Optimal. Leaf size=467 \[ -\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{693 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{693 d^3 f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.64, antiderivative size = 467, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2842, 3047,
3102, 2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{693 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {4 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{693 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right ) (c+d \sin (e+f x))^{7/2}}{11 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2842
Rule 3047
Rule 3102
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \int (a+a \sin (e+f x)) \left (a^2 (c+9 d)-2 a^2 (c-6 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2} \, dx}{11 d}\\ &=-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \int (c+d \sin (e+f x))^{5/2} \left (a^3 (c+9 d)+\left (-2 a^3 (c-6 d)+a^3 (c+9 d)\right ) \sin (e+f x)-2 a^3 (c-6 d) \sin ^2(e+f x)\right ) \, dx}{11 d}\\ &=\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 \int (c+d \sin (e+f x))^{5/2} \left (-\frac {5}{2} a^3 (c-33 d) d+\frac {1}{2} a^3 \left (4 c^2-33 c d+189 d^2\right ) \sin (e+f x)\right ) \, dx}{99 d^2}\\ &=-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {8 \int (c+d \sin (e+f x))^{3/2} \left (-\frac {15}{4} a^3 d \left (c^2-66 c d-63 d^2\right )+\frac {5}{4} a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \sin (e+f x)\right ) \, dx}{693 d^2}\\ &=-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {16 \int \sqrt {c+d \sin (e+f x)} \left (-\frac {15}{8} a^3 d \left (c^3-297 c^2 d-497 c d^2-231 d^3\right )+\frac {15}{8} a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sin (e+f x)\right ) \, dx}{3465 d^2}\\ &=-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {32 \int \frac {\frac {15}{16} a^3 d \left (c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right )+\frac {15}{16} a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{10395 d^2}\\ &=-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {\left (2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{693 d^3}-\frac {\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{693 d^3}\\ &=-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {\left (2 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{693 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (2 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{693 d^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {4 a^3 \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{693 d^2 f}-\frac {4 a^3 \left (4 c^3-33 c^2 d+182 c d^2+231 d^3\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{693 d^2 f}-\frac {4 a^3 \left (4 c^2-33 c d+189 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 a^3 (c-6 d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right ) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 a^3 (c+3 d) \left (4 c^4-45 c^3 d+309 c^2 d^2+525 c d^3+231 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{693 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {4 a^3 \left (c^2-d^2\right ) \left (4 c^4-33 c^3 d+177 c^2 d^2+561 c d^3+315 d^4\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{693 d^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.30, size = 377, normalized size = 0.81 \begin {gather*} \frac {a^3 (1+\sin (e+f x))^3 \left (-32 \left (d^2 \left (c^4+858 c^3 d+1668 c^2 d^2+1254 c d^3+315 d^4\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+\left (4 c^5-33 c^4 d+174 c^3 d^2+1452 c^2 d^3+1806 c d^4+693 d^5\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}+d (c+d \sin (e+f x)) \left (2 \left (32 c^4-264 c^3 d-8994 c^2 d^2-13926 c d^3-5859 d^4\right ) \cos (e+f x)+d^2 \left (452 c^2+2508 c d+1701 d^2\right ) \cos (3 (e+f x))-63 d^4 \cos (5 (e+f x))-4 d \left (6 c^3+990 c^2 d+2401 c d^2+1155 d^3\right ) \sin (2 (e+f x))+14 d^3 (23 c+33 d) \sin (4 (e+f x))\right )\right )}{5544 d^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sqrt {c+d \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1925\) vs.
\(2(501)=1002\).
time = 5.45, size = 1926, normalized size = 4.12
method | result | size |
default | \(\text {Expression too large to display}\) | \(1926\) |
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 = 0.23, size = 843, normalized size = 1.81 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} {\left (8 \, a^{3} c^{6} - 66 \, a^{3} c^{5} d + 345 \, a^{3} c^{4} d^{2} + 330 \, a^{3} c^{3} d^{3} - 1392 \, a^{3} c^{2} d^{4} - 2376 \, a^{3} c d^{5} - 945 \, a^{3} d^{6}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) + \sqrt {2} {\left (8 \, a^{3} c^{6} - 66 \, a^{3} c^{5} d + 345 \, a^{3} c^{4} d^{2} + 330 \, a^{3} c^{3} d^{3} - 1392 \, a^{3} c^{2} d^{4} - 2376 \, a^{3} c d^{5} - 945 \, a^{3} d^{6}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, \sqrt {2} {\left (4 i \, a^{3} c^{5} d - 33 i \, a^{3} c^{4} d^{2} + 174 i \, a^{3} c^{3} d^{3} + 1452 i \, a^{3} c^{2} d^{4} + 1806 i \, a^{3} c d^{5} + 693 i \, a^{3} d^{6}\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, \sqrt {2} {\left (-4 i \, a^{3} c^{5} d + 33 i \, a^{3} c^{4} d^{2} - 174 i \, a^{3} c^{3} d^{3} - 1452 i \, a^{3} c^{2} d^{4} - 1806 i \, a^{3} c d^{5} - 693 i \, a^{3} d^{6}\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (63 \, a^{3} d^{6} \cos \left (f x + e\right )^{5} - {\left (113 \, a^{3} c^{2} d^{4} + 627 \, a^{3} c d^{5} + 504 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, a^{3} c^{4} d^{2} - 33 \, a^{3} c^{3} d^{3} - 1209 \, a^{3} c^{2} d^{4} - 2211 \, a^{3} c d^{5} - 1071 \, a^{3} d^{6}\right )} \cos \left (f x + e\right ) - {\left (7 \, {\left (23 \, a^{3} c d^{5} + 33 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (a^{3} c^{3} d^{3} + 165 \, a^{3} c^{2} d^{4} + 427 \, a^{3} c d^{5} + 231 \, a^{3} d^{6}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}\right )}}{2079 \, d^{4} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{3} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 3 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{5}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 6 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \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 (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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