Integrand size = 19, antiderivative size = 40 \[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (3,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \] Output:
-hypergeom([3, 1/2*m],[1+1/2*m],sec(f*x+e)^2)*(b*sec(f*x+e))^m/f/m
Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (3,\frac {m}{2},\frac {2+m}{2},\sec ^2(e+f x)\right ) (b \sec (e+f x))^m}{f m} \] Input:
Integrate[Cot[e + f*x]^5*(b*Sec[e + f*x])^m,x]
Output:
-((Hypergeometric2F1[3, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m )/(f*m))
Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3086, 25, 278}
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.
\(\displaystyle \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(b \sec (e+f x))^m}{\tan (e+f x)^5}dx\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {b \int -\frac {(b \sec (e+f x))^{m-1}}{\left (1-\sec ^2(e+f x)\right )^3}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {b \int \frac {(b \sec (e+f x))^{m-1}}{\left (1-\sec ^2(e+f x)\right )^3}d\sec (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {(b \sec (e+f x))^m \operatorname {Hypergeometric2F1}\left (3,\frac {m}{2},\frac {m+2}{2},\sec ^2(e+f x)\right )}{f m}\) |
Input:
Int[Cot[e + f*x]^5*(b*Sec[e + f*x])^m,x]
Output:
-((Hypergeometric2F1[3, m/2, (2 + m)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^m )/(f*m))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
\[\int \cot \left (f x +e \right )^{5} \left (b \sec \left (f x +e \right )\right )^{m}d x\]
Input:
int(cot(f*x+e)^5*(b*sec(f*x+e))^m,x)
Output:
int(cot(f*x+e)^5*(b*sec(f*x+e))^m,x)
\[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5} \,d x } \] Input:
integrate(cot(f*x+e)^5*(b*sec(f*x+e))^m,x, algorithm="fricas")
Output:
integral((b*sec(f*x + e))^m*cot(f*x + e)^5, x)
\[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=\int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{5}{\left (e + f x \right )}\, dx \] Input:
integrate(cot(f*x+e)**5*(b*sec(f*x+e))**m,x)
Output:
Integral((b*sec(e + f*x))**m*cot(e + f*x)**5, x)
\[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5} \,d x } \] Input:
integrate(cot(f*x+e)^5*(b*sec(f*x+e))^m,x, algorithm="maxima")
Output:
integrate((b*sec(f*x + e))^m*cot(f*x + e)^5, x)
\[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=\int { \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{5} \,d x } \] Input:
integrate(cot(f*x+e)^5*(b*sec(f*x+e))^m,x, algorithm="giac")
Output:
integrate((b*sec(f*x + e))^m*cot(f*x + e)^5, x)
Timed out. \[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^5\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \] Input:
int(cot(e + f*x)^5*(b/cos(e + f*x))^m,x)
Output:
int(cot(e + f*x)^5*(b/cos(e + f*x))^m, x)
\[ \int \cot ^5(e+f x) (b \sec (e+f x))^m \, dx=b^{m} \left (\int \sec \left (f x +e \right )^{m} \cot \left (f x +e \right )^{5}d x \right ) \] Input:
int(cot(f*x+e)^5*(b*sec(f*x+e))^m,x)
Output:
b**m*int(sec(e + f*x)**m*cot(e + f*x)**5,x)