Integrand size = 30, antiderivative size = 139 \[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {4 c^3 \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {3+3 \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {3+3 \sin (e+f x)}} \]
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Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2819, 2816, 2746, 31} \[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {4 c^3 \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2819
Rubi steps \begin{align*} \text {integral}& = \frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}+(2 c) \int \frac {(c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {2 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}+\left (4 c^2\right ) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx \\ & = \frac {2 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}+\frac {\left (4 a c^3 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {2 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}}+\frac {\left (4 c^3 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 c^3 \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {2 c^2 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {c \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}
Time = 12.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00 \[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x))^2 \left (\cos (2 (e+f x))-32 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+12 \sin (e+f x)\right ) \sqrt {c-c \sin (e+f x)}}{4 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {1+\sin (e+f x)}} \]
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Time = 3.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.75
method | result | size |
default | \(-\frac {\left (\cos ^{3}\left (f x +e \right )+\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-5 \left (\cos ^{2}\left (f x +e \right )\right )+6 \sin \left (f x +e \right ) \cos \left (f x +e \right )+8 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-16 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \cos \left (f x +e \right )+8 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )-16 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-\cos \left (f x +e \right )+5 \sin \left (f x +e \right )+8 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-16 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )+5\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c^{2}}{2 f \left (1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}\) | \(243\) |
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\[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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Time = 0.33 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.92 \[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {2 \, \sqrt {a} c^{\frac {5}{2}} {\left (\frac {a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 2 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{a^{2}} + \frac {2 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]
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Timed out. \[ \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]
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