Optimal. Leaf size=84 \[ -\frac {g 2^{m+\frac {9}{4}} \sqrt {g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{4},-m-\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{c f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.27, antiderivative size = 84, 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} \[ -\frac {g 2^{m+\frac {9}{4}} \sqrt {g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a \sin (e+f x)+a)^m \, _2F_1\left (\frac {1}{4},-m-\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{c f} \]
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)} \, dx &=\frac {g^2 \int \frac {(a+a \sin (e+f x))^{1+m}}{\sqrt {g \cos (e+f x)}} \, dx}{a c}\\ &=\frac {\left (a g \sqrt {g \cos (e+f x)}\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{\frac {1}{4}+m}}{(a-a x)^{3/4}} \, dx,x,\sin (e+f x)\right )}{c f \sqrt [4]{a-a \sin (e+f x)} \sqrt [4]{a+a \sin (e+f x)}}\\ &=\frac {\left (2^{\frac {1}{4}+m} a g \sqrt {g \cos (e+f x)} (a+a \sin (e+f x))^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)^{3/4}} \, dx,x,\sin (e+f x)\right )}{c f \sqrt [4]{a-a \sin (e+f x)}}\\ &=-\frac {2^{\frac {9}{4}+m} g \sqrt {g \cos (e+f x)} \, _2F_1\left (\frac {1}{4},-\frac {1}{4}-m;\frac {5}{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))^m}{c f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 84, normalized size = 1.00 \[ -\frac {g 2^{m+\frac {9}{4}} \sqrt {g \cos (e+f x)} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (a (\sin (e+f x)+1))^m \, _2F_1\left (\frac {1}{4},-m-\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{c f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {g \cos \left (f x + e\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} g \cos \left (f x + e\right )}{c \sin \left (f x + e\right ) - c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (a +a \sin \left (f x +e \right )\right )^{m}}{c -c \sin \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{m}}{c \sin \left (f x + e\right ) - c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^m}{c-c\,\sin \left (e+f\,x\right )} \,d x \]
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]
________________________________________________________________________________________