Optimal. Leaf size=187 \[ \frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {44 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d} \]
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Rubi [A] time = 0.60, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3872, 2875, 2873, 2567, 2640, 2639, 2564, 14, 2569} \[ \frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {44 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2567
Rule 2569
Rule 2639
Rule 2640
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{5/2}}{(-a-a \cos (c+d x))^2} \, dx\\ &=\frac {e^4 \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx}{a^4}\\ &=\frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{3/2}}+\frac {a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{3/2}}\right ) \, dx}{a^4}\\ &=\frac {e^4 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a^2}\\ &=-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {\left (2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{a^2}-\frac {\left (6 e^2\right ) \int \cos ^2(c+d x) \sqrt {e \sin (c+d x)} \, dx}{a^2}-\frac {\left (2 e^3\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{x^{3/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}-\frac {\left (12 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 a^2}-\frac {\left (2 e^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}-\frac {\sqrt {x}}{e^2}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}-\frac {\left (2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{a^2 \sqrt {\sin (c+d x)}}\\ &=\frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {4 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}-\frac {\left (12 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {\sin (c+d x)}}\\ &=\frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {44 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}\\ \end {align*}
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Mathematica [C] time = 2.99, size = 249, normalized size = 1.33 \[ \frac {4 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left (\csc ^2(c+d x) \left (20 \sin (c) \cos (d x)-3 \sin (2 c) \cos (2 d x)+20 \cos (c) \sin (d x)-3 \cos (2 c) \sin (2 d x)+\sec \left (\frac {c}{2}\right ) \left (60 \sin \left (\frac {d x}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right )-36 \sin \left (\frac {3 c}{2}\right ) \sec (c)\right )-96 \tan \left (\frac {c}{2}\right ) \sec (c)\right )+\frac {352 i e^{2 i (2 c+d x)} \left (3 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};e^{2 i (c+d x)}\right )+e^{2 i d x} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (c+d x)}\right )\right )}{\left (1+e^{2 i c}\right ) \left (1-e^{2 i (c+d x)}\right )^{5/2}}\right )}{15 a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (e^{2} \cos \left (d x + c\right )^{2} - e^{2}\right )} \sqrt {e \sin \left (d x + c\right )}}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 4.03, size = 173, normalized size = 0.93 \[ -\frac {2 e^{3} \left (33 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-66 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \left (\cos ^{4}\left (d x +c \right )\right )+10 \left (\cos ^{3}\left (d x +c \right )\right )+33 \left (\cos ^{2}\left (d x +c \right )\right )-40 \cos \left (d x +c \right )\right )}{15 a^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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