Optimal. Leaf size=193 \[ \frac {10 e^4 \left (11 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 e^3 \left (11 a^2+2 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 e \left (11 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e} \]
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Rubi [A] time = 0.19, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2692, 2669, 2635, 2642, 2641} \[ -\frac {10 e^3 \left (11 a^2+2 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}+\frac {10 e^4 \left (11 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{231 d \sqrt {e \sin (c+d x)}}-\frac {2 e \left (11 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{11 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{7/2} \, dx &=\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {2}{11} \int \left (\frac {11 a^2}{2}+b^2+\frac {13}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {1}{11} \left (11 a^2+2 b^2\right ) \int (e \sin (c+d x))^{7/2} \, dx\\ &=-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {1}{77} \left (5 \left (11 a^2+2 b^2\right ) e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {1}{231} \left (5 \left (11 a^2+2 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}+\frac {\left (5 \left (11 a^2+2 b^2\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{231 \sqrt {e \sin (c+d x)}}\\ &=\frac {10 \left (11 a^2+2 b^2\right ) e^4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 \left (11 a^2+2 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 \left (11 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {26 a b (e \sin (c+d x))^{9/2}}{99 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{11 d e}\\ \end {align*}
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Mathematica [A] time = 1.67, size = 157, normalized size = 0.81 \[ \frac {(e \sin (c+d x))^{7/2} \left (\frac {1}{6} \csc ^3(c+d x) \left (-6 \left (506 a^2+71 b^2\right ) \cos (c+d x)+396 a^2 \cos (3 (c+d x))-1232 a b \cos (2 (c+d x))+308 a b \cos (4 (c+d x))+924 a b-117 b^2 \cos (3 (c+d x))+63 b^2 \cos (5 (c+d x))\right )-\frac {40 \left (11 a^2+2 b^2\right ) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac {7}{2}}(c+d x)}\right )}{924 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.13, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} e^{3} \cos \left (d x + c\right )^{4} + 2 \, a b e^{3} \cos \left (d x + c\right )^{3} - 2 \, a b e^{3} \cos \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} e^{3} \cos \left (d x + c\right )^{2} - a^{2} e^{3}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 228, normalized size = 1.18 \[ \frac {\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9 e}-\frac {e^{4} \left (-42 b^{2} \sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )+\left (-66 a^{2}+72 b^{2}\right ) \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (176 a^{2}-10 b^{2}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+55 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+10 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}\,{d x} \]
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 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \]
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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