Optimal. Leaf size=258 \[ -\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac {2 e^2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}+\frac {2 e \left (77 a^4+132 a^2 b^2+12 b^4\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e} \]
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Rubi [A] time = 0.51, antiderivative size = 258, 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} \[ \frac {2 e^2 \left (132 a^2 b^2+77 a^4+12 b^4\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d \sqrt {e \cos (c+d x)}}-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac {2 e \left (132 a^2 b^2+77 a^4+12 b^4\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{231 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e} \]
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))^{3/2} (a+b \sin (c+d x))^4 \, dx &=-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \left (\frac {11 a^2}{2}+3 b^2+\frac {17}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {4}{99} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \left (\frac {1}{4} a \left (99 a^2+122 b^2\right )+\frac {1}{4} b \left (167 a^2+54 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {8}{693} \int (e \cos (c+d x))^{3/2} \left (\frac {9}{8} \left (77 a^4+132 a^2 b^2+12 b^4\right )+\frac {13}{8} a b \left (79 a^2+74 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {1}{77} \left (77 a^4+132 a^2 b^2+12 b^4\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {1}{231} \left (\left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}+\frac {\left (\left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \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 b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}+\frac {2 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e^2 \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 \left (77 a^4+132 a^2 b^2+12 b^4\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{231 d}-\frac {2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}-\frac {34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}\\ \end {align*}
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Mathematica [A] time = 2.81, size = 189, normalized size = 0.73 \[ \frac {(e \cos (c+d x))^{3/2} \left (240 \left (77 a^4+132 a^2 b^2+12 b^4\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sqrt {\cos (c+d x)} \left (-2464 \left (9 a^3 b+4 a b^3\right ) \cos (2 (c+d x))-1848 b \left (12 a^3+7 a b^2\right )-45 b \left (264 a^2 b+31 b^3\right ) \sin (3 (c+d x))+30 \left (616 a^4+660 a^2 b^2+39 b^4\right ) \sin (c+d x)+3080 a b^3 \cos (4 (c+d x))+315 b^4 \sin (5 (c+d x))\right )\right )}{27720 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} e \cos \left (d x + c\right )^{5} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} e \cos \left (d x + c\right )^{3} + {\left (a^{4} + 6 \, a^{2} b^{2} + b^{4}\right )} e \cos \left (d x + c\right ) - 4 \, {\left (a b^{3} e \cos \left (d x + c\right )^{3} - {\left (a^{3} b + a b^{3}\right )} e \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.76, size = 639, normalized size = 2.48 \[ -\frac {2 e^{2} \left (20160 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+49280 a \,b^{3} \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-50400 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-47520 a^{2} 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 )-123200 a \,b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+41040 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-22176 a^{3} b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+71280 a^{2} 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 )+101024 a \,b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-11160 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4620 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+33264 a^{3} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27720 a^{2} 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 )-28336 a \,b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 \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^{4}+1980 \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^{2} b^{2}+180 \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 ) b^{4}-2310 a^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-16632 a^{3} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1980 a^{2} 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 )-1232 a \,b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+180 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2772 a^{3} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+1232 a \,b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3465 \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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]
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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