Optimal. Leaf size=138 \[ \frac {22 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a^3 d \sqrt {\cos (c+d x)}}+\frac {22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac {22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2680, 2682, 2635, 2640, 2639} \[ \frac {22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac {22 e^6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a^3 d \sqrt {\cos (c+d x)}}+\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2680
Rule 2682
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^3} \, dx &=\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac {\left (11 e^2\right ) \int \frac {(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx}{3 a^2}\\ &=\frac {22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac {\left (11 e^4\right ) \int (e \cos (c+d x))^{5/2} \, dx}{3 a^3}\\ &=\frac {22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^3 d}+\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac {\left (11 e^6\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a^3}\\ &=\frac {22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^3 d}+\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}+\frac {\left (11 e^6 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^3 \sqrt {\cos (c+d x)}}\\ &=\frac {22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac {22 e^6 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)}}+\frac {22 e^5 (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 a^3 d}+\frac {4 e (e \cos (c+d x))^{11/2}}{3 a d (a+a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 66, normalized size = 0.48 \[ -\frac {2\ 2^{3/4} (e \cos (c+d x))^{15/2} \, _2F_1\left (\frac {1}{4},\frac {15}{4};\frac {19}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{15 a^3 d e (\sin (c+d x)+1)^{15/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )} e^{6} \cos \left (d x + c\right )^{6}}{3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 = 1.10, size = 216, normalized size = 1.57 \[ \frac {2 e^{7} \left (-240 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+480 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+200 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-126 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-440 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+125 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {13}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{13/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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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