Integrand size = 29, antiderivative size = 125 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=-\frac {\sqrt {\frac {2}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{(c-d)^{3/2} f}+\frac {2 d \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Time = 0.17 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2858, 12, 2861, 214} \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {2 d \cos (e+f x)}{f \left (c^2-d^2\right ) \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} f (c-d)^{3/2}} \]
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Rule 12
Rule 214
Rule 2858
Rule 2861
Rubi steps \begin{align*} \text {integral}& = \frac {2 d \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\int \frac {a (c+d)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{a \left (c^2-d^2\right )} \\ & = \frac {2 d \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}+\frac {\int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{c-d} \\ & = \frac {2 d \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{2 a^2-(a c-a d) x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{(c-d) f} \\ & = -\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {a} (c-d)^{3/2} f}+\frac {2 d \cos (e+f x)}{\left (c^2-d^2\right ) f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(309\) vs. \(2(125)=250\).
Time = 3.26 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.47 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\frac {\frac {2 d \cos (e+f x)}{c+d}+\frac {\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}}}{\sqrt {3} (c-d) f \sqrt {1+\sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(443\) vs. \(2(112)=224\).
Time = 5.36 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.55
method | result | size |
default | \(-\frac {2 \left (-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )\right ) \sqrt {c +d \sin \left (f x +e \right )}\, \left (\ln \left (\frac {2 \sqrt {2 c -2 d}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )-2 c +2 d}{-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, c \sin \left (f x +e \right )+\ln \left (\frac {2 \sqrt {2 c -2 d}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )+2 c \sin \left (f x +e \right )-2 d \sin \left (f x +e \right )+2 c \cos \left (f x +e \right )-2 d \cos \left (f x +e \right )-2 c +2 d}{-\cos \left (f x +e \right )+1+\sin \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, d \sin \left (f x +e \right )-d \sqrt {2 c -2 d}\, \sin \left (f x +e \right )-d \sqrt {2 c -2 d}\, \cos \left (f x +e \right )+d \sqrt {2 c -2 d}\right )}{f \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \left (c \left (\cos ^{2}\left (f x +e \right )\right )-2 d \sin \left (f x +e \right ) \cos \left (f x +e \right )+c \left (\sin ^{2}\left (f x +e \right )\right )-2 c \cos \left (f x +e \right )+2 d \sin \left (f x +e \right )+c \right ) \left (c +d \right ) \sqrt {2 c -2 d}\, \left (c -d \right )}\) | \(444\) |
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Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (112) = 224\).
Time = 0.49 (sec) , antiderivative size = 1016, normalized size of antiderivative = 8.13 \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {1}{\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {1}{\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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