\(\int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1172]

Optimal result

Integrand size = 31, antiderivative size = 466 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x) \sin ^4(c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {8 \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^6 d}+\frac {8 a \left (480 a^2-419 b^2\right ) \cos (c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)}}{1155 b^5 d}-\frac {20 \left (32 a^2-27 b^2\right ) \cos (c+d x) \sin ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{231 b^4 d}+\frac {2 \left (40 a^2-33 b^2\right ) \cos (c+d x) \sin ^3(c+d x) \sqrt {a+b \sin (c+d x)}}{33 a b^3 d}-\frac {2 \cos (c+d x) \sin ^4(c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}-\frac {8 a \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{1155 b^7 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {8 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{1155 b^7 d \sqrt {a+b \sin (c+d x)}} \] Output:

-2*(a^2-b^2)*cos(d*x+c)*sin(d*x+c)^4/a/b^2/d/(a+b*sin(d*x+c))^(1/2)-8/1155 
*(640*a^4-592*a^2*b^2+15*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^6/d+8/11 
55*a*(480*a^2-419*b^2)*cos(d*x+c)*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2)/b^5/d- 
20/231*(32*a^2-27*b^2)*cos(d*x+c)*sin(d*x+c)^2*(a+b*sin(d*x+c))^(1/2)/b^4/ 
d+2/33*(40*a^2-33*b^2)*cos(d*x+c)*sin(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)/a/b^ 
3/d-2/11*cos(d*x+c)*sin(d*x+c)^4*(a+b*sin(d*x+c))^(1/2)/b^2/d+8/1155*a*(12 
80*a^4-1344*a^2*b^2+123*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 
)+8/1155*(1280*a^6-1664*a^4*b^2+369*a^2*b^4+15*b^6)*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 [A] (verified)

Time = 8.23 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {64 a \left (1280 a^5+1280 a^4 b-1344 a^3 b^2-1344 a^2 b^3+123 a b^4+123 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}-64 \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (-40960 a^5+40448 a^3 b^2-2728 a b^4-16 \left (160 a^3 b^2-93 a b^4\right ) \cos (2 (c+d x))+280 a b^4 \cos (4 (c+d x))-10240 a^4 b \sin (c+d x)+8672 a^2 b^3 \sin (c+d x)+330 b^5 \sin (c+d x)+800 a^2 b^3 \sin (3 (c+d x))-255 b^5 \sin (3 (c+d x))-105 b^5 \sin (5 (c+d x))\right )}{9240 b^7 d \sqrt {a+b \sin (c+d x)}} \] Input:

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

Output:

(64*a*(1280*a^5 + 1280*a^4*b - 1344*a^3*b^2 - 1344*a^2*b^3 + 123*a*b^4 + 1 
23*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c 
+ d*x])/(a + b)] - 64*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*Ell 
ipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a 
+ b)] + b*Cos[c + d*x]*(-40960*a^5 + 40448*a^3*b^2 - 2728*a*b^4 - 16*(160* 
a^3*b^2 - 93*a*b^4)*Cos[2*(c + d*x)] + 280*a*b^4*Cos[4*(c + d*x)] - 10240* 
a^4*b*Sin[c + d*x] + 8672*a^2*b^3*Sin[c + d*x] + 330*b^5*Sin[c + d*x] + 80 
0*a^2*b^3*Sin[3*(c + d*x)] - 255*b^5*Sin[3*(c + d*x)] - 105*b^5*Sin[5*(c + 
 d*x)]))/(9240*b^7*d*Sqrt[a + b*Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 3.20 (sec) , antiderivative size = 504, 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, 3371, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3371

\(\displaystyle \frac {4 \int \frac {\sin ^3(c+d x) \left (96 a^2-2 b \sin (c+d x) a-77 b^2-3 \left (40 a^2-33 b^2\right ) \sin ^2(c+d x)\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sin ^3(c+d x) \left (96 a^2-2 b \sin (c+d x) a-77 b^2-3 \left (40 a^2-33 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin (c+d x)^3 \left (96 a^2-2 b \sin (c+d x) a-77 b^2-3 \left (40 a^2-33 b^2\right ) \sin (c+d x)^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {2 \int -\frac {3 \sin ^2(c+d x) \left (-4 b \sin (c+d x) a^2-5 \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) a+3 \left (40 a^2-33 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{9 b}+\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \int \frac {\sin ^2(c+d x) \left (-4 b \sin (c+d x) a^2-5 \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) a+3 \left (40 a^2-33 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \int \frac {\sin (c+d x)^2 \left (-4 b \sin (c+d x) a^2-5 \left (32 a^2-27 b^2\right ) \sin (c+d x)^2 a+3 \left (40 a^2-33 b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {2 \int -\frac {\sin (c+d x) \left (-\left (\left (480 a^2-419 b^2\right ) \sin ^2(c+d x) a^2\right )+10 \left (32 a^2-27 b^2\right ) a^2-b \left (20 a^2-9 b^2\right ) \sin (c+d x) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}+\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \int \frac {\sin (c+d x) \left (-\left (\left (480 a^2-419 b^2\right ) \sin ^2(c+d x) a^2\right )+10 \left (32 a^2-27 b^2\right ) a^2-b \left (20 a^2-9 b^2\right ) \sin (c+d x) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \int \frac {\sin (c+d x) \left (-\left (\left (480 a^2-419 b^2\right ) \sin (c+d x)^2 a^2\right )+10 \left (32 a^2-27 b^2\right ) a^2-b \left (20 a^2-9 b^2\right ) \sin (c+d x) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 \int -\frac {2 \left (480 a^2-419 b^2\right ) a^3-b \left (160 a^2-93 b^2\right ) \sin (c+d x) a^2-3 \left (640 a^4-592 b^2 a^2+15 b^4\right ) \sin ^2(c+d x) a}{2 \sqrt {a+b \sin (c+d x)}}dx}{5 b}+\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {2 \left (480 a^2-419 b^2\right ) a^3-b \left (160 a^2-93 b^2\right ) \sin (c+d x) a^2-3 \left (640 a^4-592 b^2 a^2+15 b^4\right ) \sin ^2(c+d x) a}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\int \frac {2 \left (480 a^2-419 b^2\right ) a^3-b \left (160 a^2-93 b^2\right ) \sin (c+d x) a^2-3 \left (640 a^4-592 b^2 a^2+15 b^4\right ) \sin (c+d x)^2 a}{\sqrt {a+b \sin (c+d x)}}dx}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 \int \frac {3 \left (\left (1280 a^4-1344 b^2 a^2+123 b^4\right ) \sin (c+d x) a^2+b \left (320 a^4-246 b^2 a^2-15 b^4\right ) a\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{3 b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {\left (1280 a^4-1344 b^2 a^2+123 b^4\right ) \sin (c+d x) a^2+b \left (320 a^4-246 b^2 a^2-15 b^4\right ) a}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\int \frac {\left (1280 a^4-1344 b^2 a^2+123 b^4\right ) \sin (c+d x) a^2+b \left (320 a^4-246 b^2 a^2-15 b^4\right ) a}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {2 \left (a^2-b^2\right ) \sin ^4(c+d x) \cos (c+d x)}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {\frac {2 \left (40 a^2-33 b^2\right ) \sin ^3(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 b d}-\frac {2 \left (\frac {10 a \left (32 a^2-27 b^2\right ) \sin ^2(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{7 b d}-\frac {2 \left (\frac {2 a^2 \left (480 a^2-419 b^2\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{5 b d}-\frac {\frac {2 a \left (640 a^4-592 a^2 b^2+15 b^4\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{b d}+\frac {\frac {2 a^2 \left (1280 a^4-1344 a^2 b^2+123 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 a \left (1280 a^6-1664 a^4 b^2+369 a^2 b^4+15 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}}{b}}{5 b}\right )}{7 b}\right )}{3 b}}{11 a b^2}-\frac {2 \sin ^4(c+d x) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{11 b^2 d}\)

Input:

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

Output:

(-2*(a^2 - b^2)*Cos[c + d*x]*Sin[c + d*x]^4)/(a*b^2*d*Sqrt[a + b*Sin[c + d 
*x]]) - (2*Cos[c + d*x]*Sin[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]])/(11*b^2*d 
) + ((2*(40*a^2 - 33*b^2)*Cos[c + d*x]*Sin[c + d*x]^3*Sqrt[a + b*Sin[c + d 
*x]])/(3*b*d) - (2*((10*a*(32*a^2 - 27*b^2)*Cos[c + d*x]*Sin[c + d*x]^2*Sq 
rt[a + b*Sin[c + d*x]])/(7*b*d) - (2*((2*a^2*(480*a^2 - 419*b^2)*Cos[c + d 
*x]*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(5*b*d) - ((2*a*(640*a^4 - 592* 
a^2*b^2 + 15*b^4)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(b*d) + ((2*a^2*( 
1280*a^4 - 1344*a^2*b^2 + 123*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*(1280*a^6 - 1664*a^4*b^2 + 369*a^2*b^4 + 15*b^6)*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))/(11*a*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 3371

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[Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 2)*((d*Sin[e + f* 
x])^(n + 1)/(b^2*d*f*(m + n + 4))), x] - Simp[1/(a*b^2*(m + 1)*(m + n + 4)) 
   Int[(a + b*Sin[e + f*x])^(m + 1)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n 
+ 3) - b^2*(m + n + 2)*(m + n + 4) + a*b*(m + 1)*Sin[e + f*x] - (a^2*(n + 2 
)*(n + 3) - b^2*(m + n + 3)*(m + n + 4))*Sin[e + f*x]^2, x], x], x]) /; Fre 
eQ[{a, b, d, e, f, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && Lt 
Q[m, -1] &&  !LtQ[n, -1] && NeQ[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. \(1355\) vs. \(2(437)=874\).

Time = 4.50 (sec) , antiderivative size = 1356, normalized size of antiderivative = 2.91

method	result	size
default	\(\text {Expression too large to display}\)	\(1356\)

Input:

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

Output:

-2/1155*(60*a*b^6+10496*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/ 
(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/( 
a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2-5868*((a+b*sin(d*x+c))/(a-b))^(1/ 
2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Ellipti 
cE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^4+492*((a+b*s 
in(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c)) 
/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2) 
)*a*b^6+5120*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2 
)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2) 
,((a-b)/(a+b))^(1/2))*a^6*b-3840*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x 
+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin( 
d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^5*b^2-6656*((a+b*sin(d*x+c))/( 
a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2 
)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3+43 
92*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+s 
in(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a 
+b))^(1/2))*a^3*b^4+1476*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b 
/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/ 
(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^5-552*((a+b*sin(d*x+c))/(a-b))^(1/ 
2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*Elli...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 756, normalized size of antiderivative = 1.62 \[ \int \frac {\cos ^4(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

2/3465*(4*(2560*a^7 - 3648*a^5*b^2 + 984*a^3*b^4 + 45*a*b^6 + (2560*a^6*b 
- 3648*a^4*b^3 + 984*a^2*b^5 + 45*b^7)*sin(d*x + c))*sqrt(1/2*I*b)*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) + 4*(2560*a^7 - 3648* 
a^5*b^2 + 984*a^3*b^4 + 45*a*b^6 + (2560*a^6*b - 3648*a^4*b^3 + 984*a^2*b^ 
5 + 45*b^7)*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 + 1344*I*a^4*b^3 - 123*I*a 
^2*b^5 + (-1280*I*a^5*b^2 + 1344*I*a^3*b^4 - 123*I*a*b^6)*sin(d*x + c))*sq 
rt(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 - 1344*I*a^4*b^3 + 123*I*a^2*b^5 + (1280*I*a^5*b^2 - 13 
44*I*a^3*b^4 + 123*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, weierstrassPInv 
erse(-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*(140*a*b^6*cos(d*x + c) 
^5 - 2*(160*a^3*b^4 - 23*a*b^6)*cos(d*x + c)^3 - 2*(1280*a^5*b^2 - 1344*a^ 
3*b^4 + 123*a*b^6)*cos(d*x + c) - (105*b^7*cos(d*x + c)^5 - 5*(40*a^2*b^5 
+ 3*b^7)*cos(d*x + c)^3 + 2*(320*a^4*b^3 - 246*a^2*b^5 - 15*b^7)*cos(d*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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