Optimal. Leaf size=642 \[ \frac{2 b \left (-297 a^2 d^2+66 a b c d+b^2 \left (-\left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac{2 \left (495 a^2 b d^2 \left (3 c^2+5 d^2\right )+1848 a^3 c d^3-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (57 c^2 d^2+8 c^4+135 d^4\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3465 d^2 f}-\frac{2 \left (c^2-d^2\right ) \left (495 a^2 b d^2 \left (3 c^2+5 d^2\right )+1848 a^3 c d^3-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (57 c^2 d^2+8 c^4+135 d^4\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3465 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+231 a^3 d^3 \left (23 c^2+9 d^2\right )-33 a b^2 d \left (-279 c^2 d^2+10 c^4-147 d^4\right )+5 b^3 \left (51 c^3 d^2+8 c^5+741 c d^4\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3465 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f} \]
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Rubi [A] time = 1.40057, antiderivative size = 642, normalized size of antiderivative = 1., 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 = {2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 b \left (-297 a^2 d^2+66 a b c d+b^2 \left (-\left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac{2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac{2 \left (495 a^2 b d^2 \left (3 c^2+5 d^2\right )+1848 a^3 c d^3-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (57 c^2 d^2+8 c^4+135 d^4\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{3465 d^2 f}-\frac{2 \left (c^2-d^2\right ) \left (495 a^2 b d^2 \left (3 c^2+5 d^2\right )+1848 a^3 c d^3-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (57 c^2 d^2+8 c^4+135 d^4\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3465 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+231 a^3 d^3 \left (23 c^2+9 d^2\right )-33 a b^2 d \left (-279 c^2 d^2+10 c^4-147 d^4\right )+5 b^3 \left (51 c^3 d^2+8 c^5+741 c d^4\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{3465 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
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*}
Mathematica [A] time = 2.62406, size = 545, normalized size = 0.85 \[ \frac{d (c+d \sin (e+f x)) \left (-4 d \left (8910 a^2 b c d^2+1386 a^3 d^3+33 a b^2 d \left (150 c^2+133 d^2\right )+5 b^3 \left (6 c^3+619 c d^2\right )\right ) \sin (2 (e+f x))+5 b d^2 \left (1188 a^2 d^2+2508 a b c d+b^2 \left (452 c^2+513 d^2\right )\right ) \cos (3 (e+f x))+2 \left (-990 a^2 b d^2 \left (36 c^2+23 d^2\right )-20328 a^3 c d^3-66 a b^2 d \left (20 c^3+747 c d^2\right )+5 b^3 \left (-1866 c^2 d^2+32 c^4-1305 d^4\right )\right ) \cos (e+f x)+70 b^2 d^3 (33 a d+23 b c) \sin (4 (e+f x))-315 b^3 d^4 \cos (5 (e+f x))\right )-16 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (495 a^2 b d^2 \left (27 c^2+5 d^2\right )+231 a^3 c d \left (15 c^2+17 d^2\right )+33 a b^2 d \left (155 c^3+261 c d^2\right )+5 b^3 \left (663 c^2 d^2+2 c^4+135 d^4\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+231 a^3 d^3 \left (23 c^2+9 d^2\right )+33 a b^2 d \left (279 c^2 d^2-10 c^4+147 d^4\right )+5 b^3 \left (51 c^3 d^2+8 c^5+741 c d^4\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )}{27720 d^3 f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 7.577, size = 2728, normalized size = 4.3 \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 (b \sin \left (f x + e\right ) + a\right )}^{3}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{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 ({\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} +{\left (a^{3} + 3 \, a b^{2}\right )} c^{2} + 2 \,{\left (3 \, a^{2} b + b^{3}\right )} c d +{\left (a^{3} + 3 \, a b^{2}\right )} d^{2} -{\left (3 \, a b^{2} c^{2} + 2 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d +{\left (a^{3} + 6 \, a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} +{\left (b^{3} d^{2} \cos \left (f x + e\right )^{4} +{\left (3 \, a^{2} b + b^{3}\right )} c^{2} + 2 \,{\left (a^{3} + 3 \, a b^{2}\right )} c d +{\left (3 \, a^{2} b + b^{3}\right )} d^{2} -{\left (b^{3} c^{2} + 6 \, a b^{2} c d +{\left (3 \, a^{2} b + 2 \, b^{3}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}, 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(-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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