\(\int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx\) [1152]

Optimal result

Integrand size = 23, antiderivative size = 386 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{8 a^2 d}+\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{24 a^2 d}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}+\frac {\left (32 a^2-b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{8 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (16 a^2+21 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 d \sqrt {a+b \sin (c+d x)}}-\frac {b \left (36 a^2+b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 a d \sqrt {a+b \sin (c+d x)}} \] Output:

-1/8*b*(16*a^2+b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a^2/d+1/24*(32*a^2+b 
^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/12*b*cot(d*x+c)*csc(d*x+c)*( 
a+b*sin(d*x+c))^(5/2)/a^2/d-1/3*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^( 
5/2)/a/d-1/8*(32*a^2-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/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/8 
*(16*a^2+21*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)/d/(a+b*sin(d*x+c))^(1/2)+1/8*b*(36*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] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 6.78 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.55 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {\left (-\frac {2}{3} b \cos (c+d x)+\frac {\left (32 a^2 \cos (c+d x)-3 b^2 \cos (c+d x)\right ) \csc (c+d x)}{24 a}-\frac {7}{12} b \cot (c+d x) \csc (c+d x)-\frac {1}{3} a \cot (c+d x) \csc ^2(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {-\frac {2 \left (32 a^3-44 a b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (-40 a^2 b-3 b^3\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-32 a^2 b+b^3\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{32 a d} \] Input:

Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]
 

Output:

(((-2*b*Cos[c + d*x])/3 + ((32*a^2*Cos[c + d*x] - 3*b^2*Cos[c + d*x])*Csc[ 
c + d*x])/(24*a) - (7*b*Cot[c + d*x]*Csc[c + d*x])/12 - (a*Cot[c + d*x]*Cs 
c[c + d*x]^2)/3)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(32*a^3 - 44*a*b^2)*El 
lipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + 
 b)])/Sqrt[a + b*Sin[c + d*x]] - (2*(-40*a^2*b - 3*b^3)*EllipticPi[2, (-c 
+ Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a 
 + b*Sin[c + d*x]] - ((2*I)*(-32*a^2*b + b^3)*Cos[c + d*x]*Cos[2*(c + d*x) 
]*(2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + 
d*x]]], (a + b)/(a - b)] + b*(2*a*EllipticF[I*ArcSinh[Sqrt[-(a + b)^(-1)]* 
Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] - b*EllipticPi[(a + b)/a, I*Ar 
cSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sq 
rt[(b - b*Sin[c + d*x])/(a + b)]*Sqrt[-((b + b*Sin[c + d*x])/(a - b))])/(a 
*Sqrt[-(a + b)^(-1)]*Sqrt[1 - Sin[c + d*x]^2]*(-2*a^2 + b^2 + 4*a*(a + b*S 
in[c + d*x]) - 2*(a + b*Sin[c + d*x])^2)*Sqrt[-((a^2 - b^2 - 2*a*(a + b*Si 
n[c + d*x]) + (a + b*Sin[c + d*x])^2)/b^2)]))/(32*a*d)
 

Rubi [A] (verified)

Time = 3.10 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.01, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.043, Rules used = {3042, 3204, 27, 3042, 3526, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3204

\(\displaystyle -\frac {\int \frac {1}{4} \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (32 a^2+6 b \sin (c+d x) a+b^2-3 \left (8 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{6 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (32 a^2+6 b \sin (c+d x) a+b^2-3 \left (8 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{3/2} \left (32 a^2+6 b \sin (c+d x) a+b^2-3 \left (8 a^2+b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\int \frac {3}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 b \left (16 a^2+b^2\right ) \sin ^2(c+d x)-2 a \left (8 a^2-b^2\right ) \sin (c+d x)+b \left (36 a^2+b^2\right )\right )dx-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3}{2} \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-3 b \left (16 a^2+b^2\right ) \sin ^2(c+d x)-2 a \left (8 a^2-b^2\right ) \sin (c+d x)+b \left (36 a^2+b^2\right )\right )dx-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \int \frac {\sqrt {a+b \sin (c+d x)} \left (-3 b \left (16 a^2+b^2\right ) \sin (c+d x)^2-2 a \left (8 a^2-b^2\right ) \sin (c+d x)+b \left (36 a^2+b^2\right )\right )}{\sin (c+d x)}dx-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3528

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {2}{3} \int \frac {3 \csc (c+d x) \left (-2 \left (8 a^2-11 b^2\right ) \sin (c+d x) a^2-b \left (32 a^2-b^2\right ) \sin ^2(c+d x) a+b \left (36 a^2+b^2\right ) a\right )}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {3}{2} \left (\int \frac {\csc (c+d x) \left (-2 \left (8 a^2-11 b^2\right ) \sin (c+d x) a^2-b \left (32 a^2-b^2\right ) \sin ^2(c+d x) a+b \left (36 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \left (\int \frac {-2 \left (8 a^2-11 b^2\right ) \sin (c+d x) a^2-b \left (32 a^2-b^2\right ) \sin (c+d x)^2 a+b \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {\frac {3}{2} \left (-a \left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {3}{2} \left (-a \left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \left (-a \left (32 a^2-b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3134

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \left (-\frac {a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {\int \frac {b \left (16 a^2+21 b^2\right ) \sin (c+d x) a^2+b^2 \left (36 a^2+b^2\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {a^2 b \left (16 a^2+21 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+a b^2 \left (36 a^2+b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {a^2 b \left (16 a^2+21 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx+a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3142

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3140

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {a b^2 \left (36 a^2+b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3286

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a b^2 \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {\frac {a b^2 \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\frac {3}{2} \left (\frac {2 b \left (16 a^2+b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a \left (32 a^2-b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {2 a^2 b \left (16 a^2+21 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {2 a b^2 \left (36 a^2+b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )-\frac {\left (32 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}}{24 a^2}+\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2}}{3 a d}\)

Input:

Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]
 

Output:

(b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/(12*a^2*d) - (Cot 
[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2))/(3*a*d) - (-(((32*a^2 
 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/d) + (3*((2*b*(16*a^2 + b 
^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d - (2*a*(32*a^2 - b^2)*Ellipti 
cE[(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^2*b*(16*a^2 + 21*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]]) + (2*a*b^2*(36*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*Si 
n[c + d*x]]))/b))/2)/(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 3526

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^2*C - B*c*d + A*d^2))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - 
d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{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] && GtQ[m, 0] && LtQ[n, -1]
 

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. \(1510\) vs. \(2(360)=720\).

Time = 2.27 (sec) , antiderivative size = 1511, normalized size of antiderivative = 3.91

method	result	size
default	\(\text {Expression too large to display}\)	\(1511\)

Input:

int(cot(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),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+48*((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)^3-162*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+63*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+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*b^ 
4*sin(d*x+c)^3-96*((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+99*((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)*Ellipt 
icE(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^3*b^2*sin(d*x+c) 
^3-3*((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)^3+108*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(...
 

Fricas [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**4*(a+b*sin(d*x+c))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cot \left (d x + c\right )^{4} \,d x } \] Input:

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x, algorithm="maxima")
 

Output:

integrate((b*sin(d*x + c) + a)^(3/2)*cot(d*x + c)^4, x)
 

Giac [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:

int(cot(c + d*x)^4*(a + b*sin(c + d*x))^(3/2),x)
 

Output:

int(cot(c + d*x)^4*(a + b*sin(c + d*x))^(3/2), x)
 

Reduce [F]

\[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4} \sin \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4}d x \right ) a \] Input:

int(cot(d*x+c)^4*(a+b*sin(d*x+c))^(3/2),x)
 

Output:

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