Integrand size = 30, antiderivative size = 95 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {3 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{f (c-c \sin (e+f x))^{3/2}}+\frac {9 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2818, 2816, 2746, 31} \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {a^2 \cos (e+f x) \log (1-\sin (e+f x))}{c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f (c-c \sin (e+f x))^{3/2}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2818
Rubi steps \begin{align*} \text {integral}& = \frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f (c-c \sin (e+f x))^{3/2}}-\frac {a \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx}{c} \\ & = \frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f (c-c \sin (e+f x))^{3/2}}-\frac {\left (a^2 \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 {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f (c-c \sin (e+f x))^{3/2}}+\frac {\left (a^2 \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 {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f (c-c \sin (e+f x))^{3/2}}+\frac {a^2 \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)}} \\ \end{align*}
Time = 6.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.63 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {6 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} \left (-1-\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)\right )}{c f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \]
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Time = 3.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {\sec \left (f x +e \right ) \left (\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )-2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+2 \sin \left (f x +e \right )\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a}{f c \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(128\) |
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\[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.44 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\frac {2 \, a^{\frac {3}{2}} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{\frac {3}{2}}} - \frac {a^{\frac {3}{2}} \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{c^{\frac {3}{2}}} + \frac {4 \, a^{\frac {3}{2}} \sqrt {c} \sin \left (f x + e\right )}{{\left (c^{2} - \frac {2 \, c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
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Time = 0.37 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {2} a^{\frac {3}{2}} \sqrt {c} {\left (\frac {\sqrt {2} \log \left (-2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2\right )}{c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{2 \, f} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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