3.1.48 \(\int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx\) [48]

3.1.48.1 Optimal result

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)
 

3.1.48.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 12.60 (sec) , antiderivative size = 784, normalized size of antiderivative = 9.92 \[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=-\frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},-n,-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n}}{f (8+4 n)}+\frac {8 (-4+n) \cos ^6\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \csc ^2(e+f x) \left (n \operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-(-2+n) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \operatorname {Hypergeometric2F1}\left (-n,-\frac {n}{2},1-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )}{f (-2+n) n \left (-8 n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,1,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )-8 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,2,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^4\left (\frac {1}{2} (e+f x)\right )+(-4+n) \left (4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n-\operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sin ^2(e+f x)\right )\right )}+\frac {(d \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-n,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n} \tan ^2\left (\frac {1}{2} (e+f x)\right )}{f (8-4 n)}+\frac {(-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \cot (e+f x))^n \sin ^2\left (\frac {1}{2} (e+f x)\right )}{f (4-2 n) \left ((-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,1,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 \left (n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,1,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+\operatorname {AppellF1}\left (2-\frac {n}{2},-n,2,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]

input

Integrate[(d*Cot[e + f*x])^n*Csc[e + f*x]^3,x]
 

output

-((Cot[(e + f*x)/2]^2*(d*Cot[e + f*x])^n*Hypergeometric2F1[-1 - n/2, -n, - 
1/2*n, Tan[(e + f*x)/2]^2])/(f*(8 + 4*n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2) 
^n)) + (8*(-4 + n)*Cos[(e + f*x)/2]^6*(d*Cot[e + f*x])^n*Csc[e + f*x]^2*(n 
*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2 
] - (-2 + n)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-n, -1/2*n, 1 - n/2, Tan 
[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^4)/(f*(-2 + n)*n*(-8*n*AppellF1[2 - n/2 
, 1 - n, 1, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x 
)/2]^4 - 8*AppellF1[2 - n/2, -n, 2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + 
 f*x)/2]^2]*Sin[(e + f*x)/2]^4 + (-4 + n)*(4*Cos[(e + f*x)/2]^4*(Cos[e + f 
*x]*Sec[(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]*Sin[e + f*x]^2))) + ((d*Cot[e + f*x])^n*Hyper 
geometric2F1[1 - n/2, -n, 2 - n/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*x)/2]^2) 
/(f*(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]*(d*Cot[e + 
 f*x])^n*Sin[(e + f*x)/2]^2)/(f*(4 - 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))
 

3.1.48.3 Rubi [A] (verified)

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)))
 

3.1.48.3.1 Defintions of rubi rules used

rule 3042

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

rule 3097

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]
 

3.1.48.4 Maple [F]

\[\int \left (d \cot \left (f x +e \right )\right )^{n} \csc \left (f x +e \right )^{3}d x\]

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)
 

3.1.48.5 Fricas [F]

\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \]

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)
 

3.1.48.6 Sympy [F]

\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{n} \csc ^{3}{\left (e + f x \right )}\, dx \]

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)
 

3.1.48.7 Maxima [F]

\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \]

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)
 

3.1.48.8 Giac [F]

\[ \int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3} \,d x } \]

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)
 

3.1.48.9 Mupad [F(-1)]

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)