Optimal. Leaf size=642 \[ -\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3465 d^3 f \sqrt {c+d \sin (e+f x)}} \]
[Out]
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Rubi [A]
time = 0.92, antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2872, 3102,
2832, 2831, 2742, 2740, 2734, 2732} \begin {gather*} \frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\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 )}{3465 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\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 )}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2831
Rule 2832
Rule 2872
Rule 3102
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \int (c+d \sin (e+f x))^{5/2} \left (\frac {1}{2} \left (2 b^3 c+11 a^3 d+7 a b^2 d\right )-\frac {1}{2} b \left (2 a b c-33 a^2 d-9 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-6 a d) \sin ^2(e+f x)\right ) \, dx}{11 d}\\ &=\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 \int (c+d \sin (e+f x))^{5/2} \left (-\frac {1}{4} d \left (10 b^3 c-99 a^3 d-231 a b^2 d\right )-\frac {1}{4} b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{99 d^2}\\ &=\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {8 \int (c+d \sin (e+f x))^{3/2} \left (\frac {3}{8} d \left (231 a^3 c d+429 a b^2 c d+495 a^2 b d^2-5 b^3 \left (2 c^2-27 d^2\right )\right )+\frac {1}{8} \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \sin (e+f x)\right ) \, dx}{693 d^2}\\ &=-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {16 \int \sqrt {c+d \sin (e+f x)} \left (\frac {3}{16} d \left (3960 a^2 b c d^2+231 a^3 d \left (5 c^2+3 d^2\right )+33 a b^2 d \left (55 c^2+49 d^2\right )-10 b^3 \left (c^3-101 c d^2\right )\right )+\frac {3}{16} \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{3465 d^2}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {32 \int \frac {\frac {3}{32} d \left (495 a^2 b d^2 \left (27 c^2+5 d^2\right )+231 a^3 c d \left (15 c^2+17 d^2\right )+33 a b^2 d \left (155 c^3+261 c d^2\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right )+\frac {3}{32} \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{10395 d^2}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}-\frac {\left (\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3465 d^3}+\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3465 d^3}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {\left (\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3465 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\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}{3465 d^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\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)}}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\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}}}{3465 d^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 2.74, size = 545, normalized size = 0.85 \begin {gather*} \frac {-16 \left (d^2 \left (495 a^2 b d^2 \left (27 c^2+5 d^2\right )+231 a^3 c d \left (15 c^2+17 d^2\right )+33 a b^2 d \left (155 c^3+261 c d^2\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right ) F\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right )+\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+33 a b^2 d \left (-10 c^4+279 c^2 d^2+147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\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 (-20328 a^3 c d^3-990 a^2 b d^2 \left (36 c^2+23 d^2\right )-66 a b^2 d \left (20 c^3+747 c d^2\right )+5 b^3 \left (32 c^4-1866 c^2 d^2-1305 d^4\right )\right ) \cos (e+f x)+5 b d^2 \left (2508 a b c d+1188 a^2 d^2+b^2 \left (452 c^2+513 d^2\right )\right ) \cos (3 (e+f x))-315 b^3 d^4 \cos (5 (e+f x))-4 d \left (8910 a^2 b c d^2+1386 a^3 d^3+33 a b^2 d \left (150 c^2+133 d^2\right )+5 b^3 \left (6 c^3+619 c d^2\right )\right ) \sin (2 (e+f x))+70 b^2 d^3 (23 b c+33 a d) \sin (4 (e+f x))\right )}{27720 d^3 f \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. \(2727\) vs.
\(2(676)=1352\).
time = 43.21, size = 2728, normalized size = 4.25
method | result | size |
default | \(\text {Expression too large to display}\) | \(2728\) |
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 = 0.25, size = 1066, normalized size = 1.66 \begin {gather*} -\frac {\sqrt {2} {\left (80 \, b^{3} c^{6} - 660 \, a b^{2} c^{5} d + 30 \, {\left (99 \, a^{2} b + 16 \, b^{3}\right )} c^{4} d^{2} + 33 \, {\left (7 \, a^{3} + 93 \, a b^{2}\right )} c^{3} d^{3} - 15 \, {\left (759 \, a^{2} b + 169 \, b^{3}\right )} c^{2} d^{4} - 99 \, {\left (77 \, a^{3} + 163 \, a b^{2}\right )} c d^{5} - 675 \, {\left (11 \, a^{2} b + 3 \, b^{3}\right )} 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 (80 \, b^{3} c^{6} - 660 \, a b^{2} c^{5} d + 30 \, {\left (99 \, a^{2} b + 16 \, b^{3}\right )} c^{4} d^{2} + 33 \, {\left (7 \, a^{3} + 93 \, a b^{2}\right )} c^{3} d^{3} - 15 \, {\left (759 \, a^{2} b + 169 \, b^{3}\right )} c^{2} d^{4} - 99 \, {\left (77 \, a^{3} + 163 \, a b^{2}\right )} c d^{5} - 675 \, {\left (11 \, a^{2} b + 3 \, b^{3}\right )} 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 (40 i \, b^{3} c^{5} d - 330 i \, a b^{2} c^{4} d^{2} + 15 i \, {\left (99 \, a^{2} b + 17 \, b^{3}\right )} c^{3} d^{3} + 33 i \, {\left (161 \, a^{3} + 279 \, a b^{2}\right )} c^{2} d^{4} + 15 i \, {\left (957 \, a^{2} b + 247 \, b^{3}\right )} c d^{5} + 693 i \, {\left (3 \, a^{3} + 7 \, a b^{2}\right )} 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 (-40 i \, b^{3} c^{5} d + 330 i \, a b^{2} c^{4} d^{2} - 15 i \, {\left (99 \, a^{2} b + 17 \, b^{3}\right )} c^{3} d^{3} - 33 i \, {\left (161 \, a^{3} + 279 \, a b^{2}\right )} c^{2} d^{4} - 15 i \, {\left (957 \, a^{2} b + 247 \, b^{3}\right )} c d^{5} - 693 i \, {\left (3 \, a^{3} + 7 \, a b^{2}\right )} 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 ) + 6 \, {\left (315 \, b^{3} d^{6} \cos \left (f x + e\right )^{5} - 5 \, {\left (113 \, b^{3} c^{2} d^{4} + 627 \, a b^{2} c d^{5} + 9 \, {\left (33 \, a^{2} b + 23 \, b^{3}\right )} d^{6}\right )} \cos \left (f x + e\right )^{3} - {\left (20 \, b^{3} c^{4} d^{2} - 165 \, a b^{2} c^{3} d^{3} - 15 \, {\left (297 \, a^{2} b + 106 \, b^{3}\right )} c^{2} d^{4} - 33 \, {\left (77 \, a^{3} + 258 \, a b^{2}\right )} c d^{5} - 45 \, {\left (88 \, a^{2} b + 31 \, b^{3}\right )} d^{6}\right )} \cos \left (f x + e\right ) - {\left (35 \, {\left (23 \, b^{3} c d^{5} + 33 \, a b^{2} d^{6}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (5 \, b^{3} c^{3} d^{3} + 825 \, a b^{2} c^{2} d^{4} + 5 \, {\left (297 \, a^{2} b + 130 \, b^{3}\right )} c d^{5} + 231 \, {\left (a^{3} + 4 \, a b^{2}\right )} 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}}{10395 \, 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*} \int \left (a + b \sin {\left (e + f x \right )}\right )^{3} \left (c + d \sin {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \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+b\,\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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