\(\int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx\) [1183]

Optimal result

Integrand size = 31, antiderivative size = 346 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}-\frac {\left (4 a^2+15 b^2\right ) \cos (c+d x)}{3 a^3 b d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^2+15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 a^3 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (4 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 a^2 b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {5 b \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a^3 d \sqrt {a+b \sin (c+d x)}} \] Output:

1/3*(2*a^2-5*b^2)*cos(d*x+c)/a^2/b/d/(a+b*sin(d*x+c))^(3/2)-cot(d*x+c)/a/d 
/(a+b*sin(d*x+c))^(3/2)-1/3*(4*a^2+15*b^2)*cos(d*x+c)/a^3/b/d/(a+b*sin(d*x 
+c))^(1/2)+1/3*(4*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^2/d/((a+b*sin(d*x+c))/(a+b)) 
^(1/2)+1/3*(4*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^2/d/(a+b*sin(d*x+c))^(1/ 
2)+5*b*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 = 6.24 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.29 \[ \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\frac {2 i \left (4 a^2+15 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b^2 \sqrt {-\frac {1}{a+b}}}+\frac {40 a b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (4 a^2+45 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (4 a \left (a^2+10 b^2\right ) \cos (c+d x)+6 a^2 b \cot (c+d x)+b \left (4 a^2+15 b^2\right ) \sin (2 (c+d x))\right )}{(a+b \sin (c+d x))^{3/2}}}{12 a^3 b d} \] Input:

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

Output:

(((2*I)*(4*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*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))])/(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*(4*a^2 + 45*b^2)*EllipticP 
i[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]] - (2*(4*a*(a^2 + 10*b^2)*Cos[c + d*x] + 6*a^ 
2*b*Cot[c + d*x] + b*(4*a^2 + 15*b^2)*Sin[2*(c + d*x)]))/(a + b*Sin[c + d* 
x])^(3/2))/(12*a^3*b*d)
 

Rubi [A] (verified)

Time = 2.87 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.11, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.677, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3369

\(\displaystyle \frac {2 \int -\frac {\csc (c+d x) \left (15 b^2+6 a \sin (c+d x) b-\left (4 a^2+5 b^2\right ) \sin ^2(c+d x)\right )}{4 (a+b \sin (c+d x))^{3/2}}dx}{3 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\csc (c+d x) \left (15 b^2+6 a \sin (c+d x) b-\left (4 a^2+5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^{3/2}}dx}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {15 b^2+6 a \sin (c+d x) b-\left (4 a^2+5 b^2\right ) \sin (c+d x)^2}{\sin (c+d x) (a+b \sin (c+d x))^{3/2}}dx}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3534

\(\displaystyle -\frac {\frac {2 \int \frac {\csc (c+d x) \left (15 \left (a^2-b^2\right ) b^2+10 a \left (a^2-b^2\right ) \sin (c+d x) b+\left (4 a^4+11 b^2 a^2-15 b^4\right ) \sin ^2(c+d x)\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\int \frac {\csc (c+d x) \left (15 \left (a^2-b^2\right ) b^2+10 a \left (a^2-b^2\right ) \sin (c+d x) b+\left (4 a^4+11 b^2 a^2-15 b^4\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\int \frac {15 \left (a^2-b^2\right ) b^2+10 a \left (a^2-b^2\right ) \sin (c+d x) b+\left (4 a^4+11 b^2 a^2-15 b^4\right ) \sin (c+d x)^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {\frac {\frac {\left (4 a^4+11 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\int -\frac {\csc (c+d x) \left (15 b^3 \left (a^2-b^2\right )-a \left (4 a^4+b^2 a^2-5 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {\frac {\left (4 a^4+11 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}+\frac {\int \frac {\csc (c+d x) \left (15 b^3 \left (a^2-b^2\right )-a \left (4 a^4+b^2 a^2-5 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\left (4 a^4+11 a^2 b^2-15 b^4\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}+\frac {\int \frac {15 b^3 \left (a^2-b^2\right )-a \left (4 a^4+b^2 a^2-5 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {\frac {\left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {15 b^3 \left (a^2-b^2\right )-a \left (4 a^4+b^2 a^2-5 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {15 b^3 \left (a^2-b^2\right )-a \left (4 a^4+b^2 a^2-5 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {\frac {\int \frac {15 b^3 \left (a^2-b^2\right )-a \left (4 a^4+b^2 a^2-5 b^4\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a \left (4 a^4+a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a \left (4 a^4+a^2 b^2-5 b^4\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a \left (4 a^4+a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a \left (4 a^4+a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a \left (4 a^4+a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (4 a^4+a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\frac {\frac {15 b^3 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (4 a^4+a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{a \left (a^2-b^2\right )}+\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}}{6 a^2 b}+\frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\left (2 a^2-5 b^2\right ) \cos (c+d x)}{3 a^2 b d (a+b \sin (c+d x))^{3/2}}-\frac {\frac {2 \left (4 a^2+15 b^2\right ) \cos (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}+\frac {\frac {2 \left (4 a^4+11 a^2 b^2-15 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {30 b^3 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (4 a^4+a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{a \left (a^2-b^2\right )}}{6 a^2 b}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^{3/2}}\)

Input:

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

Output:

((2*a^2 - 5*b^2)*Cos[c + d*x])/(3*a^2*b*d*(a + b*Sin[c + d*x])^(3/2)) - Co 
t[c + d*x]/(a*d*(a + b*Sin[c + d*x])^(3/2)) - ((2*(4*a^2 + 15*b^2)*Cos[c + 
 d*x])/(a*d*Sqrt[a + b*Sin[c + d*x]]) + ((2*(4*a^4 + 11*a^2*b^2 - 15*b^4)* 
EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b* 
d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((-2*a*(4*a^4 + a^2*b^2 - 5*b^4)*E 
llipticF[(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]]) + (30*b^3*(a^2 - b^2)*EllipticPi[2, (c 
- Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqr 
t[a + b*Sin[c + d*x]]))/b)/(a*(a^2 - b^2)))/(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 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. \(2111\) vs. \(2(328)=656\).

Time = 1.61 (sec) , antiderivative size = 2112, normalized size of antiderivative = 6.10

method	result	size
default	\(\text {Expression too large to display}\)	\(2112\)

Input:

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

Output:

-1/3*(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))*b^6*sin(d*x+c)^2-4*((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)*Ellipti 
cE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^6*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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^ 
(1/2))*a^5*b*sin(d*x+c)^2-11*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)- 
1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+ 
c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^3*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 
^5*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^5*sin(d*x+c)^2+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^5*b*s 
in(d*x+c)+6*((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^2*sin(d*x+c)+5*((a+b*sin(d*x+c))/(a-b))^(1/2...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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