Integrand size = 19, antiderivative size = 79 \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2},\frac {4+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {4+n}{2}}}{d f (1+n)} \] Output:
-(d*cot(f*x+e))^(1+n)*csc(f*x+e)^3*hypergeom([2+1/2*n, 1/2+1/2*n],[3/2+1/2 *n],cos(f*x+e)^2)*(sin(f*x+e)^2)^(2+1/2*n)/d/f/(1+n)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 7.81 (sec) , antiderivative size = 741, normalized size of antiderivative = 9.38 \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx =\text {Too large to display} \] Input:
Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^3,x]
Output:
((d*Cot[e + f*x])^n*(-1/2*(Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1 - n/2, -n, -1/2*n, Tan[(e + f*x)/2]^2])/((2 + n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2 )^n) + (2*Hypergeometric2F1[1 - n/2, -n, 2 - n/2, Tan[(e + f*x)/2]^2]*Tan[ (e + f*x)/2]^2)/((8 - 4*n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) - ((-4 + n )*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^ 2]*Sin[(e + f*x)/2]^2)/((-2 + n)*((-4 + n)*AppellF1[1 - n/2, -n, 1, 2 - n/ 2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*(n*AppellF1[2 - n/2, 1 - n , 1, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[2 - n/2, -n, 2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2 ]^2)) + ((-4 + n)*Cos[(e + f*x)/2]^2*((-2 + n)*Hypergeometric2F1[-n, -1/2* n, 1 - n/2, Tan[(e + f*x)/2]^2] - n*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[ (e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2))/((-2 + n)*n*(4*( Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n - n*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n + (-4 + n)*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2 + 2*n*AppellF1[2 - n/2, 1 - n, 1, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4 + 2*AppellF1[ 2 - n/2, -n, 2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^4))))/(2*f)
Time = 0.24 (sec) , antiderivative size = 79, 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.
\(\displaystyle \int \csc ^3(e+f x) (d \cot (e+f x))^n \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sec \left (e+f x-\frac {\pi }{2}\right )^3 \left (-d \tan \left (e+f x-\frac {\pi }{2}\right )\right )^ndx\) |
\(\Big \downarrow \) 3097 |
\(\displaystyle -\frac {\csc ^3(e+f x) \sin ^2(e+f x)^{\frac {n+4}{2}} (d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2},\frac {n+4}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)}\) |
Input:
Int[(d*Cot[e + f*x])^n*Csc[e + f*x]^3,x]
Output:
-(((d*Cot[e + f*x])^(1 + n)*Csc[e + f*x]^3*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((4 + n)/2))/(d*f*(1 + n)))
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]
\[\int \left (d \cot \left (f x +e \right )\right )^{n} \csc \left (f x +e \right )^{3}d x\]
Input:
int((d*cot(f*x+e))^n*csc(f*x+e)^3,x)
Output:
int((d*cot(f*x+e))^n*csc(f*x+e)^3,x)
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \] Input:
integrate((d*cot(f*x+e))^n*csc(f*x+e)^3,x, algorithm="fricas")
Output:
integral((d*cot(f*x + e))^n*csc(f*x + e)^3, x)
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{n} \csc ^{3}{\left (e + f x \right )}\, dx \] Input:
integrate((d*cot(f*x+e))**n*csc(f*x+e)**3,x)
Output:
Integral((d*cot(e + f*x))**n*csc(e + f*x)**3, x)
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \] Input:
integrate((d*cot(f*x+e))^n*csc(f*x+e)^3,x, algorithm="maxima")
Output:
integrate((d*cot(f*x + e))^n*csc(f*x + e)^3, x)
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \] Input:
integrate((d*cot(f*x+e))^n*csc(f*x+e)^3,x, algorithm="giac")
Output:
integrate((d*cot(f*x + e))^n*csc(f*x + e)^3, x)
Timed out. \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int \frac {{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^3} \,d x \] Input:
int((d*cot(e + f*x))^n/sin(e + f*x)^3,x)
Output:
int((d*cot(e + f*x))^n/sin(e + f*x)^3, x)
\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=d^{n} \left (\int \cot \left (f x +e \right )^{n} \csc \left (f x +e \right )^{3}d x \right ) \] Input:
int((d*cot(f*x+e))^n*csc(f*x+e)^3,x)
Output:
d**n*int(cot(e + f*x)**n*csc(e + f*x)**3,x)