Integrand size = 27, antiderivative size = 278 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}-\frac {7 a e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {7 a e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (1+\cos (c+d x)+\sin (c+d x))} \]
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Time = 0.31 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2757, 2764, 2756, 2854, 209, 2912, 65, 221} \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a \sin (c+d x)+a}}-\frac {7 a e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{8 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {7 a e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}+\frac {7 a e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{8 d} \]
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Rule 65
Rule 209
Rule 221
Rule 2756
Rule 2757
Rule 2764
Rule 2854
Rule 2912
Rubi steps \begin{align*} \text {integral}& = -\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}+\frac {1}{6} (7 a) \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}+\frac {1}{8} \left (7 a^2\right ) \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}+\frac {1}{16} \left (7 a e^2\right ) \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx \\ & = -\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}+\frac {\left (7 a^2 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{16 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (7 a^2 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{16 (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = -\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}-\frac {\left (7 a^2 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{16 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (7 a^2 e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = -\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}+\frac {7 a^2 e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (7 a^2 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))} \\ & = -\frac {7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt {a+a \sin (c+d x)}}+\frac {7 a e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d}-\frac {a (e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}}{3 d e}-\frac {7 a^2 e^{3/2} \text {arcsinh}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {7 a^2 e^{3/2} \arctan \left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{8 d (a+a \cos (c+d x)+a \sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.28 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=-\frac {8\ 2^{3/4} a (e \cos (c+d x))^{5/2} \operatorname {Hypergeometric2F1}\left (-\frac {7}{4},\frac {5}{4},\frac {9}{4},\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt {a (1+\sin (c+d x))}}{5 d e (1+\sin (c+d x))^{7/4}} \]
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Time = 6.57 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.19
method | result | size |
default | \(-\frac {\sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, e a \left (8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+8 \left (\cos ^{3}\left (d x +c \right )\right )-14 \cos \left (d x +c \right ) \sin \left (d x +c \right )+22 \left (\cos ^{2}\left (d x +c \right )\right )-21 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-21 \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-21 \sin \left (d x +c \right )-7 \cos \left (d x +c \right )-21 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right )-21 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (1+\cos \left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right )-21\right )}{24 d \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}\) | \(332\) |
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Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 981, normalized size of antiderivative = 3.53 \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx=\int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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