Integrand size = 38, antiderivative size = 241 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {16 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {8 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}} \]
[Out]
Time = 0.56 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2920, 2819, 2816, 2746, 31} \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {16 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {8 a^3 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a \sin (e+f x)+a)^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Rule 31
Rule 2746
Rule 2816
Rule 2819
Rule 2920
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {(a+a \sin (e+f x))^{9/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c} \\ & = -\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}}+\frac {2 \int \frac {(a+a \sin (e+f x))^{7/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = -\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}}+\frac {(4 a) \int \frac {(a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = -\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}}+\frac {\left (8 a^2\right ) \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = -\frac {8 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}}+\frac {\left (16 a^3\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = -\frac {8 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}}+\frac {\left (16 a^4 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{c-c \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {8 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}}-\frac {\left (16 a^4 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {16 a^4 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {8 a^3 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a^2 \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{c f \sqrt {c-c \sin (e+f x)}}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{3 c f \sqrt {c-c \sin (e+f x)}}-\frac {\cos (e+f x) (a+a \sin (e+f x))^{7/2}}{4 c f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 11.27 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.65 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (-276 \cos (2 (e+f x))+3 \cos (4 (e+f x))+8 \left (384 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+195 \sin (e+f x)-5 \sin (3 (e+f x))\right )\right )}{96 c f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c-c \sin (e+f x)}} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.24
method | result | size |
default | \(\frac {\left (-3 \left (\cos ^{5}\left (f x +e \right )\right )+3 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+17 \left (\cos ^{4}\left (f x +e \right )\right )+20 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+72 \left (\cos ^{3}\left (f x +e \right )\right )-52 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+192 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \cos \left (f x +e \right )-192 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right ) \sin \left (f x +e \right )-384 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \cos \left (f x +e \right )+384 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-148 \left (\cos ^{2}\left (f x +e \right )\right )-200 \cos \left (f x +e \right ) \sin \left (f x +e \right )+192 \ln \left (\frac {2}{1+\cos \left (f x +e \right )}\right )-384 \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-69 \cos \left (f x +e \right )-131 \sin \left (f x +e \right )+131\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, a^{3}}{12 f \left (\cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c}\) | \(299\) |
[In]
[Out]
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}} \cos \left (f x + e\right )^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.82 \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {4 \, a^{\frac {7}{2}} \sqrt {c} {\left (\frac {12 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {3 \, c^{6} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 4 \, c^{6} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 6 \, c^{6} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 12 \, c^{6} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c^{8}}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{3 \, f} \]
[In]
[Out]
Timed out. \[ \int \frac {\cos ^2(e+f x) (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]