Optimal. Leaf size=106 \[ -\frac {c 2^{m+\frac {9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (c \sin (e+f x)+c)^{m-1} \, _2F_1\left (\frac {3}{4},-m-\frac {1}{4};\frac {7}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{3 f \sqrt {a-a \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2853, 2689, 70, 69} \[ -\frac {c 2^{m+\frac {9}{4}} \cos (e+f x) (g \cos (e+f x))^{3/2} (\sin (e+f x)+1)^{-m-\frac {1}{4}} (c \sin (e+f x)+c)^{m-1} \, _2F_1\left (\frac {3}{4},-m-\frac {1}{4};\frac {7}{4};\frac {1}{2} (1-\sin (e+f x))\right )}{3 f \sqrt {a-a \sin (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 69
Rule 70
Rule 2689
Rule 2853
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^m}{\sqrt {a-a \sin (e+f x)}} \, dx &=\frac {(g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} (c+c \sin (e+f x))^{\frac {1}{2}+m} \, dx}{\sqrt {a-a \sin (e+f x)} \sqrt {c+c \sin (e+f x)}}\\ &=\frac {\left (c^2 \cos (e+f x) (g \cos (e+f x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {(c+c x)^{\frac {1}{4}+m}}{\sqrt [4]{c-c x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} (c-c \sin (e+f x))^{3/4} (c+c \sin (e+f x))^{5/4}}\\ &=\frac {\left (2^{\frac {1}{4}+m} c^2 \cos (e+f x) (g \cos (e+f x))^{3/2} (c+c \sin (e+f x))^{-1+m} \left (\frac {c+c \sin (e+f x)}{c}\right )^{-\frac {1}{4}-m}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {x}{2}\right )^{\frac {1}{4}+m}}{\sqrt [4]{c-c x}} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} (c-c \sin (e+f x))^{3/4}}\\ &=-\frac {2^{\frac {9}{4}+m} c \cos (e+f x) (g \cos (e+f x))^{3/2} \, _2F_1\left (\frac {3}{4},-\frac {1}{4}-m;\frac {7}{4};\frac {1}{2} (1-\sin (e+f x))\right ) (1+\sin (e+f x))^{-\frac {1}{4}-m} (c+c \sin (e+f x))^{-1+m}}{3 f \sqrt {a-a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 180.01, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {-a \sin \left (f x + e\right ) + a} {\left (c \sin \left (f x + e\right ) + c\right )}^{m} g \cos \left (f x + e\right )}{a \sin \left (f x + e\right ) - a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt {-a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +c \sin \left (f x +e \right )\right )^{m}}{\sqrt {a -a \sin \left (f x +e \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (c \sin \left (f x + e\right ) + c\right )}^{m}}{\sqrt {-a \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+c\,\sin \left (e+f\,x\right )\right )}^m}{\sqrt {a-a\,\sin \left (e+f\,x\right )}} \,d x \]
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]
________________________________________________________________________________________