\(\int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx\) [1140]

Optimal result

Integrand size = 29, antiderivative size = 332 \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}+\frac {8 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3465 b^5 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {8 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3465 b^5 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-9 b^2-7 a b \sin (c+d x)\right )}{693 b^2 d}+\frac {4 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-69 a^2 b^2+45 b^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{3465 b^4 d} \] Output:

-2/11*cos(d*x+c)^5*(a+b*sin(d*x+c))^(1/2)/d-8/3465*a*(32*a^4-93*a^2*b^2+93 
*b^4)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)*(b/(a+b))^(1/2))*(a+b*si 
n(d*x+c))^(1/2)/b^5/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-8/3465*(32*a^6-101*a^ 
4*b^2+114*a^2*b^4-45*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^5/d/(a+b*sin(d*x+c))^(1/2)- 
2/693*cos(d*x+c)^3*(a+b*sin(d*x+c))^(1/2)*(8*a^2-9*b^2-7*a*b*sin(d*x+c))/b 
^2/d+4/3465*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)*(32*a^4-69*a^2*b^2+45*b^4-24 
*a*b*(a^2-2*b^2)*sin(d*x+c))/b^4/d
 

Mathematica [A] (verified)

Time = 4.83 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.98 \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {-64 a \left (32 a^5+32 a^4 b-93 a^3 b^2-93 a^2 b^3+93 a b^4+93 b^5\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+64 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}+b \cos (c+d x) \left (1024 a^5-2912 a^3 b^2+748 a b^4+16 \left (4 a^3 b^2-183 a b^4\right ) \cos (2 (c+d x))-700 a b^4 \cos (4 (c+d x))+256 a^4 b \sin (c+d x)-692 a^2 b^3 \sin (c+d x)+990 b^5 \sin (c+d x)-20 a^2 b^3 \sin (3 (c+d x))-765 b^5 \sin (3 (c+d x))-315 b^5 \sin (5 (c+d x))\right )}{27720 b^5 d \sqrt {a+b \sin (c+d x)}} \] Input:

Integrate[Cos[c + d*x]^4*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(-64*a*(32*a^5 + 32*a^4*b - 93*a^3*b^2 - 93*a^2*b^3 + 93*a*b^4 + 93*b^5)*E 
llipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/( 
a + b)] + 64*(32*a^6 - 101*a^4*b^2 + 114*a^2*b^4 - 45*b^6)*EllipticF[(-2*c 
 + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Co 
s[c + d*x]*(1024*a^5 - 2912*a^3*b^2 + 748*a*b^4 + 16*(4*a^3*b^2 - 183*a*b^ 
4)*Cos[2*(c + d*x)] - 700*a*b^4*Cos[4*(c + d*x)] + 256*a^4*b*Sin[c + d*x] 
- 692*a^2*b^3*Sin[c + d*x] + 990*b^5*Sin[c + d*x] - 20*a^2*b^3*Sin[3*(c + 
d*x)] - 765*b^5*Sin[3*(c + d*x)] - 315*b^5*Sin[5*(c + d*x)]))/(27720*b^5*d 
*Sqrt[a + b*Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.05, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3341, 27, 3042, 3344, 27, 3042, 3344, 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.

\(\displaystyle \int \sin (c+d x) \cos ^4(c+d x) \sqrt {a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin (c+d x) \cos (c+d x)^4 \sqrt {a+b \sin (c+d x)}dx\)

\(\Big \downarrow \) 3341

\(\displaystyle \frac {2}{11} \int \frac {\cos ^4(c+d x) (b+a \sin (c+d x))}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \int \frac {\cos ^4(c+d x) (b+a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \int \frac {\cos (c+d x)^4 (b+a \sin (c+d x))}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3344

\(\displaystyle \frac {1}{11} \left (\frac {4 \int -\frac {\cos ^2(c+d x) \left (b \left (a^2-9 b^2\right )+8 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (-\frac {2 \int \frac {\cos ^2(c+d x) \left (b \left (a^2-9 b^2\right )+8 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (-\frac {2 \int \frac {\cos (c+d x)^2 \left (b \left (a^2-9 b^2\right )+8 a \left (a^2-2 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3344

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (\frac {4 \int -\frac {b \left (8 a^4-21 b^2 a^2+45 b^4\right )+a \left (32 a^4-93 b^2 a^2+93 b^4\right ) \sin (c+d x)}{2 \sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \int \frac {b \left (8 a^4-21 b^2 a^2+45 b^4\right )+a \left (32 a^4-93 b^2 a^2+93 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \int \frac {b \left (8 a^4-21 b^2 a^2+45 b^4\right )+a \left (32 a^4-93 b^2 a^2+93 b^4\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3231

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{11} \left (-\frac {2 \left (-\frac {2 \left (\frac {2 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{b \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}\right )}{21 b^2}-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {1}{11} \left (-\frac {2 \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (8 a^2-7 a b \sin (c+d x)-9 b^2\right )}{63 b^2 d}-\frac {2 \left (-\frac {2 \cos (c+d x) \sqrt {a+b \sin (c+d x)} \left (32 a^4-24 a b \left (a^2-2 b^2\right ) \sin (c+d x)-69 a^2 b^2+45 b^4\right )}{15 b^2 d}-\frac {2 \left (\frac {2 a \left (32 a^4-93 a^2 b^2+93 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (32 a^6-101 a^4 b^2+114 a^2 b^4-45 b^6\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \sin (c+d x)}}\right )}{15 b^2}\right )}{21 b^2}\right )-\frac {2 \cos ^5(c+d x) \sqrt {a+b \sin (c+d x)}}{11 d}\)

Input:

Int[Cos[c + d*x]^4*Sin[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(-2*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) + ((-2*Cos[c + d*x]^3* 
Sqrt[a + b*Sin[c + d*x]]*(8*a^2 - 9*b^2 - 7*a*b*Sin[c + d*x]))/(63*b^2*d) 
- (2*((-2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^4 - 69*a^2*b^2 + 45* 
b^4 - 24*a*b*(a^2 - 2*b^2)*Sin[c + d*x]))/(15*b^2*d) - (2*((2*a*(32*a^4 - 
93*a^2*b^2 + 93*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*(32*a^6 - 
101*a^4*b^2 + 114*a^2*b^4 - 45*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]])) 
)/(15*b^2)))/(21*b^2))/11
 

Defintions of rubi rules used

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 3341

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* 
(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S 
imp[1/(m + p + 1)   Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim 
p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] 
 &&  !LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S 
implerQ[c + d*x, a + b*x])
 

rule 3344

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x 
_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[g*(g* 
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c*(m + p + 1) - a*d* 
p + b*d*(m + p)*Sin[e + f*x])/(b^2*f*(m + p)*(m + p + 1))), x] + Simp[g^2*( 
(p - 1)/(b^2*(m + p)*(m + p + 1)))   Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si 
n[e + f*x])^m*Simp[b*(a*d*m + b*c*(m + p + 1)) + (a*b*c*(m + p + 1) - d*(a^ 
2*p - b^2*(m + p)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && NeQ[m + p + 1 
, 0] && IntegerQ[2*m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1355\) vs. \(2(313)=626\).

Time = 2.73 (sec) , antiderivative size = 1356, normalized size of antiderivative = 4.08

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

Input:

int(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

-2/3465*(180*a*b^6-500*((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+744*((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^3*b^4-372*((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*b^6-128*((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+96*((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+404*((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 
pticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b^3-288*((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^3*b^4-456*((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+192*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si 
n(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.72 \[ \int \cos ^4(c+d x) \sin (c+d x) \sqrt {a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2),x, algorithm="fri 
cas")
 

Output:

-2/10395*(4*(64*a^6 - 210*a^4*b^2 + 249*a^2*b^4 - 135*b^6)*sqrt(1/2*I*b)*w 
eierstrassPInverse(-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*(64*a^6 - 2 
10*a^4*b^2 + 249*a^2*b^4 - 135*b^6)*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*(32*I*a^5*b - 93*I*a^3*b^3 + 93 
*I*a*b^5)*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*(-32*I*a^5*b + 93*I*a^3*b^3 - 93*I*a*b^5)*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)) + 3*(315*b 
^6*cos(d*x + c)^5 + 5*(8*a^2*b^4 - 9*b^6)*cos(d*x + c)^3 - 2*(32*a^4*b^2 - 
 69*a^2*b^4 + 45*b^6)*cos(d*x + c) - (35*a*b^5*cos(d*x + c)^3 - 48*(a^3*b^ 
3 - 2*a*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt(b*sin(d*x + c) + a))/(b^6*d)
 

Sympy [F(-1)]

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

integrate(cos(d*x+c)**4*sin(d*x+c)*(a+b*sin(d*x+c))**(1/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2),x, algorithm="max 
ima")
 

Output:

integrate(sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^4*sin(d*x + c), x)
 

Giac [F]

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

integrate(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2),x, algorithm="gia 
c")
 

Output:

integrate(sqrt(b*sin(d*x + c) + a)*cos(d*x + c)^4*sin(d*x + c), x)
 

Mupad [F(-1)]

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

int(cos(c + d*x)^4*sin(c + d*x)*(a + b*sin(c + d*x))^(1/2),x)
 

Output:

int(cos(c + d*x)^4*sin(c + d*x)*(a + b*sin(c + d*x))^(1/2), x)
 

Reduce [F]

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

int(cos(d*x+c)^4*sin(d*x+c)*(a+b*sin(d*x+c))^(1/2),x)
 

Output:

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