Optimal. Leaf size=168 \[ \frac {130 a^2 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {130 a^2 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}+\frac {26 a^2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d} \]
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Rubi [A] time = 0.14, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {130 a^2 e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}+\frac {130 a^2 e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 \left (a^2 \sin (c+d x)+a^2\right ) (e \cos (c+d x))^{9/2}}{11 d e}+\frac {26 a^2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d} \]
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))^2 \, dx &=-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{11} (13 a) \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx\\ &=-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{11} \left (13 a^2\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{77} \left (65 a^2 e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {1}{231} \left (65 a^2 e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}+\frac {\left (65 a^2 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 {26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}+\frac {130 a^2 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 {130 a^2 e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {26 a^2 e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {2 (e \cos (c+d x))^{9/2} \left (a^2+a^2 \sin (c+d x)\right )}{11 d e}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 66, normalized size = 0.39 \[ -\frac {32 \sqrt [4]{2} a^2 (e \cos (c+d x))^{9/2} \, _2F_1\left (-\frac {13}{4},\frac {9}{4};\frac {13}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{9 d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} e^{3} \cos \left (d x + c\right )^{5} - 2 \, a^{2} e^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} e^{3} \cos \left (d x + c\right )^{3}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.75, size = 295, normalized size = 1.76 \[ -\frac {2 a^{2} e^{4} \left (-4032 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10080 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4928 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-8208 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12320 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2232 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-12320 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+924 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+6160 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-498 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1540 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+154 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{693 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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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