Integrand size = 27, antiderivative size = 154 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {32 (a+a \sin (c+d x))^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}} \]
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Time = 0.21 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {32 (a \sin (c+d x)+a)^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}}-\frac {16 \sqrt {a \sin (c+d x)+a}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {4}{15 a d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac {2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}} \]
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Rule 2750
Rule 2751
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}+\frac {2 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}} \, dx}{3 a} \\ & = -\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}+\frac {8 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{15 a^2} \\ & = -\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {16 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{15 a^3} \\ & = -\frac {2}{9 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {4}{15 a d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {16 \sqrt {a+a \sin (c+d x)}}{15 a^2 d e (e \cos (c+d x))^{3/2}}+\frac {32 (a+a \sin (c+d x))^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 (7+12 \cos (2 (c+d x))-6 \sin (c+d x)+4 \sin (3 (c+d x)))}{45 d e (e \cos (c+d x))^{3/2} (a (1+\sin (c+d x)))^{3/2}} \]
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Time = 2.67 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {2 \left (16 \cos \left (d x +c \right ) \sin \left (d x +c \right )+24 \cos \left (d x +c \right )-10 \tan \left (d x +c \right )-5 \sec \left (d x +c \right )\right )}{45 d a \,e^{2} \left (1+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {e \cos \left (d x +c \right )}}\) | \(83\) |
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Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (24 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 5\right )} \sin \left (d x + c\right ) - 5\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45 \, {\left (a^{2} d e^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{2} d e^{3} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 2 \, a^{2} d e^{3} \cos \left (d x + c\right )^{2}\right )}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (130) = 260\).
Time = 0.34 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (19 \, \sqrt {a} \sqrt {e} + \frac {12 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {58 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {116 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {116 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {58 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {12 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {19 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{45 \, {\left (a^{2} e^{3} + \frac {4 \, a^{2} e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a^{2} e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a^{2} e^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a^{2} e^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Time = 10.55 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.49 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx=-\frac {14\,\sqrt {a+a\,\sin \left (c+d\,x\right )}-12\,\sin \left (c+d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+24\,\cos \left (2\,c+2\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}+8\,\sin \left (3\,c+3\,d\,x\right )\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\frac {225\,a^2\,d\,e^2\,\cos \left (c+d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}-\frac {45\,a^2\,d\,e^2\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}{4}+45\,a^2\,d\,e^2\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {\frac {e\,{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {e\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}} \]
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