Integrand size = 27, antiderivative size = 193 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}} \]
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Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=\frac {256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac {128 \sqrt {a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac {32}{195 a^2 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac {16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac {2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}} \]
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Rule 2750
Rule 2751
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}+\frac {8 \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx}{13 a} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}+\frac {16 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}} \, dx}{39 a^2} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}+\frac {64 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^3} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {128 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^4} \\ & = -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 (77+136 \cos (2 (c+d x))-16 \cos (4 (c+d x))-40 \sin (c+d x)+80 \sin (3 (c+d x)))}{585 d e (e \cos (c+d x))^{3/2} (a (1+\sin (c+d x)))^{5/2}} \]
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Time = 2.81 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.54
method | result | size |
default | \(-\frac {2 \left (-128 \left (\cos ^{3}\left (d x +c \right )\right )+320 \cos \left (d x +c \right ) \sin \left (d x +c \right )+400 \cos \left (d x +c \right )-120 \tan \left (d x +c \right )-75 \sec \left (d x +c \right )\right )}{585 d \,a^{2} e^{2} \left (2 \sin \left (d x +c \right )-\left (\cos ^{2}\left (d x +c \right )\right )+2\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \sqrt {e \cos \left (d x +c \right )}}\) | \(105\) |
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Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (128 \, \cos \left (d x + c\right )^{4} - 400 \, \cos \left (d x + c\right )^{2} - 40 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 75\right )} \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{585 \, {\left (3 \, a^{3} d e^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} d e^{3} \cos \left (d x + c\right )^{2} + {\left (a^{3} d e^{3} \cos \left (d x + c\right )^{4} - 4 \, a^{3} d e^{3} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (163) = 326\).
Time = 0.33 (sec) , antiderivative size = 451, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (197 \, \sqrt {a} \sqrt {e} + \frac {400 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1760 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2230 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2230 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1760 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {400 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {197 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{585 \, {\left (a^{3} e^{3} + \frac {5 \, a^{3} e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} e^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} e^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} e^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {15}{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. \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Time = 11.51 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx=-\frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}\,\left (\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,2464{}\mathrm {i}}{585\,a^3\,d\,e^2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,4352{}\mathrm {i}}{585\,a^3\,d\,e^2}-\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,512{}\mathrm {i}}{585\,a^3\,d\,e^2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,512{}\mathrm {i}}{117\,a^3\,d\,e^2}-\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,256{}\mathrm {i}}{117\,a^3\,d\,e^2}\right )}{\cos \left (c+d\,x\right )\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,28{}\mathrm {i}-{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,12{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,28{}\mathrm {i}-{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (4\,c+4\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,2{}\mathrm {i}} \]
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