\(\int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\) [478]

Optimal result

Integrand size = 23, antiderivative size = 144 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 a^{3/2} d}-\frac {\cot (c+d x)}{8 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \cot (c+d x) \csc (c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 a^2 d} \] Output:

-1/8*arctanh(a^(1/2)*cos(d*x+c)/(a+a*sin(d*x+c))^(1/2))/a^(3/2)/d-1/8*cot( 
d*x+c)/a/d/(a+a*sin(d*x+c))^(1/2)+11/12*cot(d*x+c)*csc(d*x+c)/a/d/(a+a*sin 
(d*x+c))^(1/2)-1/3*cot(d*x+c)*csc(d*x+c)^2*(a+a*sin(d*x+c))^(1/2)/a^2/d
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(144)=288\).

Time = 1.46 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.04 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\csc ^9\left (\frac {1}{2} (c+d x)\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-132 \cos \left (\frac {1}{2} (c+d x)\right )+62 \cos \left (\frac {3}{2} (c+d x)\right )+6 \cos \left (\frac {5}{2} (c+d x)\right )+132 \sin \left (\frac {1}{2} (c+d x)\right )-9 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+9 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+62 \sin \left (\frac {3}{2} (c+d x)\right )-6 \sin \left (\frac {5}{2} (c+d x)\right )+3 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-3 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))\right )}{24 d \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3 (a (1+\sin (c+d x)))^{3/2}} \] Input:

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

Output:

(Csc[(c + d*x)/2]^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-132*Cos[(c + 
 d*x)/2] + 62*Cos[(3*(c + d*x))/2] + 6*Cos[(5*(c + d*x))/2] + 132*Sin[(c + 
 d*x)/2] - 9*Log[1 + Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 9 
*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 62*Sin[(3*(c 
+ d*x))/2] - 6*Sin[(5*(c + d*x))/2] + 3*Log[1 + Cos[(c + d*x)/2] - Sin[(c 
+ d*x)/2]]*Sin[3*(c + d*x)] - 3*Log[1 - Cos[(c + d*x)/2] + Sin[(c + d*x)/2 
]]*Sin[3*(c + d*x)]))/(24*d*(Csc[(c + d*x)/4]^2 - Sec[(c + d*x)/4]^2)^3*(a 
*(1 + Sin[c + d*x]))^(3/2))
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.82, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.783, Rules used = {3042, 3196, 3042, 3251, 3042, 3251, 3042, 3252, 219, 3523, 27, 3042, 3459, 3042, 3251, 3042, 3252, 219}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (c+d x)^4 (a \sin (c+d x)+a)^{3/2}}dx\)

\(\Big \downarrow \) 3196

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3251

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3251

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3252

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\int \frac {\sqrt {\sin (c+d x) a+a} \left (\sin (c+d x)^2+1\right )}{\sin (c+d x)^4}dx}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3523

\(\displaystyle \frac {\frac {\int \frac {1}{2} \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} (9 \sin (c+d x) a+a)dx}{3 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \csc ^3(c+d x) \sqrt {\sin (c+d x) a+a} (9 \sin (c+d x) a+a)dx}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin (c+d x) a+a} (9 \sin (c+d x) a+a)}{\sin (c+d x)^3}dx}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3459

\(\displaystyle \frac {\frac {\frac {39}{4} a \int \csc ^2(c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {39}{4} a \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)^2}dx-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3251

\(\displaystyle \frac {\frac {\frac {39}{4} a \left (\frac {1}{2} \int \csc (c+d x) \sqrt {\sin (c+d x) a+a}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {39}{4} a \left (\frac {1}{2} \int \frac {\sqrt {\sin (c+d x) a+a}}{\sin (c+d x)}dx-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 3252

\(\displaystyle \frac {\frac {\frac {39}{4} a \left (-\frac {a \int \frac {1}{a-\frac {a^2 \cos ^2(c+d x)}{\sin (c+d x) a+a}}d\frac {a \cos (c+d x)}{\sqrt {\sin (c+d x) a+a}}}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {39}{4} a \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a^2 \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}}{6 a}-\frac {\cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}}{a^2}-\frac {2 \left (\frac {3}{4} \left (-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {a \cot (c+d x)}{d \sqrt {a \sin (c+d x)+a}}\right )-\frac {a \cot (c+d x) \csc (c+d x)}{2 d \sqrt {a \sin (c+d x)+a}}\right )}{a^2}\)

Input:

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

Output:

(-2*(-1/2*(a*Cot[c + d*x]*Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (3* 
(-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - 
 (a*Cot[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]])))/4))/a^2 + (-1/3*(Cot[c + 
d*x]*Csc[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/d + (-1/2*(a^2*Cot[c + d*x]* 
Csc[c + d*x])/(d*Sqrt[a + a*Sin[c + d*x]]) + (39*a*(-((Sqrt[a]*ArcTanh[(Sq 
rt[a]*Cos[c + d*x])/Sqrt[a + a*Sin[c + d*x]]])/d) - (a*Cot[c + d*x])/(d*Sq 
rt[a + a*Sin[c + d*x]])))/4)/(6*a))/a^2
 

Defintions of rubi rules used

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

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

rule 3196

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

rule 3251

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e 
+ f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim 
p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2)))   Int[Sqrt[a + b*Sin[e 
+ f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x 
] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 
-1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
 

rule 3252

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

rule 3459

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

rule 3523

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (3 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}-8 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-3 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \sin \left (d x +c \right )^{3}\right )}{24 a^{\frac {11}{2}} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\)	\(144\)

Input:

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

Output:

1/24*(1+sin(d*x+c))*(-a*(sin(d*x+c)-1))^(1/2)*(3*(-a*(sin(d*x+c)-1))^(1/2) 
*a^(7/2)-8*(-a*(sin(d*x+c)-1))^(3/2)*a^(5/2)-3*(-a*(sin(d*x+c)-1))^(5/2)*a 
^(3/2)-3*arctanh((-a*(sin(d*x+c)-1))^(1/2)/a^(1/2))*a^4*sin(d*x+c)^3)/a^(1 
1/2)/sin(d*x+c)^3/cos(d*x+c)/(a+a*sin(d*x+c))^(1/2)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 383 vs. \(2 (124) = 248\).

Time = 0.10 (sec) , antiderivative size = 383, normalized size of antiderivative = 2.66 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {3 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (d x + c\right )^{3} + 17 \, \cos \left (d x + c\right )^{2} - {\left (3 \, \cos \left (d x + c\right )^{2} - 14 \, \cos \left (d x + c\right ) - 25\right )} \sin \left (d x + c\right ) - 11 \, \cos \left (d x + c\right ) - 25\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d - {\left (a^{2} d \cos \left (d x + c\right )^{3} + a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - a^{2} d\right )} \sin \left (d x + c\right )\right )}} \] Input:

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

Output:

1/96*(3*(cos(d*x + c)^4 - 2*cos(d*x + c)^2 - (cos(d*x + c)^3 + cos(d*x + c 
)^2 - cos(d*x + c) - 1)*sin(d*x + c) + 1)*sqrt(a)*log((a*cos(d*x + c)^3 - 
7*a*cos(d*x + c)^2 - 4*(cos(d*x + c)^2 + (cos(d*x + c) + 3)*sin(d*x + c) - 
 2*cos(d*x + c) - 3)*sqrt(a*sin(d*x + c) + a)*sqrt(a) - 9*a*cos(d*x + c) + 
 (a*cos(d*x + c)^2 + 8*a*cos(d*x + c) - a)*sin(d*x + c) - a)/(cos(d*x + c) 
^3 + cos(d*x + c)^2 + (cos(d*x + c)^2 - 1)*sin(d*x + c) - cos(d*x + c) - 1 
)) + 4*(3*cos(d*x + c)^3 + 17*cos(d*x + c)^2 - (3*cos(d*x + c)^2 - 14*cos( 
d*x + c) - 25)*sin(d*x + c) - 11*cos(d*x + c) - 25)*sqrt(a*sin(d*x + c) + 
a))/(a^2*d*cos(d*x + c)^4 - 2*a^2*d*cos(d*x + c)^2 + a^2*d - (a^2*d*cos(d* 
x + c)^3 + a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - a^2*d)*sin(d*x + c) 
)
 

Sympy [F]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral(cot(c + d*x)**4/(a*(sin(c + d*x) + 1))**(3/2), x)
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {4 \, {\left (12 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{96 \, d} \] Input:

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

Output:

-1/96*sqrt(2)*sqrt(a)*(3*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(-1/4*pi + 1/2* 
d*x + 1/2*c))/abs(2*sqrt(2) + 4*sin(-1/4*pi + 1/2*d*x + 1/2*c)))/(a^2*sgn( 
cos(-1/4*pi + 1/2*d*x + 1/2*c))) + 4*(12*sin(-1/4*pi + 1/2*d*x + 1/2*c)^5 
+ 16*sin(-1/4*pi + 1/2*d*x + 1/2*c)^3 - 3*sin(-1/4*pi + 1/2*d*x + 1/2*c))/ 
((2*sin(-1/4*pi + 1/2*d*x + 1/2*c)^2 - 1)^3*a^2*sgn(cos(-1/4*pi + 1/2*d*x 
+ 1/2*c))))/d
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\sin \left (d x +c \right )+1}\, \cot \left (d x +c \right )^{4}}{\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1}d x \right )}{a^{2}} \] Input:

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

Output:

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