\(\int \frac {\cot ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx\) [1170]

Optimal result

Integrand size = 23, antiderivative size = 353 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{24 a^3 d}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}+\frac {\left (32 a^2-15 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{24 a^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (16 a^2+5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{24 a^2 d \sqrt {a+b \sin (c+d x)}}+\frac {b \left (12 a^2-5 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a^3 d \sqrt {a+b \sin (c+d x)}} \] Output:

1/24*(32*a^2-15*b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^3/d+5/12*b*cot(d* 
x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d-1/3*cot(d*x+c)*csc(d*x+c)^2*( 
a+b*sin(d*x+c))^(1/2)/a/d-1/24*(32*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/d/((a+b*sin(d 
*x+c))/(a+b))^(1/2)+1/24*(16*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/d/(a+b*sin( 
d*x+c))^(1/2)-1/8*b*(12*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] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.14 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.35 \[ \int \frac {\cot ^4(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {-\frac {4 \cot (c+d x) \left (-32 a^2+15 b^2-10 a b \csc (c+d x)+8 a^2 \csc ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{a^3}+\frac {\frac {2 i \left (32 a^2-15 b^2\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}} \left (-2+\csc ^2(c+d x)\right )}-\frac {8 a \left (24 a^2-5 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (-104 a^2+45 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}}{a^3}}{96 d} \] Input:

Integrate[Cot[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

((-4*Cot[c + d*x]*(-32*a^2 + 15*b^2 - 10*a*b*Csc[c + d*x] + 8*a^2*Csc[c + 
d*x]^2)*Sqrt[a + b*Sin[c + d*x]])/a^3 + (((2*I)*(32*a^2 - 15*b^2)*Cos[2*(c 
 + d*x)]*Csc[c + d*x]^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*ArcSin 
h[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*Elli 
pticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], 
 (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]* 
Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]*(-2 + Cs 
c[c + d*x]^2)) - (8*a*(24*a^2 - 5*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2 
*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] 
+ (2*b*(-104*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)/(9 
6*d)
 

Rubi [A] (verified)

Time = 2.67 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.02, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.913, Rules used = {3042, 3204, 27, 3042, 3534, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3204

\(\displaystyle -\frac {\int \frac {\csc ^2(c+d x) \left (32 a^2-2 b \sin (c+d x) a-15 b^2-\left (24 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{4 \sqrt {a+b \sin (c+d x)}}dx}{6 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\csc ^2(c+d x) \left (32 a^2-2 b \sin (c+d x) a-15 b^2-\left (24 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {32 a^2-2 b \sin (c+d x) a-15 b^2-\left (24 a^2-5 b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^2 \sqrt {a+b \sin (c+d x)}}dx}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3534

\(\displaystyle -\frac {\frac {\int -\frac {\csc (c+d x) \left (b \left (32 a^2-15 b^2\right ) \sin ^2(c+d x)+2 a \left (24 a^2-5 b^2\right ) \sin (c+d x)+3 b \left (12 a^2-5 b^2\right )\right )}{2 \sqrt {a+b \sin (c+d x)}}dx}{a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\csc (c+d x) \left (b \left (32 a^2-15 b^2\right ) \sin ^2(c+d x)+2 a \left (24 a^2-5 b^2\right ) \sin (c+d x)+3 b \left (12 a^2-5 b^2\right )\right )}{\sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int \frac {b \left (32 a^2-15 b^2\right ) \sin (c+d x)^2+2 a \left (24 a^2-5 b^2\right ) \sin (c+d x)+3 b \left (12 a^2-5 b^2\right )}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {-\frac {\left (32 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (3 \left (12 a^2-5 b^2\right ) b^2+a \left (16 a^2+5 b^2\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\left (32 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (3 \left (12 a^2-5 b^2\right ) b^2+a \left (16 a^2+5 b^2\right ) \sin (c+d x) b\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\left (32 a^2-15 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {3 \left (12 a^2-5 b^2\right ) b^2+a \left (16 a^2+5 b^2\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {-\frac {\frac {\int \frac {3 \left (12 a^2-5 b^2\right ) b^2+a \left (16 a^2+5 b^2\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {\left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {3 \left (12 a^2-5 b^2\right ) b^2+a \left (16 a^2+5 b^2\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {-\frac {\frac {\int \frac {3 \left (12 a^2-5 b^2\right ) b^2+a \left (16 a^2+5 b^2\right ) \sin (c+d x) b}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {-\frac {\frac {a b \left (16 a^2+5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+3 b^2 \left (12 a^2-5 b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {3 b^2 \left (12 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+a b \left (16 a^2+5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {-\frac {\frac {3 b^2 \left (12 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a b \left (16 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {3 b^2 \left (12 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a b \left (16 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {-\frac {\frac {3 b^2 \left (12 a^2-5 b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b \left (16 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {-\frac {\frac {\frac {3 b^2 \left (12 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b \left (16 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\frac {3 b^2 \left (12 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a b \left (16 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}}{2 a}-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {-\frac {\left (32 a^2-15 b^2\right ) \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{a d}-\frac {\frac {2 \left (32 a^2-15 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {2 a b \left (16 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {6 b^2 \left (12 a^2-5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}}{2 a}}{24 a^2}+\frac {5 b \cot (c+d x) \csc (c+d x) \sqrt {a+b \sin (c+d x)}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+b \sin (c+d x)}}{3 a d}\)

Input:

Int[Cot[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]
 

Output:

(5*b*Cot[c + d*x]*Csc[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(12*a^2*d) - (Cot 
[c + d*x]*Csc[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]])/(3*a*d) - (-(((32*a^2 - 
 15*b^2)*Cot[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(a*d)) - ((2*(32*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*(16*a^2 + 5*b^2)*Ellipt 
icF[(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*(12*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))/(24*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 3204

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, 
x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ 
e + f*x]^3)), x] + (-Simp[b*(m - 2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 
 1)/(6*a^2*f*Sin[e + f*x]^2)), x] - Simp[1/(6*a^2)   Int[((a + b*Sin[e + f* 
x])^m/Sin[e + f*x]^2)*Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] 
 - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f 
, m}, x] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1] && IntegerQ[2*m]
 

rule 3284

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

rule 3286

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

rule 3481

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

rule 3534

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) 
 + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && 
NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

rule 3538

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 
2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d)   Int[Sqrt[a + b*Sin[e + f*x]], x] 
, x] - Simp[1/(b*d)   Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ 
e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre 
eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 
] && NeQ[c^2 - d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1495\) vs. \(2(331)=662\).

Time = 0.93 (sec) , antiderivative size = 1496, normalized size of antiderivative = 4.24

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

Input:

int(cot(d*x+c)^4/(a+b*sin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/24*(48*a^5*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2 
)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2) 
,((a-b)/(a+b))^(1/2))*sin(d*x+c)^3-16*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si 
n(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b 
*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b*sin(d*x+c)^3-42*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)^3-5*b^3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c 
)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b*sin(d* 
x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*sin(d*x+c)^3+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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a*b^4 
*sin(d*x+c)^3-32*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^ 
(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticE(((a+b*sin(d*x+c))/(a-b))^( 
1/2),((a-b)/(a+b))^(1/2))*a^5*sin(d*x+c)^3+47*((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^3*b^2*sin(d*x+c)^ 
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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/ 
(a+b))^(1/2))*a*b^4*sin(d*x+c)^3-36*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(s...
 

Fricas [F(-1)]

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

integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(cot(d*x+c)**4/(a+b*sin(d*x+c))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")
 

Output:

integrate(cot(d*x + c)^4/sqrt(b*sin(d*x + c) + a), x)
 

Giac [F]

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

integrate(cot(d*x+c)^4/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")
 

Output:

integrate(cot(d*x + c)^4/sqrt(b*sin(d*x + c) + a), x)
 

Mupad [F(-1)]

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

int(cot(c + d*x)^4/(a + b*sin(c + d*x))^(1/2),x)
 

Output:

int(cot(c + d*x)^4/(a + b*sin(c + d*x))^(1/2), x)
 

Reduce [F]

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

int(cot(d*x+c)^4/(a+b*sin(d*x+c))^(1/2),x)
 

Output:

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