Optimal. Leaf size=242 \[ \frac {10 a \left (11 a^2+6 b^2\right ) e^4 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{231 d \sqrt {e \sin (c+d x)}}-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e} \]
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Rubi [A]
time = 0.21, antiderivative size = 242, 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 = {2771, 2941,
2748, 2715, 2721, 2720} \begin {gather*} \frac {10 a e^4 \left (11 a^2+6 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 a e^3 \left (11 a^2+6 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}-\frac {2 a e \left (11 a^2+6 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))^2}{13 d e}+\frac {34 a b (e \sin (c+d x))^{9/2} (a+b \cos (c+d x))}{143 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rule 2941
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{7/2} \, dx &=\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {2}{13} \int (a+b \cos (c+d x)) \left (\frac {13 a^2}{2}+2 b^2+\frac {17}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {4}{143} \int \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 ) \cos (c+d x)\right ) (e \sin (c+d x))^{7/2} \, dx\\ &=\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {1}{11} \left (a \left (11 a^2+6 b^2\right )\right ) \int (e \sin (c+d x))^{7/2} \, dx\\ &=-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {1}{77} \left (5 a \left (11 a^2+6 b^2\right ) e^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{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 \sin (c+d x)}} \, dx\\ &=-\frac {10 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}+\frac {\left (5 a \left (11 a^2+6 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 a \left (11 a^2+6 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 a \left (11 a^2+6 b^2\right ) e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{231 d}-\frac {2 a \left (11 a^2+6 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{5/2}}{77 d}+\frac {2 b \left (177 a^2+44 b^2\right ) (e \sin (c+d x))^{9/2}}{1287 d e}+\frac {34 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{9/2}}{143 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{9/2}}{13 d e}\\ \end {align*}
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Mathematica [A]
time = 2.58, size = 205, normalized size = 0.85 \begin {gather*} \frac {\left (154 b \left (78 a^2+11 b^2\right ) \csc ^3(c+d x)+\frac {1}{3} \left (-156 a \left (506 a^2+213 b^2\right ) \cos (c+d x)-77 b \left (624 a^2+73 b^2\right ) \cos (2 (c+d x))+234 a \left (44 a^2-39 b^2\right ) \cos (3 (c+d x))-154 b \left (-78 a^2+b^2\right ) \cos (4 (c+d x))+4914 a b^2 \cos (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right ) \csc ^3(c+d x)-\frac {2080 a \left (11 a^2+6 b^2\right ) F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )}{\sin ^{\frac {7}{2}}(c+d x)}\right ) (e \sin (c+d x))^{7/2}}{48048 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 276, normalized size = 1.14
method | result | size |
default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}} \left (9 \left (\cos ^{2}\left (d x +c \right )\right ) b^{2}+39 a^{2}+4 b^{2}\right )}{117 e}-\frac {e^{4} a \left (-126 b^{2} \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )-66 a^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+216 b^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+55 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+30 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}+176 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-30 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )\right )}{231 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(276\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.16, size = 238, normalized size = 0.98 \begin {gather*} \frac {195 \, \sqrt {2} \sqrt {-i} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} \sqrt {i} {\left (11 \, a^{3} + 6 \, a b^{2}\right )} e^{\frac {7}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (693 \, b^{3} \cos \left (d x + c\right )^{6} e^{\frac {7}{2}} + 2457 \, a b^{2} \cos \left (d x + c\right )^{5} e^{\frac {7}{2}} + 77 \, {\left (39 \, a^{2} b - 14 \, b^{3}\right )} \cos \left (d x + c\right )^{4} e^{\frac {7}{2}} + 117 \, {\left (11 \, a^{3} - 36 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} e^{\frac {7}{2}} - 77 \, {\left (78 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2} e^{\frac {7}{2}} - 39 \, {\left (88 \, a^{3} - 15 \, a b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {7}{2}} + 77 \, {\left (39 \, a^{2} b + 4 \, b^{3}\right )} e^{\frac {7}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{9009 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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