Optimal. Leaf size=118 \[ \frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a^3 d \sqrt {\cos (c+d x)}}+\frac {6 e (e \cos (c+d x))^{3/2}}{5 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2} \]
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Rubi [A] time = 0.13, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2680, 2683, 2640, 2639} \[ \frac {6 e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 a^3 d \sqrt {\cos (c+d x)}}+\frac {6 e (e \cos (c+d x))^{3/2}}{5 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2680
Rule 2683
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx &=-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a+a \sin (c+d x))^2}-\frac {\left (3 e^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx}{5 a^2}\\ &=-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {6 e (e \cos (c+d x))^{3/2}}{5 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {\left (3 e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 a^3}\\ &=-\frac {4 e (e \cos (c+d x))^{3/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {6 e (e \cos (c+d x))^{3/2}}{5 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {\left (3 e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^3 \sqrt {\cos (c+d x)}}\\ &=\frac {6 e^2 \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 {4 e (e \cos (c+d x))^{3/2}}{5 a d (a+a \sin (c+d x))^2}+\frac {6 e (e \cos (c+d x))^{3/2}}{5 d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 66, normalized size = 0.56 \[ -\frac {(e \cos (c+d x))^{7/2} \, _2F_1\left (\frac {7}{4},\frac {9}{4};\frac {11}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{7 \sqrt [4]{2} a^3 d e (\sin (c+d x)+1)^{7/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e \cos \left (d x + c\right )} e^{2} \cos \left (d x + c\right )^{2}}{3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 [B] time = 2.14, size = 330, normalized size = 2.80 \[ \frac {2 \left (12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-24 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12 \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+20 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-20 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e^{3}}{5 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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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