Optimal. Leaf size=93 \[ \frac {g^5 2^{m+\frac {9}{4}} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac {7}{4},-m-\frac {1}{4};-\frac {3}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{7 a^2 c^3 f (g \cos (e+f x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.28, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2840, 2689, 70, 69} \[ \frac {g^5 2^{m+\frac {9}{4}} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^{m+2} \, _2F_1\left (-\frac {7}{4},-m-\frac {1}{4};-\frac {3}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{7 a^2 c^3 f (g \cos (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 2689
Rule 2840
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^m}{(c-c \sin (e+f x))^3} \, dx &=\frac {g^6 \int \frac {(a+a \sin (e+f x))^{3+m}}{(g \cos (e+f x))^{9/2}} \, dx}{a^3 c^3}\\ &=\frac {\left (g^5 (a-a \sin (e+f x))^{7/4} (a+a \sin (e+f x))^{7/4}\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{\frac {1}{4}+m}}{(a-a x)^{11/4}} \, dx,x,\sin (e+f x)\right )}{a c^3 f (g \cos (e+f x))^{7/2}}\\ &=\frac {\left (2^{\frac {1}{4}+m} g^5 (a-a \sin (e+f x))^{7/4} (a+a \sin (e+f x))^{2+m} \left (\frac {a+a \sin (e+f x)}{a}\right )^{-\frac {1}{4}-m}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{4}+m}}{(a-a x)^{11/4}} \, dx,x,\sin (e+f x)\right )}{a c^3 f (g \cos (e+f x))^{7/2}}\\ &=\frac {2^{\frac {9}{4}+m} g^5 \, _2F_1\left (-\frac {7}{4},-\frac {1}{4}-m;-\frac {3}{4};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (a+a \sin (e+f x))^{2+m}}{7 a^2 c^3 f (g \cos (e+f x))^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.21, size = 96, normalized size = 1.03 \[ \frac {g 2^{m+\frac {9}{4}} \sqrt {g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a (\sin (e+f x)+1))^m \, _2F_1\left (-\frac {7}{4},-m-\frac {1}{4};-\frac {3}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{7 c^3 f (\sin (e+f x)-1)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {g \cos \left (f x + e\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} g \cos \left (f x + e\right )}{3 \, c^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{3} - {\left (c^{3} \cos \left (f x + e\right )^{2} - 4 \, c^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.40, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m}}{\left (c -c \sin \left (f x +e \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (c \sin \left (f x + e\right ) - c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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]
________________________________________________________________________________________