\(\int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1178]

Optimal result

Integrand size = 23, antiderivative size = 416 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}+\frac {5 \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^4 d}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^3 b d}+\frac {5 \left (16 a^2-21 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{24 a^4 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (32 a^2-35 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{24 a^3 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (36 a^2-35 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^4 d \sqrt {a+b \sin (c+d x)}} \] Output:

1/3*(6*a^2-7*b^2)*cot(d*x+c)*csc(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(1/2)-1/3 
*cot(d*x+c)*csc(d*x+c)^2/a/d/(a+b*sin(d*x+c))^(1/2)+5/24*(16*a^2-21*b^2)*c 
ot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^4/d-1/12*(24*a^2-35*b^2)*cot(d*x+c)*csc 
(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/b/d-5/24*(16*a^2-21*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^4/ 
d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/24*(32*a^2-35*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 
^3/d/(a+b*sin(d*x+c))^(1/2)-1/8*b*(36*a^2-35*b^2)*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^4 
/d/(a+b*sin(d*x+c))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.96 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.12 \[ \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {-\frac {4 \left (\left (-80 a^2 b+105 b^3\right ) \cos (c+d x)+a \cot (c+d x) \left (-32 a^2+35 b^2-14 a b \csc (c+d x)+8 a^2 \csc ^2(c+d x)\right )\right )}{a^4 \sqrt {a+b \sin (c+d x)}}+\frac {\frac {10 i \left (-16 a^2+21 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-\frac {8 a \left (24 a^2-35 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (-296 a^2+315 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^4}}{96 d} \] Input:

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

Output:

((-4*((-80*a^2*b + 105*b^3)*Cos[c + d*x] + a*Cot[c + d*x]*(-32*a^2 + 35*b^ 
2 - 14*a*b*Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2)))/(a^4*Sqrt[a + b*Sin[c + 
d*x]]) + (((10*I)*(-16*a^2 + 21*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqr 
t[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*Ell 
ipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a 
 - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b* 
Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d* 
x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b) 
^(-1)]) - (8*a*(24*a^2 - 35*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a 
 + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*b 
*(-296*a^2 + 315*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]* 
Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]])/a^4)/(96*d)
 

Rubi [A] (verified)

Time = 3.37 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.05, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.043, Rules used = {3042, 3203, 27, 3042, 3534, 27, 3042, 3534, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a+b \sin (c+d x))^{3/2}}dx\)

\(\Big \downarrow \) 3203

\(\displaystyle \frac {2 \int \frac {\csc ^3(c+d x) \left (24 a^2-2 b \sin (c+d x) a-35 b^2-3 \left (4 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{3 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (24 a^2-2 b \sin (c+d x) a-35 b^2-3 \left (4 a^2-7 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {24 a^2-2 b \sin (c+d x) a-35 b^2-3 \left (4 a^2-7 b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^3 \sqrt {a+b \sin (c+d x)}}dx}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\int -\frac {\csc ^2(c+d x) \left (-14 a \sin (c+d x) b^2-\left (24 a^2-35 b^2\right ) \sin ^2(c+d x) b+5 \left (16 a^2-21 b^2\right ) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-14 a \sin (c+d x) b^2-\left (24 a^2-35 b^2\right ) \sin ^2(c+d x) b+5 \left (16 a^2-21 b^2\right ) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {-14 a \sin (c+d x) b^2-\left (24 a^2-35 b^2\right ) \sin (c+d x)^2 b+5 \left (16 a^2-21 b^2\right ) b}{\sin (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3534

\(\displaystyle \frac {-\frac {\frac {\int -\frac {\csc (c+d x) \left (5 \left (16 a^2-21 b^2\right ) \sin ^2(c+d x) b^2+3 \left (36 a^2-35 b^2\right ) b^2+2 a \left (24 a^2-35 b^2\right ) \sin (c+d x) b\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {-\frac {\int \frac {\csc (c+d x) \left (5 \left (16 a^2-21 b^2\right ) \sin ^2(c+d x) b^2+3 \left (36 a^2-35 b^2\right ) b^2+2 a \left (24 a^2-35 b^2\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\int \frac {5 \left (16 a^2-21 b^2\right ) \sin (c+d x)^2 b^2+3 \left (36 a^2-35 b^2\right ) b^2+2 a \left (24 a^2-35 b^2\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {-\frac {-\frac {5 b \left (16 a^2-21 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (3 b^3 \left (36 a^2-35 b^2\right )-a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {-\frac {5 b \left (16 a^2-21 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (3 b^3 \left (36 a^2-35 b^2\right )-a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {5 b \left (16 a^2-21 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {3 b^3 \left (36 a^2-35 b^2\right )-a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {-\frac {-\frac {\frac {5 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {3 b^3 \left (36 a^2-35 b^2\right )-a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\frac {5 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {3 b^3 \left (36 a^2-35 b^2\right )-a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {3 b^3 \left (36 a^2-35 b^2\right )-a b^2 \left (32 a^2-35 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {-\frac {-\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a b^2 \left (32 a^2-35 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b^2 \left (32 a^2-35 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {-\frac {-\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b^2 \left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b^2 \left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {-\frac {-\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a b^2 \left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {-\frac {-\frac {\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {-\frac {\frac {\frac {3 b^3 \left (36 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a}-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}}{6 a^2 b}+\frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\left (6 a^2-7 b^2\right ) \cot (c+d x) \csc (c+d x)}{3 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {\left (24 a^2-35 b^2\right ) \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a d}-\frac {-\frac {5 b \left (16 a^2-21 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {10 b \left (16 a^2-21 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {6 b^3 \left (36 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a b^2 \left (32 a^2-35 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{2 a}}{4 a}}{6 a^2 b}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d \sqrt {a+b \sin (c+d x)}}\)

Input:

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

Output:

((6*a^2 - 7*b^2)*Cot[c + d*x]*Csc[c + d*x])/(3*a^2*b*d*Sqrt[a + b*Sin[c + 
d*x]]) - (Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d*Sqrt[a + b*Sin[c + d*x]]) + 
(-1/2*((24*a^2 - 35*b^2)*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x] 
])/(a*d) - ((-5*b*(16*a^2 - 21*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]]) 
/(a*d) - ((10*b*(16*a^2 - 21*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*(32*a^2 - 35*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sq 
rt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]) + (6*b^3*(3 
6*a^2 - 35*b^2)*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*a))/(4*a))/ 
(6*a^2*b)
 

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 3203

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, 
x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ 
e + f*x]^3)), x] + (-Simp[(3*a^2 + b^2*(m - 2))*Cos[e + f*x]*((a + b*Sin[e 
+ f*x])^(m + 1)/(3*a^2*b*f*(m + 1)*Sin[e + f*x]^2)), x] - Simp[1/(3*a^2*b*( 
m + 1))   Int[((a + b*Sin[e + f*x])^(m + 1)/Sin[e + f*x]^3)*Simp[6*a^2 - b^ 
2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[ 
e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && L 
tQ[m, -1] && IntegerQ[2*m]
 

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 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 3534

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

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. \(1495\) vs. \(2(390)=780\).

Time = 1.95 (sec) , antiderivative size = 1496, normalized size of antiderivative = 3.60

method	result	size
default	\(\text {Expression too large to display}\)	\(1496\)

Input:

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

Output:

1/24*(48*a^5*((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))*sin(d*x+c)^3+32*((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(((a+b 
*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b*sin(d*x+c)^3-150*b^2* 
((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*sin(d*x+c)^3-35*b^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^2*sin(d*x+c)^3+105*((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)^3-80*((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*sin(d*x+c)^3+185*((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)*Ell 
ipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x 
+c)^3-105*((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^4*sin(d*x+c)^3-108*((a+b*sin(d*x+c))/(a-b))^(1/2...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(sin(c + d*x)*b + a)*cot(c + d*x)**4)/(sin(c + d*x)**2*b**2 + 2*s 
in(c + d*x)*a*b + a**2),x)