Optimal. Leaf size=101 \[ \frac {2 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d \sqrt {e \cos (c+d x)}}+\frac {2 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 a d}+\frac {2 e (e \cos (c+d x))^{5/2}}{5 a d} \]
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Rubi [A] time = 0.09, antiderivative size = 101, 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 = {2682, 2635, 2642, 2641} \[ \frac {2 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 a d}+\frac {2 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d \sqrt {e \cos (c+d x)}}+\frac {2 e (e \cos (c+d x))^{5/2}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2682
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{7/2}}{a+a \sin (c+d x)} \, dx &=\frac {2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac {e^2 \int (e \cos (c+d x))^{3/2} \, dx}{a}\\ &=\frac {2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac {2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {e^4 \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{3 a}\\ &=\frac {2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac {2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {\left (e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 a \sqrt {e \cos (c+d x)}}\\ &=\frac {2 e (e \cos (c+d x))^{5/2}}{5 a d}+\frac {2 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d \sqrt {e \cos (c+d x)}}+\frac {2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 a d}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 66, normalized size = 0.65 \[ -\frac {4 \sqrt [4]{2} (e \cos (c+d x))^{9/2} \, _2F_1\left (-\frac {1}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{9 a d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \cos \left (d x + c\right )} e^{3} \cos \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.09, size = 181, normalized size = 1.79 \[ -\frac {2 e^{4} \left (24 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-36 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-10 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+18 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
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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