Optimal. Leaf size=258 \[ \frac{2 e^2 \left (132 a^2 b^2+77 a^4+12 b^4\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{26 a b \left (79 a^2+74 b^2\right ) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac{2 b \left (167 a^2+54 b^2\right ) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac{2 e \left (132 a^2 b^2+77 a^4+12 b^4\right ) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac{34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e} \]
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Rubi [A] time = 0.509408, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, 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 2692
Rule 2862
Rule 2669
Rule 2635
Rule 2642
Rule 2641
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*}
Mathematica [A] time = 2.72029, size = 189, normalized size = 0.73 \[ \frac{(e \cos (c+d x))^{3/2} \left (240 \left (132 a^2 b^2+77 a^4+12 b^4\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\sqrt{\cos (c+d x)} \left (-45 b \left (264 a^2 b+31 b^3\right ) \sin (3 (c+d x))+30 \left (660 a^2 b^2+616 a^4+39 b^4\right ) \sin (c+d x)-2464 \left (9 a^3 b+4 a b^3\right ) \cos (2 (c+d x))-1848 b \left (12 a^3+7 a b^2\right )+3080 a b^3 \cos (4 (c+d x))+315 b^4 \sin (5 (c+d x))\right )\right )}{27720 d \cos ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.106, size = 639, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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