Optimal. Leaf size=224 \[ \frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}} \]
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Rubi [A] time = 0.66, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3872, 2875, 2873, 2567, 2636, 2640, 2639, 2564, 14} \[ \frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2567
Rule 2636
Rule 2639
Rule 2640
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx &=\int \frac {\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx\\ &=\frac {e^4 \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{11/2}} \, dx}{a^4}\\ &=\frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{11/2}}-\frac {2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{11/2}}+\frac {a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{11/2}}\right ) \, dx}{a^4}\\ &=\frac {e^4 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}\\ &=-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {\left (2 e^2\right ) \int \frac {1}{(e \sin (c+d x))^{7/2}} \, dx}{9 a^2}-\frac {\left (2 e^2\right ) \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{3 a^2}-\frac {\left (2 e^3\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{x^{11/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{15 a^2}+\frac {4 \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{15 a^2}-\frac {\left (2 e^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{11/2}}-\frac {1}{e^2 x^{7/2}}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}\\ &=\frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}+\frac {2 \int \sqrt {e \sin (c+d x)} \, dx}{15 a^2 e^2}-\frac {4 \int \sqrt {e \sin (c+d x)} \, dx}{15 a^2 e^2}\\ &=\frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}+\frac {\left (2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2 e^2 \sqrt {\sin (c+d x)}}-\frac {\left (4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2 e^2 \sqrt {\sin (c+d x)}}\\ &=\frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d e^2 \sqrt {\sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 1.44, size = 163, normalized size = 0.73 \[ \frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) (\cos (c+d x)+i \sin (c+d x)) \left (e^{-2 i (c+d x)} \sqrt {1-e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^4 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (c+d x)}\right )+16 i \sin (c+d x)+13 i \sin (2 (c+d x))-40 \cos (c+d x)-19 \cos (2 (c+d x))-31\right )}{180 a^2 d e \sqrt {e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \sin \left (d x + c\right )}}{a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2} + {\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )^{2} + 2 \, {\left (a^{2} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sec \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 4.42, size = 213, normalized size = 0.95 \[ \frac {\frac {4 e^{3} \left (9 \left (\cos ^{2}\left (d x +c \right )\right )-4\right )}{45 a^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}+\frac {\frac {4 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {11}{2}}\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sin ^{\frac {11}{2}}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \left (\sin ^{7}\left (d x +c \right )\right )}{15}-\frac {38 \left (\sin ^{5}\left (d x +c \right )\right )}{45}+\frac {46 \left (\sin ^{3}\left (d x +c \right )\right )}{45}-\frac {4 \sin \left (d x +c \right )}{9}}{e \,a^{2} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
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 = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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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