Optimal. Leaf size=237 \[ \frac{10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]
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Rubi [A] time = 0.307995, antiderivative size = 237, 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} \[ \frac{10 a e^3 \left (11 a^2+6 b^2\right ) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 a e^4 \left (11 a^2+6 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{231 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left (177 a^2+44 b^2\right ) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left (11 a^2+6 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e} \]
Antiderivative was successfully verified.
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Rule 2692
Rule 2862
Rule 2669
Rule 2635
Rule 2642
Rule 2641
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*}
Mathematica [A] time = 2.01581, size = 205, normalized size = 0.86 \[ \frac{(e \cos (c+d x))^{7/2} \left (-154 b \left (78 a^2+11 b^2\right ) \sqrt{\cos (c+d x)}+2080 \left (11 a^3+6 a b^2\right ) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{1}{3} \sqrt{\cos (c+d x)} \left (156 a \left (506 a^2+213 b^2\right ) \sin (c+d x)+234 a \left (44 a^2-39 b^2\right ) \sin (3 (c+d x))-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+154 b \left (b^2-78 a^2\right ) \cos (4 (c+d x))-4914 a b^2 \sin (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right )\right )}{48048 d \cos ^{\frac{7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 2.566, size = 618, normalized size = 2.6 \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{7}{2}}{\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{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 (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 ) \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{7}{2}}{\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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