\(\int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+b \sin (c+d x)} \, dx\) [1143]

Optimal result

Integrand size = 29, antiderivative size = 345 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=-\frac {\left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}+\frac {\left (8 a^2-3 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{12 a b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (8 a^2+31 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{12 b d \sqrt {a+b \sin (c+d x)}}-\frac {\left (12 a^2+b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{4 a d \sqrt {a+b \sin (c+d x)}} \] Output:

-1/12*(8*a^2+3*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/4*b*cot(d*x+ 
c)*(a+b*sin(d*x+c))^(3/2)/a^2/d-1/2*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c)) 
^(3/2)/a/d-1/12*(8*a^2-3*b^2)*EllipticE(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2)* 
(b/(a+b))^(1/2))*(a+b*sin(d*x+c))^(1/2)/a/b/d/((a+b*sin(d*x+c))/(a+b))^(1/ 
2)-1/12*(8*a^2+31*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)/b/d/(a+b*sin(d*x+c))^(1/2)+1/4*( 
12*a^2+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/d/(a+b*sin(d*x+c))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.88 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.30 \[ \int \cos (c+d x) \cot ^3(c+d x) \sqrt {a+b \sin (c+d x)} \, dx=\frac {\frac {2 i \left (8 a^2-3 b^2\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a^2 b^2 \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {4 (8 a \cos (c+d x)+3 \cot (c+d x) (b+2 a \csc (c+d x))) \sqrt {a+b \sin (c+d x)}}{a}+\frac {136 b \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 \left (64 a^2+9 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{a \sqrt {a+b \sin (c+d x)}}}{48 d} \] Input:

Integrate[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(((2*I)*(8*a^2 - 3*b^2)*Cos[2*(c + d*x)]*Csc[c + d*x]^2*(2*a*(a - b)*Ellip 
ticE[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[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^2*b^2*Sqrt[-(a + b)^(-1)]*(-2 + Csc[c + d*x]^2)) - (4*(8*a*Cos[c + d*x] 
+ 3*Cot[c + d*x]*(b + 2*a*Csc[c + d*x]))*Sqrt[a + b*Sin[c + d*x]])/a + (13 
6*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*(64*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)]) 
/(a*Sqrt[a + b*Sin[c + d*x]]))/(48*d)
 

Rubi [A] (verified)

Time = 2.54 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.02, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {3042, 3372, 27, 3042, 3528, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3372

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {2}{3} \int \frac {\csc (c+d x) \left (34 b \sin (c+d x) a^2-\left (8 a^2-3 b^2\right ) \sin ^2(c+d x) a+3 \left (12 a^2+b^2\right ) a\right )}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {1}{3} \int \frac {\csc (c+d x) \left (34 b \sin (c+d x) a^2-\left (8 a^2-3 b^2\right ) \sin ^2(c+d x) a+3 \left (12 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{3} \int \frac {34 b \sin (c+d x) a^2-\left (8 a^2-3 b^2\right ) \sin (c+d x)^2 a+3 \left (12 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {\frac {1}{3} \left (-\frac {a \left (8 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}-\frac {\int -\frac {\csc (c+d x) \left (\left (8 a^2+31 b^2\right ) \sin (c+d x) a^2+3 b \left (12 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\int \frac {\csc (c+d x) \left (\left (8 a^2+31 b^2\right ) \sin (c+d x) a^2+3 b \left (12 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a \left (8 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\int \frac {\left (8 a^2+31 b^2\right ) \sin (c+d x) a^2+3 b \left (12 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a \left (8 a^2-3 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx}{b}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\int \frac {\left (8 a^2+31 b^2\right ) \sin (c+d x) a^2+3 b \left (12 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\int \frac {\left (8 a^2+31 b^2\right ) \sin (c+d x) a^2+3 b \left (12 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\int \frac {\left (8 a^2+31 b^2\right ) \sin (c+d x) a^2+3 b \left (12 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {a^2 \left (8 a^2+31 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+3 a b \left (12 a^2+b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {a^2 \left (8 a^2+31 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+3 a b \left (12 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\frac {a^2 \left (8 a^2+31 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+3 a b \left (12 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\frac {a^2 \left (8 a^2+31 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+3 a b \left (12 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {3 a b \left (12 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 \left (8 a^2+31 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\frac {3 a b \left (12 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 \left (8 a^2+31 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {1}{3} \left (\frac {\frac {3 a b \left (12 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 \left (8 a^2+31 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\frac {2 \left (8 a^2+3 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {\frac {2 a^2 \left (8 a^2+31 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {6 a b \left (12 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )}{8 a^2}+\frac {b \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{4 a^2 d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{3/2}}{2 a d}\)

Input:

Int[Cos[c + d*x]*Cot[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(b*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(4*a^2*d) - (Cot[c + d*x]*Csc[ 
c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(2*a*d) - ((2*(8*a^2 + 3*b^2)*Cos[c + 
 d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*d) + ((-2*a*(8*a^2 - 3*b^2)*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^2*(8*a^2 + 31*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*a*b*(12*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*Sqrt[a + b*Sin[c + d 
*x]]))/b)/3)/(8*a^2)
 

Defintions of rubi rules used

rule 25

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

rule 27

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

rule 3042

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

rule 3132

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

rule 3134

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

rule 3140

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

rule 3142

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

rule 3284

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

rule 3286

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

rule 3372

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

rule 3481

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

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*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] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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. \(2(323)=646\).

Time = 1.70 (sec) , antiderivative size = 1365, normalized size of antiderivative = 3.96

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

Input:

int(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

-1/12*(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)^2-11*((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)*EllipticE(((a+b 
*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c)^2+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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1 
/2))*a*b^4*sin(d*x+c)^2-8*((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)*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+42*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)^2-31*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)^2-3*((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)*Ellip 
ticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+c)^ 
2-36*((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+36*((a+b*sin(d*x+c))/(a-b))...
 

Fricas [F(-1)]

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

integrate(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(1/2),x, algorithm="fri 
cas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(cos(d*x+c)*cot(d*x+c)**3*(a+b*sin(d*x+c))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(1/2),x, algorithm="max 
ima")
 

Output:

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

Giac [F(-1)]

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

integrate(cos(d*x+c)*cot(d*x+c)^3*(a+b*sin(d*x+c))^(1/2),x, algorithm="gia 
c")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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