Optimal. Leaf size=203 \[ \frac {170 a^3 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {170 a^3 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {34 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{9/2}}{143 d e}+\frac {34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e} \]
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Rubi [A] time = 0.21, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2678, 2669, 2635, 2642, 2641} \[ \frac {170 a^3 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}+\frac {170 a^3 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {34 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{9/2}}{143 d e}+\frac {34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2678
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^3 \, dx &=-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}+\frac {1}{13} (17 a) \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac {34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac {1}{11} \left (17 a^2\right ) \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac {34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac {1}{11} \left (17 a^3\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac {34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac {1}{77} \left (85 a^3 e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {170 a^3 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac {34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac {1}{231} \left (85 a^3 e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {170 a^3 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac {34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}+\frac {\left (85 a^3 e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{231 \sqrt {e \cos (c+d x)}}\\ &=-\frac {34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {170 a^3 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {170 a^3 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {34 a^3 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 a (e \cos (c+d x))^{9/2} (a+a \sin (c+d x))^2}{13 d e}-\frac {34 (e \cos (c+d x))^{9/2} \left (a^3+a^3 \sin (c+d x)\right )}{143 d e}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 66, normalized size = 0.33 \[ -\frac {64 \sqrt [4]{2} a^3 (e \cos (c+d x))^{9/2} \, _2F_1\left (-\frac {17}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a^{3} e^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} e^{3} \cos \left (d x + c\right )^{3} + {\left (a^{3} e^{3} \cos \left (d x + c\right )^{5} - 4 \, a^{3} e^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.94, size = 321, normalized size = 1.58 \[ -\frac {2 a^{3} e^{4} \left (88704 \left (\sin ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-157248 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-310464 \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+393120 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+337568 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-361296 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-67760 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+148824 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-126280 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-12012 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+101948 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3315 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-5694 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-30338 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3311 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9009 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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