Integrand size = 30, antiderivative size = 92 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {18 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {3 \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.15 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2819, 2816, 2746, 31} \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {2 a^2 \cos (e+f x) \log (1-\sin (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{f \sqrt {c-c \sin (e+f x)}} \]
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Rule 31
Rule 2746
Rule 2816
Rule 2819
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}+(2 a) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c-c \sin (e+f x)}} \, dx \\ & = -\frac {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}}+\frac {\left (2 a^2 c \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 \sqrt {c-c \sin (e+f x)}}-\frac {\left (2 a^2 \cos (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+x} \, dx,x,-c \sin (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {2 a^2 \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 {a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{f \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 1.63 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.26 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {3 \sqrt {3} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{3/2} \left (4 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+\sin (e+f x)\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c-c \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(202\) vs. \(2(88)=176\).
Time = 2.65 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.21
method | result | size |
default | \(-\frac {\left (4 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \cos \left (f x +e \right )-4 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sin \left (f x +e \right )-2 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+2 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )+\cos ^{2}\left (f x +e \right )+\sin \left (f x +e \right ) \cos \left (f x +e \right )+4 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )-2 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a}{f \left (1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(203\) |
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\[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}\, dx \]
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\[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
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none
Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04 \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {2 \, a^{\frac {3}{2}} \sqrt {c} {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {\log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}{c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{f} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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