Optimal. Leaf size=124 \[ \frac {10 a e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d} \]
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Rubi [A] time = 0.08, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2669, 2635, 2642, 2641} \[ \frac {10 a e^3 \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}+\frac {10 a e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx &=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+a \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{7} \left (5 a e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {1}{21} \left (5 a e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}+\frac {\left (5 a e^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {e \cos (c+d x)}}\\ &=-\frac {2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac {10 a e^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a e (e \cos (c+d x))^{5/2} \sin (c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 98, normalized size = 0.79 \[ -\frac {a e^3 \sqrt {e \cos (c+d x)} \left (\sqrt {\cos (c+d x)} (-138 \sin (c+d x)-18 \sin (3 (c+d x))+28 \cos (2 (c+d x))+7 \cos (4 (c+d x))+21)-120 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{252 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e^{3} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a e^{3} \cos \left (d x + c\right )^{3}\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.66, size = 249, normalized size = 2.01 \[ -\frac {2 a \,e^{4} \left (-224 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+144 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+560 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-216 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-560 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+168 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+280 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-48 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-70 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+a\,\sin \left (c+d\,x\right )\right ) \,d x \]
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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