\(\int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) [1153]

Optimal result

Integrand size = 29, antiderivative size = 408 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{32 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}+\frac {b \left (236 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{64 a^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (20 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 a d \sqrt {a+b \sin (c+d x)}}+\frac {3 \left (16 a^4-24 a^2 b^2+b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 a^2 d \sqrt {a+b \sin (c+d x)}} \] Output:

1/64*b*(68*a^2-3*b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/32*(20*a^2 
-b^2)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/8*b*cot(d*x+c)* 
csc(d*x+c)^2*(a+b*sin(d*x+c))^(5/2)/a^2/d-1/4*cot(d*x+c)*csc(d*x+c)^3*(a+b 
*sin(d*x+c))^(5/2)/a/d-1/64*b*(236*a^2+3*b^2)*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)/a^2/d/((a+b*sin(d* 
x+c))/(a+b))^(1/2)-1/64*b*(20*a^2+b^2)*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)/a/d/(a+b*sin(d*x 
+c))^(1/2)-3/64*(16*a^4-24*a^2*b^2+b^4)*EllipticPi(cos(1/2*c+1/4*Pi+1/2*d* 
x),2,2^(1/2)*(b/(a+b))^(1/2))*((a+b*sin(d*x+c))/(a+b))^(1/2)/a^2/d/(a+b*si 
n(d*x+c))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 7.89 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.57 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\left (\frac {3 \left (36 a^2 b \cos (c+d x)+b^3 \cos (c+d x)\right ) \csc (c+d x)}{64 a^2}+\frac {\left (20 a^2 \cos (c+d x)-b^2 \cos (c+d x)\right ) \csc ^2(c+d x)}{32 a}-\frac {3}{8} b \cot (c+d x) \csc ^2(c+d x)-\frac {1}{4} a \cot (c+d x) \csc ^3(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {-\frac {2 \left (432 a^3 b+4 a b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (96 a^4+92 a^2 b^2+9 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-236 a^2 b^2-3 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{256 a^2 d} \] Input:

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

Output:

(((3*(36*a^2*b*Cos[c + d*x] + b^3*Cos[c + d*x])*Csc[c + d*x])/(64*a^2) + ( 
(20*a^2*Cos[c + d*x] - b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(32*a) - (3*b*Cot 
[c + d*x]*Csc[c + d*x]^2)/8 - (a*Cot[c + d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + 
b*Sin[c + d*x]])/d + ((-2*(432*a^3*b + 4*a*b^3)*EllipticF[(-c + Pi/2 - d*x 
)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + 
 d*x]] - (2*(96*a^4 + 92*a^2*b^2 + 9*b^4)*EllipticPi[2, (-c + Pi/2 - d*x)/ 
2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d 
*x]] - ((2*I)*(-236*a^2*b^2 - 3*b^4)*Cos[c + d*x]*Cos[2*(c + d*x)]*(2*a*(a 
 - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], ( 
a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + 
b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*ArcSinh[Sqr 
t[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sqrt[(b - b 
*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a*Sqrt[-(a 
 + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*Sin[c + d* 
x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Sin[c + d*x 
]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(256*a^2*d)
 

Rubi [A] (verified)

Time = 3.35 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.01, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.828, Rules used = {3042, 3372, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3372

\(\displaystyle -\frac {\int \frac {3}{4} \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (20 a^2+2 b \sin (c+d x) a-b^2-\left (16 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^{3/2} \left (20 a^2+2 b \sin (c+d x) a-b^2-\left (16 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{3/2} \left (20 a^2+2 b \sin (c+d x) a-b^2-\left (16 a^2+b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\frac {1}{2} \int \frac {1}{2} \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 b \left (28 a^2+b^2\right ) \sin ^2(c+d x)-2 a \left (12 a^2-b^2\right ) \sin (c+d x)+b \left (68 a^2-3 b^2\right )\right )dx-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{4} \int \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 b \left (28 a^2+b^2\right ) \sin ^2(c+d x)-2 a \left (12 a^2-b^2\right ) \sin (c+d x)+b \left (68 a^2-3 b^2\right )\right )dx-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \int \frac {\sqrt {a+b \sin (c+d x)} \left (-3 b \left (28 a^2+b^2\right ) \sin (c+d x)^2-2 a \left (12 a^2-b^2\right ) \sin (c+d x)+b \left (68 a^2-3 b^2\right )\right )}{\sin (c+d x)^2}dx-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\frac {1}{4} \left (\int -\frac {\csc (c+d x) \left (b^2 \left (236 a^2+3 b^2\right ) \sin ^2(c+d x)+2 a b \left (108 a^2+b^2\right ) \sin (c+d x)+3 \left (16 a^4-24 b^2 a^2+b^4\right )\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {1}{2} \int \frac {\csc (c+d x) \left (b^2 \left (236 a^2+3 b^2\right ) \sin ^2(c+d x)+2 a b \left (108 a^2+b^2\right ) \sin (c+d x)+3 \left (16 a^4-24 b^2 a^2+b^4\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {1}{2} \int \frac {b^2 \left (236 a^2+3 b^2\right ) \sin (c+d x)^2+2 a b \left (108 a^2+b^2\right ) \sin (c+d x)+3 \left (16 a^4-24 b^2 a^2+b^4\right )}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (\frac {\int -\frac {\csc (c+d x) \left (3 b \left (16 a^4-24 b^2 a^2+b^4\right )-a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-b \left (236 a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-b \left (236 a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int \frac {\csc (c+d x) \left (3 b \left (16 a^4-24 b^2 a^2+b^4\right )-a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-b \left (236 a^2+3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int \frac {3 b \left (16 a^4-24 b^2 a^2+b^4\right )-a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\int \frac {3 b \left (16 a^4-24 b^2 a^2+b^4\right )-a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\int \frac {3 b \left (16 a^4-24 b^2 a^2+b^4\right )-a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\int \frac {3 b \left (16 a^4-24 b^2 a^2+b^4\right )-a b^2 \left (20 a^2+b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a b^2 \left (20 a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b^2 \left (20 a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b^2 \left (20 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b^2 \left (20 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a b^2 \left (20 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (20 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {\frac {3 b \left (16 a^4-24 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (20 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{8 a^2 d}-\frac {\frac {1}{4} \left (\frac {1}{2} \left (-\frac {2 b \left (236 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\frac {6 b \left (16 a^4-24 a^2 b^2+b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (20 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )-\frac {b \left (68 a^2-3 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (20 a^2-b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 d}}{16 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2}}{4 a d}\)

Input:

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

Output:

(b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(8*a^2*d) - (Co 
t[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2))/(4*a*d) - (-1/2*((20 
*a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/d + (-(( 
b*(68*a^2 - 3*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d) + ((-2*b*(236 
*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[ 
c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - ((-2*a*b^2*(20*a^2 + b 
^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x]) 
/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]) + (6*b*(16*a^4 - 24*a^2*b^2 + b^4) 
*EllipticPi[2, (c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x] 
)/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]))/b)/2)/4)/(16*a^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 3284

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

rule 3286

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

rule 3372

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

rule 3481

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

rule 3526

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^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d 
, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
 

rule 3538

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1758\) vs. \(2(382)=764\).

Time = 2.42 (sec) , antiderivative size = 1759, normalized size of antiderivative = 4.31

method	result	size
default	\(\text {Expression too large to display}\)	\(1759\)

Input:

int(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(3/2),x,method=_RETURNVERBOSE 
)
 

Output:

1/64*(-16*a^5+20*((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*sin(d*x+c)^4-234*((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^3*b^2*sin(d*x+ 
c)^4+((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^3*sin(d*x+c)^4-3*((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*b^4*sin(d*x+c)^4+48*((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)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a 
+b))^(1/2))*a^4*b*sin(d*x+c)^4+72*((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)*EllipticPi(((a+b*si 
n(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^4-7 
2*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+si 
n(d*x+c))/(a-b))^(1/2)*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,( 
(a-b)/(a+b))^(1/2))*a^2*b^3*sin(d*x+c)^4-3*((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)*EllipticPi 
(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a*b^4*sin(...
 

Fricas [F(-1)]

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

integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="fri 
cas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F(-1)]

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

integrate(cot(d*x+c)^4*csc(d*x+c)*(a+b*sin(d*x+c))^(3/2),x, algorithm="gia 
c")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int \frac {{\left ({\sin \left (c+d\,x\right )}^2-1\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^5} \,d x \] Input:

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

Output:

int(((sin(c + d*x)^2 - 1)^2*(a + b*sin(c + d*x))^(3/2))/sin(c + d*x)^5, x)
 

Reduce [F]

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

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

Output:

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