\(\int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [1182]

Optimal result

Integrand size = 29, antiderivative size = 313 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {2 \left (a^2-b^2\right ) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {2 \left (5 a^2+3 b^2\right ) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \left (8 a^2+3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 a^2 b^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {2 \left (8 a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 a b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {2 \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a^2 d \sqrt {a+b \sin (c+d x)}} \] Output:

-2/3*(a^2-b^2)*cos(d*x+c)/a/b^2/d/(a+b*sin(d*x+c))^(3/2)+2/3*(5*a^2+3*b^2) 
*cos(d*x+c)/a^2/b^2/d/(a+b*sin(d*x+c))^(1/2)-2/3*(8*a^2+3*b^2)*EllipticE(c 
os(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/b^3/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-2/3*(8*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/b^3/d/(a+b*sin(d*x+c))^(1/2)-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^2/d/(a+b*sin(d*x+ 
c))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.62 (sec) , antiderivative size = 443, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {\frac {i \left (8 a^2+3 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}}}{b^2 \sqrt {-\frac {1}{a+b}}}+\frac {2 a^2 \left (a^2-b^2\right ) \cos (c+d x)}{(a+b \sin (c+d x))^{3/2}}-\frac {2 a \left (5 a^2+3 b^2\right ) \cos (c+d x)}{\sqrt {a+b \sin (c+d x)}}+\frac {4 a^2 b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {a \left (8 a^2+9 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{3 a^3 b^2 d} \] Input:

Integrate[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^(5/2),x]
 

Output:

-1/3*((I*(8*a^2 + 3*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*Arc 
Sinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*E 
llipticPi[(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)])/(b^2*Sqrt[-(a + b)^(-1)]) + (2*a 
^2*(a^2 - b^2)*Cos[c + d*x])/(a + b*Sin[c + d*x])^(3/2) - (2*a*(5*a^2 + 3* 
b^2)*Cos[c + d*x])/Sqrt[a + b*Sin[c + d*x]] + (4*a^2*b*EllipticF[(-2*c + P 
i - 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*(8*a^2 + 9*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*b^2*d)
 

Rubi [A] (verified)

Time = 2.18 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.01, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {3042, 3370, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

\(\Big \downarrow \) 3286

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

\(\displaystyle \frac {2 \left (5 a^2+3 b^2\right ) \cos (c+d x)}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) \cos (c+d x)}{3 a b^2 d (a+b \sin (c+d x))^{3/2}}+\frac {\frac {2 \left (8 a^2+3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {6 b^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (8 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{3 a^2 b^2}\)

Input:

Int[(Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x])^(5/2),x]
 

Output:

(-2*(a^2 - b^2)*Cos[c + d*x])/(3*a*b^2*d*(a + b*Sin[c + d*x])^(3/2)) + (2* 
(5*a^2 + 3*b^2)*Cos[c + d*x])/(3*a^2*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + ((2 
*(8*a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*S 
in[c + d*x]])/(b*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((-2*a*(8*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^3*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)/(3*a^2*b^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3132

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

rule 3134

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

rule 3140

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

rule 3142

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

rule 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 3370

Int[cos[(e_.) + (f_.)*(x_)]^4*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + 
(b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a^2 - b^2)*Cos[e + 
f*x]*(a + b*Sin[e + f*x])^(m + 1)*((d*Sin[e + f*x])^(n + 1)/(a*b^2*d*f*(m + 
 1))), x] + (Simp[(a^2*(n - m + 1) - b^2*(m + n + 2))*Cos[e + f*x]*(a + b*S 
in[e + f*x])^(m + 2)*((d*Sin[e + f*x])^(n + 1)/(a^2*b^2*d*f*(m + 1)*(m + 2) 
)), x] - Simp[1/(a^2*b^2*(m + 1)*(m + 2))   Int[(a + b*Sin[e + f*x])^(m + 2 
)*(d*Sin[e + f*x])^n*Simp[a^2*(n + 1)*(n + 3) - b^2*(m + n + 2)*(m + n + 3) 
 + a*b*(m + 2)*Sin[e + f*x] - (a^2*(n + 2)*(n + 3) - b^2*(m + n + 2)*(m + n 
 + 4))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, d, e, f, n}, x] && NeQ[a 
^2 - b^2, 0] && IntegersQ[2*m, 2*n] && LtQ[m, -1] &&  !LtQ[n, -1] && (LtQ[m 
, -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 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. \(1364\) vs. \(2(297)=594\).

Time = 1.75 (sec) , antiderivative size = 1365, normalized size of antiderivative = 4.36

method	result	size
default	\(\text {Expression too large to display}\)	\(1365\)

Input:

int(cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE 
)
 

Output:

(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)*(-2/a^3*(1/b*a-1)*((a+b*sin(d*x+c) 
)/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1 
/2)/(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)*b*EllipticPi(((a+b*sin(d*x+c)) 
/(a-b))^(1/2),-(-1/b*a+1)*b/a,((a-b)/(a+b))^(1/2))+1/b^3*(2*b*(1/b*a-1)*(( 
a+b*sin(d*x+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*(-b*(1+sin(d*x 
+c))/(a-b))^(1/2)/(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)*((-1/b*a-1)*Elli 
pticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b* 
sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2)))-4*a*(1/b*a-1)*((a+b*sin(d*x 
+c))/(a-b))^(1/2)*((1-sin(d*x+c))*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b)) 
^(1/2)/(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)*EllipticF(((a+b*sin(d*x+c)) 
/(a-b))^(1/2),((a-b)/(a+b))^(1/2)))-1/b^3*(a^4-2*a^2*b^2+b^4)/a*(2/3/(a^2- 
b^2)/b*(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)/(sin(d*x+c)+1/b*a)^2+8/3*b* 
cos(d*x+c)^2/(a^2-b^2)^2*a/(-(-a-b*sin(d*x+c))*cos(d*x+c)^2)^(1/2)+2*(3*a^ 
2+b^2)/(3*a^4-6*a^2*b^2+3*b^4)*(1/b*a-1)*((a+b*sin(d*x+c))/(a-b))^(1/2)*(( 
1-sin(d*x+c))*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)/(-(-a-b*sin(d 
*x+c))*cos(d*x+c)^2)^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b) 
/(a+b))^(1/2))+8/3*a*b/(a^2-b^2)^2*(1/b*a-1)*((a+b*sin(d*x+c))/(a-b))^(1/2 
)*((1-sin(d*x+c))*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)/(-(-a-b*s 
in(d*x+c))*cos(d*x+c)^2)^(1/2)*((-1/b*a-1)*EllipticE(((a+b*sin(d*x+c))/(a- 
b))^(1/2),((a-b)/(a+b))^(1/2))+EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2)...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c))^(5/2),x, algorithm="fri 
cas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**3*cot(d*x+c)/(a+b*sin(d*x+c))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{3} \cot \left (d x + c\right )}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c))^(5/2),x, algorithm="max 
ima")
 

Output:

integrate(cos(d*x + c)^3*cot(d*x + c)/(b*sin(d*x + c) + a)^(5/2), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c))^(5/2),x, algorithm="gia 
c")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cos ^3(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\sqrt {\sin \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )}{\sin \left (d x +c \right )^{3} b^{3}+3 \sin \left (d x +c \right )^{2} a \,b^{2}+3 \sin \left (d x +c \right ) a^{2} b +a^{3}}d x \] Input:

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

Output:

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