Optimal. Leaf size=145 \[ \frac {6 e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {6 e^5 \sin (c+d x) \sqrt {e \cos (c+d x)}}{7 a^2 d}+\frac {18 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^2 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2680, 2635, 2642, 2641} \[ \frac {6 e^5 \sin (c+d x) \sqrt {e \cos (c+d x)}}{7 a^2 d}+\frac {18 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^2 d}+\frac {6 e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2 \sin (c+d x)+a^2\right )} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2680
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx &=\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (9 e^2\right ) \int (e \cos (c+d x))^{7/2} \, dx}{5 a^2}\\ &=\frac {18 e^3 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^2 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (9 e^4\right ) \int (e \cos (c+d x))^{3/2} \, dx}{7 a^2}\\ &=\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^2 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (3 e^6\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{7 a^2}\\ &=\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^2 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2+a^2 \sin (c+d x)\right )}+\frac {\left (3 e^6 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 a^2 \sqrt {e \cos (c+d x)}}\\ &=\frac {6 e^6 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 a^2 d \sqrt {e \cos (c+d x)}}+\frac {6 e^5 \sqrt {e \cos (c+d x)} \sin (c+d x)}{7 a^2 d}+\frac {18 e^3 (e \cos (c+d x))^{5/2} \sin (c+d x)}{35 a^2 d}+\frac {4 e (e \cos (c+d x))^{9/2}}{5 d \left (a^2+a^2 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 66, normalized size = 0.46 \[ -\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left (-\frac {1}{4},\frac {13}{4};\frac {17}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{13 a^2 d e (\sin (c+d x)+1)^{13/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )} e^{5} \cos \left (d x + c\right )^{5}}{a^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.92, size = 203, normalized size = 1.40 \[ -\frac {2 e^{6} \left (-80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+112 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-168 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-20 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+84 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a^{2} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {11}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{11/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2} \,d x \]
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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