\(\int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx\) [1177]

Optimal result

Integrand size = 29, antiderivative size = 366 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d}-\frac {\left (8 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{4 a^3 b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (8 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {3 \left (4 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a^3 d \sqrt {a+b \sin (c+d x)}} \] Output:

1/2*(4*a^2-5*b^2)*cot(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(1/2)-1/2*cot(d*x+c) 
*csc(d*x+c)/a/d/(a+b*sin(d*x+c))^(1/2)-1/4*(8*a^2-15*b^2)*cot(d*x+c)*(a+b* 
sin(d*x+c))^(1/2)/a^3/b/d+1/4*(8*a^2-15*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^3/b/d/((a+b*sin(d 
*x+c))/(a+b))^(1/2)+1/4*(8*a^2-5*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^2/b/d/(a+b*sin( 
d*x+c))^(1/2)+3/4*(4*a^2-5*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^3/d/(a+b*sin(d*x+c) 
)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.54 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx=\frac {\frac {\left (-32 a^2+60 b^2\right ) \cos (c+d x)+4 a \cot (c+d x) (5 b-2 a \csc (c+d x))}{a^3 \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {2 i \left (-8 a^2+15 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}}}-\frac {40 a b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (32 a^2-45 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^3}}{16 d} \] Input:

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

Output:

(((-32*a^2 + 60*b^2)*Cos[c + d*x] + 4*a*Cot[c + d*x]*(5*b - 2*a*Csc[c + d* 
x]))/(a^3*Sqrt[a + b*Sin[c + d*x]]) + (((-2*I)*(-8*a^2 + 15*b^2)*(-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)]))*Sec[c + d*x 
]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/( 
a - b))])/(a*b^2*Sqrt[-(a + b)^(-1)]) - (40*a*b*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*(32*a^2 - 45*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^ 
3)/(16*d)
 

Rubi [A] (verified)

Time = 2.74 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3369, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3369

\(\displaystyle \frac {\int \frac {\csc ^2(c+d x) \left (8 a^2-2 b \sin (c+d x) a-15 b^2+5 b^2 \sin ^2(c+d x)\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\csc ^2(c+d x) \left (8 a^2-2 b \sin (c+d x) a-15 b^2+5 b^2 \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {8 a^2-2 b \sin (c+d x) a-15 b^2+5 b^2 \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

\(\displaystyle \frac {-\frac {\left (8 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\left (8 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\left (8 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {-\frac {\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {\left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {-\frac {\frac {\int \frac {3 b^2 \left (4 a^2-5 b^2\right )-a b \left (8 a^2-5 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3481

\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a b \left (8 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a b \left (8 a^2-5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3140

\(\displaystyle \frac {-\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3286

\(\displaystyle \frac {-\frac {\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\frac {\frac {3 b^2 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{4 a^2 b}+\frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\left (4 a^2-5 b^2\right ) \cot (c+d x)}{2 a^2 b d \sqrt {a+b \sin (c+d x)}}+\frac {-\frac {\left (8 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {2 \left (8 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {6 b^2 \left (4 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a b \left (8 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{2 a}}{4 a^2 b}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d \sqrt {a+b \sin (c+d x)}}\)

Input:

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

Output:

((4*a^2 - 5*b^2)*Cot[c + d*x])/(2*a^2*b*d*Sqrt[a + b*Sin[c + d*x]]) - (Cot 
[c + d*x]*Csc[c + d*x])/(2*a*d*Sqrt[a + b*Sin[c + d*x]]) + (-(((8*a^2 - 15 
*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d)) - ((2*(8*a^2 - 15*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*(8*a^2 - 5*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^2*(4*a^2 - 5*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^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 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 3369

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]*(d*Sin[ 
e + f*x])^(n + 1)*((a + b*Sin[e + f*x])^(m + 1)/(a*d*f*(n + 1))), x] + (-Si 
mp[(a^2*(n + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(d*Sin[e + f*x])^(n + 2)*(( 
a + b*Sin[e + f*x])^(m + 1)/(a^2*b*d^2*f*(n + 1)*(m + 1))), x] + Simp[1/(a^ 
2*b*d*(n + 1)*(m + 1))   Int[(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^ 
(m + 1)*Simp[a^2*(n + 1)*(n + 2) - b^2*(m + n + 2)*(m + n + 3) + a*b*(m + 1 
)*Sin[e + f*x] - (a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 4))*Sin[e 
+ f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && 
IntegersQ[2*m, 2*n] && LtQ[m, -1] && LtQ[n, -1]
 

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. \(1348\) vs. \(2(344)=688\).

Time = 1.57 (sec) , antiderivative size = 1349, normalized size of antiderivative = 3.69

method	result	size
default	\(\text {Expression too large to display}\)	\(1349\)

Input:

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

Output:

1/4*(8*((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)^2-23*((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*s 
in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^2+15*((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)^2+12*((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)*b^2*EllipticPi(((a+b*sin(d* 
x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^3*sin(d*x+c)^2-12*((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)*b^3*EllipticPi(((a+b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b 
)/(a+b))^(1/2))*a^2*sin(d*x+c)^2-15*((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(d*x+c)^2+1 
5*((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))*b^5*sin(d*x+c)^2-8*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(s 
in(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)^2+18*b...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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