Optimal. Leaf size=237 \[ \frac {10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]
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Rubi [A] time = 0.31, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2692, 2862, 2669, 2635, 2642, 2641} \[ \frac {10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}+\frac {10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {2}{13} \int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \left (\frac {13 a^2}{2}+2 b^2+\frac {17}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {4}{143} \int (e \cos (c+d x))^{7/2} \left (\frac {13}{4} a \left (11 a^2+6 b^2\right )+\frac {1}{4} b \left (177 a^2+44 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{11} \left (a \left (11 a^2+6 b^2\right )\right ) \int (e \cos (c+d x))^{7/2} \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{77} \left (5 a \left (11 a^2+6 b^2\right ) e^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {1}{231} \left (5 a \left (11 a^2+6 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}+\frac {\left (5 a \left (11 a^2+6 b^2\right ) 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 {2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac {10 a \left (11 a^2+6 b^2\right ) 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 {10 a \left (11 a^2+6 b^2\right ) e^3 \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 a \left (11 a^2+6 b^2\right ) e (e \cos (c+d x))^{5/2} \sin (c+d x)}{77 d}-\frac {34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}-\frac {2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}\\ \end {align*}
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Mathematica [A] time = 2.09, size = 205, normalized size = 0.86 \[ \frac {(e \cos (c+d x))^{7/2} \left (2080 \left (11 a^3+6 a b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )-154 b \left (78 a^2+11 b^2\right ) \sqrt {\cos (c+d x)}+\frac {1}{3} \sqrt {\cos (c+d x)} \left (156 a \left (506 a^2+213 b^2\right ) \sin (c+d x)+234 a \left (44 a^2-39 b^2\right ) \sin (3 (c+d x))-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+154 b \left (b^2-78 a^2\right ) \cos (4 (c+d x))-4914 a b^2 \sin (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right )\right )}{48048 d \cos ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} e^{3} \cos \left (d x + c\right )^{5} - {\left (a^{3} + 3 \, a b^{2}\right )} e^{3} \cos \left (d x + c\right )^{3} + {\left (b^{3} e^{3} \cos \left (d x + c\right )^{5} - {\left (3 \, a^{2} b + b^{3}\right )} e^{3} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.47, size = 618, normalized size = 2.61 \[ -\frac {2 e^{4} \left (3003 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-308 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+433664 b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-310464 b^{3} \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+120120 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240240 a^{2} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30030 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6864 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240240 a^{2} b \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+24024 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96096 a^{2} b \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20592 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+308 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-18172 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30888 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2145 \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-381888 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+179712 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-36036 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1170 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-157248 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+393120 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+88704 b^{3} \left (\sin ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-308000 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+113960 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1170 \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}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}\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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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