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx &=\frac {\left (g (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac {1-p}{2}} (1+\sin (e+f x))^{\frac {1-p}{2}}\right ) \operatorname {Subst}\left (\int (1-x)^{\frac {1}{2} (-1+p)} (1+x)^{-2+\frac {1}{2} (-1+p)} (c+d x)^n \, dx,x,\sin (e+f x)\right )}{a^2 f}\\ &=\frac {\left (g (g \cos (e+f x))^{-1+p} (1-\sin (e+f x))^{\frac {1-p}{2}} (1+\sin (e+f x))^{\frac {1-p}{2}} (c+d \sin (e+f x))^n \left (-\frac {c+d \sin (e+f x)}{-c-d}\right )^{-n}\right ) \operatorname {Subst}\left (\int (1-x)^{\frac {1}{2} (-1+p)} (1+x)^{-2+\frac {1}{2} (-1+p)} \left (-\frac {c}{-c-d}-\frac {d x}{-c-d}\right )^n \, dx,x,\sin (e+f x)\right )}{a^2 f}\\ &=-\frac {2^{\frac {1}{2} (-3+p)} g F_1\left (\frac {1+p}{2};\frac {5-p}{2},-n;\frac {3+p}{2};\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)) (1+\sin (e+f x))^{\frac {1-p}{2}} (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n}}{a^2 f (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 12.06, size = 0, normalized size = 0.00 \[ \int \frac {(g \cos (e+f x))^p (c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (g \cos \left (f x + e\right )\right )^{p} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{p} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 4.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x +e \right )\right )^{p} \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{p} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^p\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\left (g \cos {\left (e + f x \right )}\right )^{p} \left (c + d \sin {\left (e + f x \right )}\right )^{n}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________