Integrand size = 29, antiderivative size = 127 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {(c-3 d) \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} \sqrt {c-d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{6 \sqrt {6} (c-d)^{3/2} f}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 (c-d) f (3+3 \sin (e+f x))^{3/2}} \]
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Time = 0.17 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2845, 12, 2861, 214} \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=-\frac {(c-3 d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c-d} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f (c-d)^{3/2}}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 f (c-d) (a \sin (e+f x)+a)^{3/2}} \]
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Rule 12
Rule 214
Rule 2845
Rule 2861
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 (c-d) f (a+a \sin (e+f x))^{3/2}}-\frac {\int -\frac {a (c-3 d)}{2 \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{2 a^2 (c-d)} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 (c-d) f (a+a \sin (e+f x))^{3/2}}+\frac {(c-3 d) \int \frac {1}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{4 a (c-d)} \\ & = -\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 (c-d) f (a+a \sin (e+f x))^{3/2}}-\frac {(c-3 d) \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 )}{2 (c-d) f} \\ & = -\frac {(c-3 d) \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 )}{2 \sqrt {2} a^{3/2} (c-d)^{3/2} f}-\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{2 (c-d) f (a+a \sin (e+f x))^{3/2}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(384\) vs. \(2(127)=254\).
Time = 3.51 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.02 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {(c-3 d) \left (\log \left (1+\tan \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )}{\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{2+2 \tan \left (\frac {1}{2} (e+f x)\right )}-\frac {-\frac {1}{2} (c-d) \sec ^2\left (\frac {1}{2} (e+f x)\right )+\frac {\sqrt {c-d} \left (\frac {1}{1+\cos (e+f x)}\right )^{3/2} (d+d \cos (e+f x)+c \sin (e+f x))}{\sqrt {c+d \sin (e+f x)}}}{c-d+2 \sqrt {c-d} \sqrt {\frac {1}{1+\cos (e+f x)}} \sqrt {c+d \sin (e+f x)}+(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}}\right )}{12 \sqrt {3} (c-d) f (1+\sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(642\) vs. \(2(112)=224\).
Time = 4.91 (sec) , antiderivative size = 643, normalized size of antiderivative = 5.06
method | result | size |
default | \(-\frac {\left (\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}\, c -3 \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}\, d +\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}\, c -3 \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}\, d +\sqrt {2 c -2 d}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \cos \left (f x +e \right )-\sin \left (f x +e \right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {2 c -2 d}+\sqrt {2 c -2 d}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\right ) \sqrt {c +d \sin \left (f x +e \right )}}{2 f \left (1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right ) a \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2 c -2 d}\, \left (c -d \right )}\) | \(643\) |
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (112) = 224\).
Time = 0.52 (sec) , antiderivative size = 1008, normalized size of antiderivative = 7.94 \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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\[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int { \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {d \sin \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {1}{(3+3 \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx=\int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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