\(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [1179]

Optimal result

Integrand size = 31, antiderivative size = 469 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (13 a^2-5 b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {128 a \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^6 d}-\frac {8 \left (480 a^2-203 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{315 b^5 d}+\frac {4 \left (160 a^2-63 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{63 a b^4 d}-\frac {10 \left (8 a^2-3 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{9 a^2 b^3 d}+\frac {8 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{315 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 a \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{315 b^7 d \sqrt {a+b \sin (c+d x)}} \] Output:

-2/3*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))^(3/2)+2/3* 
(13*a^2-5*b^2)*cos(d*x+c)*sin(d*x+c)^4/a^2/b^2/d/(a+b*sin(d*x+c))^(1/2)+12 
8/315*a*(40*a^2-19*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^6/d-8/315*(480 
*a^2-203*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d+4/63*(160 
*a^2-63*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/a/b^4/d-10/9*( 
8*a^2-3*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/a^2/b^3/d-8/31 
5*(1280*a^4-768*a^2*b^2+21*b^4)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2 
)*(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/b^7/d/((a+b*sin(d*x+c))/(a+b))^( 
1/2)-8/315*a*(1280*a^4-1088*a^2*b^2+123*b^4)*InverseJacobiAM(1/2*c-1/4*Pi+ 
1/2*d*x,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/b^7/d/(a+b 
*sin(d*x+c))^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1044\) vs. \(2(469)=938\).

Time = 11.44 (sec) , antiderivative size = 1044, normalized size of antiderivative = 2.23 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx =\text {Too large to display} \] Input:

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]
 

Output:

(315*((((a^2 + 3*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a* 
(-a + b)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + 
d*x])/(a + b))^(3/2))/((a - b)^2*b) - (Cos[c + d*x]*(2*a*(a^2 + b^2) + b*( 
a^2 + 3*b^2)*Sin[c + d*x]))/(a^2 - b^2)^2) + (315*((((32*a^4 - 57*a^2*b^2 
+ 21*b^4)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(-32*a^3 + 3 
2*a^2*b + 33*a*b^2 - 33*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b 
)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/(a - b)^2 - (b*(4*a*(8*a^4 - 13* 
a^2*b^2 + 3*b^4)*Cos[c + d*x] + b*(20*a^4 - 33*a^2*b^2 + 9*b^4)*Sin[2*(c + 
 d*x)]))/(2*(a^2 - b^2)^2)))/b^3 - (21*((((-2048*a^6 + 4192*a^4*b^2 - 2355 
*a^2*b^4 + 231*b^6)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + a*(2 
048*a^5 - 2048*a^4*b - 2656*a^3*b^2 + 2656*a^2*b^3 + 603*a*b^4 - 603*b^5)* 
EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a 
+ b))^(3/2))/(a - b)^2 + (b*Cos[c + d*x]*(-64*a*b^2*(a^2 - b^2)^2*Cos[2*(c 
 + d*x)] + b*(1280*a^6 - 2536*a^4*b^2 + 1347*a^2*b^4 - 111*b^6)*Sin[c + d* 
x] + 2*(512*a^7 - 952*a^5*b^2 + 423*a^3*b^4 + 7*a*b^6 + 6*b^3*(a^2 - b^2)^ 
2*Sin[3*(c + d*x)])))/(a^2 - b^2)^2))/b^5 - (5*(a + b*Sin[c + d*x])*(((-4* 
b*(-4096*a^7*b + 8960*a^5*b^3 - 5884*a^3*b^5 + 1041*a*b^7)*EllipticF[(-2*c 
 + Pi - 2*d*x)/4, (2*b)/(a + b)] + (65536*a^8 - 161792*a^6*b^2 + 129664*a^ 
4*b^4 - 35109*a^2*b^6 + 1617*b^8)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4 
, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*...
 

Rubi [A] (verified)

Time = 3.14 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.08, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.774, Rules used = {3042, 3370, 27, 3042, 3528, 27, 3042, 3528, 25, 3042, 3528, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^4}{(a+b \sin (c+d x))^{5/2}}dx\)

\(\Big \downarrow \) 3370

\(\displaystyle -\frac {4 \int \frac {\sin ^3(c+d x) \left (96 a^2-2 b \sin (c+d x) a-35 b^2-15 \left (8 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sin ^3(c+d x) \left (96 a^2-2 b \sin (c+d x) a-35 b^2-15 \left (8 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sin (c+d x)^3 \left (96 a^2-2 b \sin (c+d x) a-35 b^2-15 \left (8 a^2-3 b^2\right ) \sin (c+d x)^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {2 \int -\frac {3 \sin ^2(c+d x) \left (-4 b \sin (c+d x) a^2-\left (160 a^2-63 b^2\right ) \sin ^2(c+d x) a+15 \left (8 a^2-3 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{9 b}+\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \int \frac {\sin ^2(c+d x) \left (-4 b \sin (c+d x) a^2-\left (160 a^2-63 b^2\right ) \sin ^2(c+d x) a+15 \left (8 a^2-3 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \int \frac {\sin (c+d x)^2 \left (-4 b \sin (c+d x) a^2-\left (160 a^2-63 b^2\right ) \sin (c+d x)^2 a+15 \left (8 a^2-3 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 \int -\frac {\sin (c+d x) \left (-20 b \sin (c+d x) a^3-\left (480 a^2-203 b^2\right ) \sin ^2(c+d x) a^2+2 \left (160 a^2-63 b^2\right ) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}+\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \int \frac {\sin (c+d x) \left (-20 b \sin (c+d x) a^3-\left (480 a^2-203 b^2\right ) \sin ^2(c+d x) a^2+2 \left (160 a^2-63 b^2\right ) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \int \frac {\sin (c+d x) \left (-20 b \sin (c+d x) a^3-\left (480 a^2-203 b^2\right ) \sin (c+d x)^2 a^2+2 \left (160 a^2-63 b^2\right ) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 \int -\frac {-48 \left (40 a^2-19 b^2\right ) \sin ^2(c+d x) a^3+2 \left (480 a^2-203 b^2\right ) a^3-b \left (160 a^2-21 b^2\right ) \sin (c+d x) a^2}{2 \sqrt {a+b \sin (c+d x)}}dx}{5 b}+\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {-48 \left (40 a^2-19 b^2\right ) \sin ^2(c+d x) a^3+2 \left (480 a^2-203 b^2\right ) a^3-b \left (160 a^2-21 b^2\right ) \sin (c+d x) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {-48 \left (40 a^2-19 b^2\right ) \sin (c+d x)^2 a^3+2 \left (480 a^2-203 b^2\right ) a^3-b \left (160 a^2-21 b^2\right ) \sin (c+d x) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3502

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 \int \frac {3 \left (2 b \left (160 a^2-51 b^2\right ) a^3+\left (1280 a^4-768 b^2 a^2+21 b^4\right ) \sin (c+d x) a^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {2 b \left (160 a^2-51 b^2\right ) a^3+\left (1280 a^4-768 b^2 a^2+21 b^4\right ) \sin (c+d x) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {2 b \left (160 a^2-51 b^2\right ) a^3+\left (1280 a^4-768 b^2 a^2+21 b^4\right ) \sin (c+d x) a^2}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3231

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}+\frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {2 \left (13 a^2-5 b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}-\frac {\frac {10 \left (8 a^2-3 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 a \left (160 a^2-63 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-203 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {32 a^3 \left (40 a^2-19 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}+\frac {\frac {2 a^2 \left (1280 a^4-768 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a^3 \left (1280 a^4-1088 a^2 b^2+123 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}}{b}}{5 b}\right )}{7 b}\right )}{3 b}}{3 a^2 b^2}\)

Input:

Int[(Cos[c + d*x]^4*Sin[c + d*x]^3)/(a + b*Sin[c + d*x])^(5/2),x]
 

Output:

(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a*b^2*d*(a + b*Sin[c + d*x 
])^(3/2)) + (2*(13*a^2 - 5*b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(3*a^2*b^2*d* 
Sqrt[a + b*Sin[c + d*x]]) - ((10*(8*a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x] 
^3*Sqrt[a + b*Sin[c + d*x]])/(3*b*d) - (2*((2*a*(160*a^2 - 63*b^2)*Cos[c + 
 d*x]*Sin[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(7*b*d) - (2*((2*a^2*(480*a 
^2 - 203*b^2)*Cos[c + d*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(5*b*d) 
- ((32*a^3*(40*a^2 - 19*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(b*d) 
+ ((2*a^2*(1280*a^4 - 768*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, 
(2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a 
 + b)]) - (2*a^3*(1280*a^4 - 1088*a^2*b^2 + 123*b^4)*EllipticF[(c - Pi/2 + 
 d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b*d*Sqrt[a + 
b*Sin[c + d*x]]))/b)/(5*b)))/(7*b)))/(3*b))/(3*a^2*b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 3231

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( 
f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b   Int[1/Sqrt[a + b*Sin[e + f*x 
]], x], x] + Simp[d/b   Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b 
, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 3370

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + 
f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 
 1))), x] + (Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*S 
in[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2) 
)), x] - Simp[1/(a^2*b^2*(m + 1)*(m + 2))   Int[(a + b*Sin[e + f*x])^(m + 2 
)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) 
 + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n 
 + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a 
^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m 
, -2] || EqQ[m + n + 4, 0])
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2032\) vs. \(2(438)=876\).

Time = 4.15 (sec) , antiderivative size = 2033, normalized size of antiderivative = 4.33

method	result	size
default	\(\text {Expression too large to display}\)	\(2033\)

Input:

int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBO 
SE)
 

Output:

2/315*(-35*b^7*cos(d*x+c)^6*sin(d*x+c)+60*a*b^6*cos(d*x+c)^6+(120*a^2*b^5- 
7*b^7)*cos(d*x+c)^4*sin(d*x+c)+(-320*a^3*b^4+102*a*b^6)*cos(d*x+c)^4+(3200 
*a^4*b^3-1740*a^2*b^5+42*b^7)*cos(d*x+c)^2*sin(d*x+c)+(2560*a^5*b^2-896*a^ 
3*b^4-162*a*b^6)*cos(d*x+c)^2+4*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(b/(a- 
b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*b*(1280*E 
llipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b-960 
*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^2 
-1088*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^ 
3*b^3+666*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2) 
)*a^2*b^4+123*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^( 
1/2))*a*b^5-21*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^ 
(1/2))*b^6-1280*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b)) 
^(1/2))*a^6+2048*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b) 
)^(1/2))*a^4*b^2-789*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/( 
a+b))^(1/2))*a^2*b^4+21*EllipticE((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b 
)/(a+b))^(1/2))*b^6)*sin(d*x+c)+5120*(b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(- 
b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*Elli 
pticF((b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*b-3840*( 
b/(a-b)*sin(d*x+c)+a/(a-b))^(1/2)*(-b/(a+b)*sin(d*x+c)+b/(a+b))^(1/2)*(-b/ 
(a-b)*sin(d*x+c)-b/(a-b))^(1/2)*EllipticF((b/(a-b)*sin(d*x+c)+a/(a-b))^...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 911, normalized size of antiderivative = 1.94 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="f 
ricas")
 

Output:

2/945*(16*(640*a^7 + 16*a^5*b^2 - 537*a^3*b^4 + 87*a*b^6 - (640*a^5*b^2 - 
624*a^3*b^4 + 87*a*b^6)*cos(d*x + c)^2 + 2*(640*a^6*b - 624*a^4*b^3 + 87*a 
^2*b^5)*sin(d*x + c))*sqrt(1/2*I*b)*weierstrassPInverse(-4/3*(4*a^2 - 3*b^ 
2)/b^2, -8/27*(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin 
(d*x + c) - 2*I*a)/b) + 16*(640*a^7 + 16*a^5*b^2 - 537*a^3*b^4 + 87*a*b^6 
- (640*a^5*b^2 - 624*a^3*b^4 + 87*a*b^6)*cos(d*x + c)^2 + 2*(640*a^6*b - 6 
24*a^4*b^3 + 87*a^2*b^5)*sin(d*x + c))*sqrt(-1/2*I*b)*weierstrassPInverse( 
-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d 
*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b) + 12*(1280*I*a^6*b + 512*I*a^4*b^ 
3 - 747*I*a^2*b^5 + 21*I*b^7 + (-1280*I*a^4*b^3 + 768*I*a^2*b^5 - 21*I*b^7 
)*cos(d*x + c)^2 + 2*(1280*I*a^5*b^2 - 768*I*a^3*b^4 + 21*I*a*b^6)*sin(d*x 
 + c))*sqrt(1/2*I*b)*weierstrassZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*I* 
a^3 - 9*I*a*b^2)/b^3, weierstrassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27* 
(8*I*a^3 - 9*I*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) - 2* 
I*a)/b)) + 12*(-1280*I*a^6*b - 512*I*a^4*b^3 + 747*I*a^2*b^5 - 21*I*b^7 + 
(1280*I*a^4*b^3 - 768*I*a^2*b^5 + 21*I*b^7)*cos(d*x + c)^2 + 2*(-1280*I*a^ 
5*b^2 + 768*I*a^3*b^4 - 21*I*a*b^6)*sin(d*x + c))*sqrt(-1/2*I*b)*weierstra 
ssZeta(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, weierst 
rassPInverse(-4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(-8*I*a^3 + 9*I*a*b^2)/b^3, 1 
/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*I*a)/b)) - 3*(60*a*b^6*co...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**4*sin(d*x+c)**3/(a+b*sin(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="m 
axima")
 

Output:

integrate(cos(d*x + c)^4*sin(d*x + c)^3/(b*sin(d*x + c) + a)^(5/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x, algorithm="g 
iac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \] Input:

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(5/2),x)
 

Output:

int((cos(c + d*x)^4*sin(c + d*x)^3)/(a + b*sin(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{3}}{\sin \left (d x +c \right )^{3} b^{3}+3 \sin \left (d x +c \right )^{2} a \,b^{2}+3 \sin \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:

int(cos(d*x+c)^4*sin(d*x+c)^3/(a+b*sin(d*x+c))^(5/2),x)
 

Output:

int((sqrt(sin(c + d*x)*b + a)*cos(c + d*x)**4*sin(c + d*x)**3)/(sin(c + d* 
x)**3*b**3 + 3*sin(c + d*x)**2*a*b**2 + 3*sin(c + d*x)*a**2*b + a**3),x)