Integrand size = 30, antiderivative size = 49 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {c \cos (e+f x) \log (1+\sin (e+f x))}{f \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2816, 2746, 31} \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {c \cos (e+f x) \log (\sin (e+f x)+1)}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]
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Rule 31
Rule 2746
Rule 2816
Rubi steps \begin{align*} \text {integral}& = \frac {(a c \cos (e+f x)) \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 {(c \cos (e+f x)) \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 {c \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)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 2.43 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\sqrt {\frac {2}{3}} \left (i+e^{i (e+f x)}\right ) \left (f x+2 i \log \left (i+e^{i (e+f x)}\right )\right ) \sqrt {c-c \sin (e+f x)}}{\left (-i+e^{i (e+f x)}\right ) \sqrt {-i e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2} f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(45)=90\).
Time = 2.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.94
method | result | size |
default | \(\frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )\right ) \left (1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right )}{f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}}\) | \(95\) |
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\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int { \frac {\sqrt {-c \sin \left (f x + e\right ) + c}}{\sqrt {a \sin \left (f x + e\right ) + a}} \,d x } \]
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\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]
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none
Time = 0.30 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=-\frac {\frac {2 \, \sqrt {c} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{\sqrt {a}} - \frac {\sqrt {c} \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}{\sqrt {a}}}{f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (45) = 90\).
Time = 0.37 (sec) , antiderivative size = 219, normalized size of antiderivative = 4.47 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\frac {\sqrt {2} \sqrt {c} {\left (\frac {\sqrt {2} \log \left ({\left | -\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1} - \frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 2 \right |}\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} \log \left ({\left | -\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1} - \frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} - 2 \right |}\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{2 \, f} \]
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Timed out. \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {3+3 \sin (e+f x)}} \, dx=\int \frac {\sqrt {c-c\,\sin \left (e+f\,x\right )}}{\sqrt {a+a\,\sin \left (e+f\,x\right )}} \,d x \]
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