Integrand size = 19, antiderivative size = 63 \[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=-\frac {\cot ^5(e+f x) (b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {5+m}{2},\frac {7}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {5+m}{2}}}{5 f} \] Output:
-1/5*cot(f*x+e)^5*(b*csc(f*x+e))^m*hypergeom([5/2, 5/2+1/2*m],[7/2],cos(f* x+e)^2)*(sin(f*x+e)^2)^(5/2+1/2*m)/f
Time = 0.48 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.68 \[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=-\frac {\cot (e+f x) (b \csc (e+f x))^m \left (\operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1+m}{2},\frac {3}{2},\cos ^2(e+f x)\right )-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3+m}{2},\frac {3}{2},\cos ^2(e+f x)\right )+\operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {5+m}{2},\frac {3}{2},\cos ^2(e+f x)\right )\right ) \sin ^2(e+f x)^{\frac {1+m}{2}}}{f} \] Input:
Integrate[Cot[e + f*x]^4*(b*Csc[e + f*x])^m,x]
Output:
-((Cot[e + f*x]*(b*Csc[e + f*x])^m*(Hypergeometric2F1[1/2, (1 + m)/2, 3/2, Cos[e + f*x]^2] - 2*Hypergeometric2F1[1/2, (3 + m)/2, 3/2, Cos[e + f*x]^2 ] + Hypergeometric2F1[1/2, (5 + m)/2, 3/2, Cos[e + f*x]^2])*(Sin[e + f*x]^ 2)^((1 + m)/2))/f)
Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3042, 3097}
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.
\(\displaystyle \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan \left (e+f x-\frac {\pi }{2}\right )^4 \left (b \sec \left (e+f x-\frac {\pi }{2}\right )\right )^mdx\) |
\(\Big \downarrow \) 3097 |
\(\displaystyle -\frac {\cot ^5(e+f x) \sin ^2(e+f x)^{\frac {m+5}{2}} (b \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+5}{2},\frac {7}{2},\cos ^2(e+f x)\right )}{5 f}\) |
Input:
Int[Cot[e + f*x]^4*(b*Csc[e + f*x])^m,x]
Output:
-1/5*(Cot[e + f*x]^5*(b*Csc[e + f*x])^m*Hypergeometric2F1[5/2, (5 + m)/2, 7/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((5 + m)/2))/f
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[(a*Sec[e + f*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[(n - 1)/2] && !IntegerQ[m/2]
\[\int \cot \left (f x +e \right )^{4} \left (b \csc \left (f x +e \right )\right )^{m}d x\]
Input:
int(cot(f*x+e)^4*(b*csc(f*x+e))^m,x)
Output:
int(cot(f*x+e)^4*(b*csc(f*x+e))^m,x)
\[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cot(f*x+e)^4*(b*csc(f*x+e))^m,x, algorithm="fricas")
Output:
integral((b*csc(f*x + e))^m*cot(f*x + e)^4, x)
\[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=\int \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \] Input:
integrate(cot(f*x+e)**4*(b*csc(f*x+e))**m,x)
Output:
Integral((b*csc(e + f*x))**m*cot(e + f*x)**4, x)
\[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cot(f*x+e)^4*(b*csc(f*x+e))^m,x, algorithm="maxima")
Output:
integrate((b*csc(f*x + e))^m*cot(f*x + e)^4, x)
\[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=\int { \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4} \,d x } \] Input:
integrate(cot(f*x+e)^4*(b*csc(f*x+e))^m,x, algorithm="giac")
Output:
integrate((b*csc(f*x + e))^m*cot(f*x + e)^4, x)
Timed out. \[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \] Input:
int(cot(e + f*x)^4*(b/sin(e + f*x))^m,x)
Output:
int(cot(e + f*x)^4*(b/sin(e + f*x))^m, x)
\[ \int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx=b^{m} \left (\int \csc \left (f x +e \right )^{m} \cot \left (f x +e \right )^{4}d x \right ) \] Input:
int(cot(f*x+e)^4*(b*csc(f*x+e))^m,x)
Output:
b**m*int(csc(e + f*x)**m*cot(e + f*x)**4,x)