3.2.55 \(\int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx\) [155]

3.2.55.1 Optimal result

Integrand size = 21, antiderivative size = 349 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\frac {\left (2-n+n^2\right ) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) \left (1-4 n^2\right ) (1-\cos (c+d x))^2}-\frac {a^4 \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d (3-2 n) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))^2}-\frac {a^3 (4-n) \cos (c+d x) (a+a \sec (c+d x))^n \sin (c+d x)}{d \left (3-8 n+4 n^2\right ) (a-a \cos (c+d x))^2 (a+a \cos (c+d x))}+\frac {n \left (7-3 n-n^2\right ) \cos (c+d x) \left (\frac {1+\cos (c+d x)}{1-\cos (c+d x)}\right )^{-\frac {1}{2}-n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-n,1-n,2-n,-\frac {2 \cos (c+d x)}{1-\cos (c+d x)}\right ) (a+a \sec (c+d x))^n \sin (c+d x)}{d (1-2 n) (3-2 n) (1-n) (1+2 n) (1-\cos (c+d x))^2} \]

output

(n^2-n+2)*cos(d*x+c)*(a+a*sec(d*x+c))^n*sin(d*x+c)/d/(3-2*n)/(-4*n^2+1)/(1 
-cos(d*x+c))^2-a^4*cos(d*x+c)*(a+a*sec(d*x+c))^n*sin(d*x+c)/d/(3-2*n)/(a-a 
*cos(d*x+c))^2/(a+a*cos(d*x+c))^2-a^3*(4-n)*cos(d*x+c)*(a+a*sec(d*x+c))^n* 
sin(d*x+c)/d/(4*n^2-8*n+3)/(a-a*cos(d*x+c))^2/(a+a*cos(d*x+c))+n*(-n^2-3*n 
+7)*cos(d*x+c)*((1+cos(d*x+c))/(1-cos(d*x+c)))^(-1/2-n)*hypergeom([1-n, -1 
/2-n],[2-n],-2*cos(d*x+c)/(1-cos(d*x+c)))*(a+a*sec(d*x+c))^n*sin(d*x+c)/d/ 
(-8*n^4+20*n^3-10*n^2-5*n+3)/(1-cos(d*x+c))^2
 

3.2.55.2 Mathematica [A] (verified)

Time = 4.57 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.00 \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\frac {(a (1+\sec (c+d x)))^n \left (-2 \cot ^2\left (\frac {1}{2} (c+d x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},n,\frac {1}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^n (1+\sec (c+d x))^{-n} \left (3\ 2^n \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n+2 (1+\sec (c+d x))^n+n (1+\sec (c+d x))^n\right )+\frac {-\cos (c+d x) (4 n \cos (c+d x)+(-3+n) (3+\cos (2 (c+d x)))) \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )+24 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},n,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^n (1+\sec (c+d x))^{-n} \left (-3 2^n \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n-2 (1+\sec (c+d x))^n+n \left (2^{1+n} \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n+(1+\sec (c+d x))^n\right )\right )}{4 (-3+2 n)}\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d} \]

input

Integrate[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]
 

output

((a*(1 + Sec[c + d*x]))^n*((-2*Cot[(c + d*x)/2]^2*Hypergeometric2F1[-1/2, 
n, 1/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(3*2^n*(Co 
s[(c + d*x)/2]^2*Sec[c + d*x])^n + 2*(1 + Sec[c + d*x])^n + n*(1 + Sec[c + 
 d*x])^n))/(1 + Sec[c + d*x])^n + (-(Cos[c + d*x]*(4*n*Cos[c + d*x] + (-3 
+ n)*(3 + Cos[2*(c + d*x)]))*Csc[(c + d*x)/2]^4*Sec[(c + d*x)/2]^2) + (24* 
Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + 
d*x)/2]^2)^n*(-3*2^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n - 2*(1 + Sec[c + 
d*x])^n + n*(2^(1 + n)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n + (1 + Sec[c + 
d*x])^n)))/(1 + Sec[c + d*x])^n)/(4*(-3 + 2*n)))*Tan[(c + d*x)/2])/(24*d)
 

3.2.55.3 Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.26, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4364, 3042, 3362, 144, 25, 27, 172, 27, 172, 27, 142}

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.

\(\displaystyle \int \csc ^4(c+d x) (a \sec (c+d x)+a)^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^n}{\cos \left (c+d x-\frac {\pi }{2}\right )^4}dx\)

\(\Big \downarrow \) 4364

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \int (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^n \csc ^4(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n} (a \sec (c+d x)+a)^n \int \frac {\left (-\sin \left (c+d x+\frac {\pi }{2}\right )\right )^{-n} \left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^n}{\cos \left (c+d x+\frac {\pi }{2}\right )^4}dx\)

\(\Big \downarrow \) 3362

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \int \frac {(-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {5}{2}}}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 144

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (\frac {\int -\frac {a^2 (-n-2 \cos (c+d x)+2) (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {3}{2}}}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a^3 (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\int \frac {a^2 (-n-2 \cos (c+d x)+2) (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {3}{2}}}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a^3 (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\int \frac {(-n-2 \cos (c+d x)+2) (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {3}{2}}}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 172

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\frac {\int \frac {a^2 (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {1}{2}} \left (-n^2+(4-n) \cos (c+d x)+2\right )}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a^3 (1-2 n)}-\frac {(4-n) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {1}{2}}}{a^2 (1-2 n) (a \cos (c+d x)-a)^{3/2}}}{a (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\frac {\int \frac {(-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n-\frac {1}{2}} \left (-n^2+(4-n) \cos (c+d x)+2\right )}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a (1-2 n)}-\frac {(4-n) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {1}{2}}}{a^2 (1-2 n) (a \cos (c+d x)-a)^{3/2}}}{a (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 172

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\frac {-\frac {\int \frac {a^2 n \left (-n^2-3 n+7\right ) (-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n+\frac {1}{2}}}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a^3 (2 n+1)}-\frac {\left (n^2-n+2\right ) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n+\frac {1}{2}}}{a^2 (2 n+1) (a \cos (c+d x)-a)^{3/2}}}{a (1-2 n)}-\frac {(4-n) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {1}{2}}}{a^2 (1-2 n) (a \cos (c+d x)-a)^{3/2}}}{a (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\frac {-\frac {n \left (-n^2-3 n+7\right ) \int \frac {(-\cos (c+d x))^{-n} (-\cos (c+d x) a-a)^{n+\frac {1}{2}}}{(a \cos (c+d x)-a)^{5/2}}d\cos (c+d x)}{a (2 n+1)}-\frac {\left (n^2-n+2\right ) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n+\frac {1}{2}}}{a^2 (2 n+1) (a \cos (c+d x)-a)^{3/2}}}{a (1-2 n)}-\frac {(4-n) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {1}{2}}}{a^2 (1-2 n) (a \cos (c+d x)-a)^{3/2}}}{a (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

\(\Big \downarrow \) 142

\(\displaystyle -\frac {a^6 \sin (c+d x) (-\cos (c+d x))^n (a (-\cos (c+d x))-a)^{-n-\frac {1}{2}} (a \sec (c+d x)+a)^n \left (-\frac {\frac {-\frac {n \left (-n^2-3 n+7\right ) (-\cos (c+d x))^{1-n} \left (\frac {\cos (c+d x)+1}{1-\cos (c+d x)}\right )^{-n-\frac {1}{2}} (a (-\cos (c+d x))-a)^{n+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (-n-\frac {1}{2},1-n,2-n,-\frac {2 \cos (c+d x)}{1-\cos (c+d x)}\right )}{a^2 (1-n) (2 n+1) (a \cos (c+d x)-a)^{3/2}}-\frac {\left (n^2-n+2\right ) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n+\frac {1}{2}}}{a^2 (2 n+1) (a \cos (c+d x)-a)^{3/2}}}{a (1-2 n)}-\frac {(4-n) (-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {1}{2}}}{a^2 (1-2 n) (a \cos (c+d x)-a)^{3/2}}}{a (3-2 n)}-\frac {(-\cos (c+d x))^{1-n} (a (-\cos (c+d x))-a)^{n-\frac {3}{2}}}{a^2 (3-2 n) (a \cos (c+d x)-a)^{3/2}}\right )}{d \sqrt {a \cos (c+d x)-a}}\)

input

Int[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]
 

output

-((a^6*(-Cos[c + d*x])^n*(-a - a*Cos[c + d*x])^(-1/2 - n)*(-(((-Cos[c + d* 
x])^(1 - n)*(-a - a*Cos[c + d*x])^(-3/2 + n))/(a^2*(3 - 2*n)*(-a + a*Cos[c 
 + d*x])^(3/2))) - (-(((4 - n)*(-Cos[c + d*x])^(1 - n)*(-a - a*Cos[c + d*x 
])^(-1/2 + n))/(a^2*(1 - 2*n)*(-a + a*Cos[c + d*x])^(3/2))) + (-(((2 - n + 
 n^2)*(-Cos[c + d*x])^(1 - n)*(-a - a*Cos[c + d*x])^(1/2 + n))/(a^2*(1 + 2 
*n)*(-a + a*Cos[c + d*x])^(3/2))) - (n*(7 - 3*n - n^2)*(-Cos[c + d*x])^(1 
- n)*((1 + Cos[c + d*x])/(1 - Cos[c + d*x]))^(-1/2 - n)*(-a - a*Cos[c + d* 
x])^(1/2 + n)*Hypergeometric2F1[-1/2 - n, 1 - n, 2 - n, (-2*Cos[c + d*x])/ 
(1 - Cos[c + d*x])])/(a^2*(1 - n)*(1 + 2*n)*(-a + a*Cos[c + d*x])^(3/2)))/ 
(a*(1 - 2*n)))/(a*(3 - 2*n)))*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*Sqrt 
[-a + a*Cos[c + d*x]]))
 

3.2.55.3.1 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 142

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((b*e 
 - a*f)*(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f))*((a + 
b*x)/((b*c - a*d)*(e + f*x)))])/((b*e - a*f)*((c + d*x)/((b*c - a*d)*(e + f 
*x))))^n, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m + n + p + 2, 
 0] &&  !IntegerQ[n]
 

rule 144

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[b*(a + b*x)^(m + 1)*( 
c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] 
+ Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) 
- b*d*f*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))] /; FreeQ[{a, b, c, d, e, f 
, m, n, p}, x] && NeQ[m, -1]
 

rule 172

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> With[{mnp = Simplify[m + n + p]}, Simp[ 
(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1) 
*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f 
)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g 
 - a*h)*(mnp + 3)*x, x], x], x] /; ILtQ[mnp + 2, 0] && (SumSimplerQ[m, 1] | 
| ( !(NeQ[n, -1] && SumSimplerQ[n, 1]) &&  !(NeQ[p, -1] && SumSimplerQ[p, 1 
])))] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && NeQ[m, -1]
 

rule 3042

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

rule 3362

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) 
 + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[Cos[e + f*x]/(a^( 
p - 2)*f*Sqrt[a + b*Sin[e + f*x]]*Sqrt[a - b*Sin[e + f*x]])   Subst[Int[(d* 
x)^n*(a + b*x)^(m + p/2 - 1/2)*(a - b*x)^(p/2 - 1/2), x], x, Sin[e + f*x]], 
 x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[p/2 
] &&  !IntegerQ[m]
 

rule 4364

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_), x_Symbol] :> Simp[Sin[e + f*x]^FracPart[m]*((a + b*Csc[e + f*x] 
)^FracPart[m]/(b + a*Sin[e + f*x])^FracPart[m])   Int[(g*Cos[e + f*x])^p*(( 
b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x], x] /; FreeQ[{a, b, e, f, g, m, p 
}, x] && (EqQ[a^2 - b^2, 0] || IntegersQ[2*m, p])
 

3.2.55.4 Maple [F]

\[\int \csc \left (d x +c \right )^{4} \left (a +a \sec \left (d x +c \right )\right )^{n}d x\]

input

int(csc(d*x+c)^4*(a+a*sec(d*x+c))^n,x)
 

output

int(csc(d*x+c)^4*(a+a*sec(d*x+c))^n,x)
 

3.2.55.5 Fricas [F]

\[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4} \,d x } \]

input

integrate(csc(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="fricas")
 

output

integral((a*sec(d*x + c) + a)^n*csc(d*x + c)^4, x)
 

3.2.55.6 Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\text {Timed out} \]

input

integrate(csc(d*x+c)**4*(a+a*sec(d*x+c))**n,x)
 

output

Timed out
 

3.2.55.7 Maxima [F]

\[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4} \,d x } \]

input

integrate(csc(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="maxima")
 

output

integrate((a*sec(d*x + c) + a)^n*csc(d*x + c)^4, x)
 

3.2.55.8 Giac [F]

\[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4} \,d x } \]

input

integrate(csc(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="giac")
 

output

integrate((a*sec(d*x + c) + a)^n*csc(d*x + c)^4, x)
 

3.2.55.9 Mupad [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^4} \,d x \]

input

int((a + a/cos(c + d*x))^n/sin(c + d*x)^4,x)
 

output

int((a + a/cos(c + d*x))^n/sin(c + d*x)^4, x)