Integrand size = 21, antiderivative size = 106 \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=-\frac {2^{-3+n} \operatorname {AppellF1}\left (-\frac {3}{2},-4+n,1,-\frac {1}{2},-\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac {a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \cot ^3(c+d x) \left (\frac {1}{1+\sec (c+d x)}\right )^{-3+n} (a+a \sec (c+d x))^n}{3 d} \] Output:
-1/3*2^(-3+n)*AppellF1(-3/2,1,-4+n,-1/2,(a-a*sec(d*x+c))/(a+a*sec(d*x+c)), -(a-a*sec(d*x+c))/(a+a*sec(d*x+c)))*cot(d*x+c)^3*(1/(1+sec(d*x+c)))^(-3+n) *(a+a*sec(d*x+c))^n/d
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx \] Input:
Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]
Output:
Integrate[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n, x]
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3042, 4377}
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.
\(\displaystyle \int \cot ^4(c+d x) (a \sec (c+d x)+a)^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^n}{\cot \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
\(\Big \downarrow \) 4377 |
\(\displaystyle -\frac {2^{n-3} \cot ^3(c+d x) \left (\frac {1}{\sec (c+d x)+1}\right )^{n-3} (a \sec (c+d x)+a)^n \operatorname {AppellF1}\left (-\frac {3}{2},n-4,1,-\frac {1}{2},-\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac {a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{3 d}\) |
Input:
Int[Cot[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]
Output:
-1/3*(2^(-3 + n)*AppellF1[-3/2, -4 + n, 1, -1/2, -((a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])), (a - a*Sec[c + d*x])/(a + a*Sec[c + d*x])]*Cot[c + d* x]^3*((1 + Sec[c + d*x])^(-1))^(-3 + n)*(a + a*Sec[c + d*x])^n)/d
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[(-2^(m + n + 1))*(e*Cot[c + d*x])^(m + 1)*((a + b*Csc[c + d*x])^n/(d*e*(m + 1)))*(a/(a + b*Csc[c + d*x]))^(m + n + 1)*App ellF1[(m + 1)/2, m + n, 1, (m + 3)/2, -(a - b*Csc[c + d*x])/(a + b*Csc[c + d*x]), (a - b*Csc[c + d*x])/(a + b*Csc[c + d*x])], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[n]
\[\int \cot \left (d x +c \right )^{4} \left (a +a \sec \left (d x +c \right )\right )^{n}d x\]
Input:
int(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x)
Output:
int(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x)
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="fricas")
Output:
integral((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*(a+a*sec(d*x+c))**n,x)
Output:
Timed out
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="maxima")
Output:
integrate((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x, algorithm="giac")
Output:
integrate((a*sec(d*x + c) + a)^n*cot(d*x + c)^4, x)
Timed out. \[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n \,d x \] Input:
int(cot(c + d*x)^4*(a + a/cos(c + d*x))^n,x)
Output:
int(cot(c + d*x)^4*(a + a/cos(c + d*x))^n, x)
\[ \int \cot ^4(c+d x) (a+a \sec (c+d x))^n \, dx=\int \cot \left (d x +c \right )^{4} \left (\sec \left (d x +c \right ) a +a \right )^{n}d x \] Input:
int(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x)
Output:
int(cot(d*x+c)^4*(a+a*sec(d*x+c))^n,x)