Optimal. Leaf size=716 \[ -\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.43605, antiderivative size = 716, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2792, 3021, 2754, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \cos (e+f x)}{105 d^2 f \left (c^2-d^2\right )^4 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (-17 c^2 d^2+8 c^4+105 d^4\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^3 \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (-9 a^2 b d^2 \left (102 c^2 d^2+5 c^4+21 d^4\right )+16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (-62 c^2 d^2+3 c^4-133 d^4\right )+b^3 \left (-\left (-23 c^4 d^2+294 c^2 d^4+8 c^6+105 d^6\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \left (c^2-d^2\right )^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) (b c-a d)^2 \cos (e+f x)}{35 d^2 f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^{5/2}}+\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2792
Rule 3021
Rule 2754
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{(a+b \sin (e+f x))^3}{(c+d \sin (e+f x))^{9/2}} \, dx &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}-\frac{2 \int \frac{\frac{1}{2} \left (2 b (b c-a d)^2-7 a d \left (\left (a^2+b^2\right ) c-2 a b d\right )\right )+\frac{1}{2} \left (5 a (b c-a d)^2-7 b \left (a b c^2+\left (a^2+b^2\right ) c d-3 a b d^2\right )\right ) \sin (e+f x)-\frac{1}{2} b \left (6 a b c d-3 a^2 d^2+b^2 \left (4 c^2-7 d^2\right )\right ) \sin ^2(e+f x)}{(c+d \sin (e+f x))^{7/2}} \, dx}{7 d \left (c^2-d^2\right )}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}+\frac{4 \int \frac{-\frac{5}{4} d \left (36 a^2 b c d^2-a^3 d \left (7 c^2+5 d^2\right )-3 a b^2 d \left (5 c^2+7 d^2\right )-2 b^3 \left (c^3-7 c d^2\right )\right )-\frac{1}{4} \left (36 a^3 c d^3-18 a b^2 c d \left (c^2-7 d^2\right )-9 a^2 b d^2 \left (5 c^2+7 d^2\right )-b^3 \left (8 c^4-7 c^2 d^2+35 d^4\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{5/2}} \, dx}{35 d^2 \left (c^2-d^2\right )^2}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}-\frac{8 \int \frac{\frac{3}{8} d \left (9 a^2 b d^2 \left (25 c^2+7 d^2\right )-a^3 c d \left (35 c^2+61 d^2\right )-3 a b^2 d \left (19 c^3+77 c d^2\right )-b^3 \left (2 c^4-63 c^2 d^2-35 d^4\right )\right )-\frac{1}{8} (b c-a d) \left (8 b^2 c^4+26 a b c^3 d+71 a^2 c^2 d^2-17 b^2 c^2 d^2-218 a b c d^3+25 a^2 d^4+105 b^2 d^4\right ) \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx}{105 d^2 \left (c^2-d^2\right )^3}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{16 \int \frac{-\frac{1}{16} d \left (144 a^2 b c d^2 \left (5 c^2+3 d^2\right )-a^3 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-3 a b^2 d \left (51 c^4+298 c^2 d^2+35 d^4\right )+2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )\right )+\frac{1}{16} \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^2 \left (c^2-d^2\right )^4}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{105 d^3 \left (c^2-d^2\right )^4}--\frac{\left (16 \left (-\frac{1}{16} d^2 \left (144 a^2 b c d^2 \left (5 c^2+3 d^2\right )-a^3 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-3 a b^2 d \left (51 c^4+298 c^2 d^2+35 d^4\right )+2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )\right )-\frac{1}{16} c \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^3 \left (c^2-d^2\right )^4}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{\left (\left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\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}{105 d^3 \left (c^2-d^2\right )^4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}--\frac{\left (16 \left (-\frac{1}{16} d^2 \left (144 a^2 b c d^2 \left (5 c^2+3 d^2\right )-a^3 d \left (105 c^4+254 c^2 d^2+25 d^4\right )-3 a b^2 d \left (51 c^4+298 c^2 d^2+35 d^4\right )+2 b^3 \left (c^5+86 c^3 d^2+105 c d^4\right )\right )-\frac{1}{16} c \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\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}{105 d^3 \left (c^2-d^2\right )^4 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 (b c-a d)^2 \cos (e+f x) (a+b \sin (e+f x))}{7 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^{7/2}}+\frac{8 (b c-a d)^2 \left (3 a c d+b \left (c^2-4 d^2\right )\right ) \cos (e+f x)}{35 d^2 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (a^2 d^2 \left (71 c^2+25 d^2\right )+a b \left (26 c^3 d-218 c d^3\right )+b^2 \left (8 c^4-17 c^2 d^2+105 d^4\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))^{3/2}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\right )\right ) \cos (e+f x)}{105 d^2 \left (c^2-d^2\right )^4 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (16 a^3 c d^3 \left (11 c^2+13 d^2\right )-6 a b^2 c d \left (3 c^4-62 c^2 d^2-133 d^4\right )-9 a^2 b d^2 \left (5 c^4+102 c^2 d^2+21 d^4\right )-b^3 \left (8 c^6-23 c^4 d^2+294 c^2 d^4+105 d^6\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)}}{105 d^3 \left (c^2-d^2\right )^4 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 (b c-a d) \left (8 b^2 c^4+26 a b c^3 d+71 a^2 c^2 d^2-17 b^2 c^2 d^2-218 a b c d^3+25 a^2 d^4+105 b^2 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}}}{105 d^3 \left (c^2-d^2\right )^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.94125, size = 1127, normalized size = 1.57 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-\frac{2 \left (-b^3 \cos (e+f x) c^3+3 a b^2 d \cos (e+f x) c^2-3 a^2 b d^2 \cos (e+f x) c+a^3 d^3 \cos (e+f x)\right )}{7 d^2 \left (d^2-c^2\right ) (c+d \sin (e+f x))^4}-\frac{2 \left (8 b^3 \cos (e+f x) c^6+18 a b^2 d \cos (e+f x) c^5-23 b^3 d^2 \cos (e+f x) c^4+45 a^2 b d^2 \cos (e+f x) c^4-176 a^3 d^3 \cos (e+f x) c^3-372 a b^2 d^3 \cos (e+f x) c^3+294 b^3 d^4 \cos (e+f x) c^2+918 a^2 b d^4 \cos (e+f x) c^2-208 a^3 d^5 \cos (e+f x) c-798 a b^2 d^5 \cos (e+f x) c+105 b^3 d^6 \cos (e+f x)+189 a^2 b d^6 \cos (e+f x)\right )}{105 d^2 \left (d^2-c^2\right )^4 (c+d \sin (e+f x))}-\frac{2 \left (-8 b^3 \cos (e+f x) c^5-18 a b^2 d \cos (e+f x) c^4+17 b^3 d^2 \cos (e+f x) c^3-45 a^2 b d^2 \cos (e+f x) c^3+71 a^3 d^3 \cos (e+f x) c^2+201 a b^2 d^3 \cos (e+f x) c^2-105 b^3 d^4 \cos (e+f x) c-243 a^2 b d^4 \cos (e+f x) c+25 a^3 d^5 \cos (e+f x)+105 a b^2 d^5 \cos (e+f x)\right )}{105 d^2 \left (d^2-c^2\right )^3 (c+d \sin (e+f x))^2}-\frac{6 \left (-3 b^3 \cos (e+f x) c^4+2 a b^2 d \cos (e+f x) c^3+7 b^3 d^2 \cos (e+f x) c^2+5 a^2 b d^2 \cos (e+f x) c^2-4 a^3 d^3 \cos (e+f x) c-14 a b^2 d^3 \cos (e+f x) c+7 a^2 b d^4 \cos (e+f x)\right )}{35 d^2 \left (d^2-c^2\right )^2 (c+d \sin (e+f x))^3}\right )}{f}-\frac{-\frac{2 \left (-25 a^3 d^6-105 a b^2 d^6+210 b^3 c d^5+432 a^2 b c d^5-254 a^3 c^2 d^4-894 a b^2 c^2 d^4+172 b^3 c^3 d^3+720 a^2 b c^3 d^3-105 a^3 c^4 d^2-153 a b^2 c^4 d^2+2 b^3 c^5 d\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 )}{\sqrt{c+d \sin (e+f x)}}-\frac{\left (8 b^3 c^6+18 a b^2 d c^5-23 b^3 d^2 c^4+45 a^2 b d^2 c^4-176 a^3 d^3 c^3-372 a b^2 d^3 c^3+294 b^3 d^4 c^2+918 a^2 b d^4 c^2-208 a^3 d^5 c-798 a b^2 d^5 c+105 b^3 d^6+189 a^2 b d^6\right ) \left (\frac{2 (c+d) E\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}-\frac{2 c F\left (\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{\sqrt{c+d \sin (e+f x)}}\right )}{d}}{105 (c-d)^4 d^2 (c+d)^4 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 12.731, size = 2111, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}}{5 \, c d^{4} \cos \left (f x + e\right )^{4} + c^{5} + 10 \, c^{3} d^{2} + 5 \, c d^{4} - 10 \,{\left (c^{3} d^{2} + c d^{4}\right )} \cos \left (f x + e\right )^{2} +{\left (d^{5} \cos \left (f x + e\right )^{4} + 5 \, c^{4} d + 10 \, c^{2} d^{3} + d^{5} - 2 \,{\left (5 \, c^{2} d^{3} + d^{5}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]