\(\int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx\) [51]

Optimal result

Integrand size = 19, antiderivative size = 79 \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2+n),\frac {1+n}{2},\frac {3+n}{2},\cos ^2(e+f x)\right ) \sin ^3(e+f x) \sin ^2(e+f x)^{\frac {1}{2} (-2+n)}}{d f (1+n)} \] Output:

-(d*cot(f*x+e))^(1+n)*hypergeom([-1+1/2*n, 1/2+1/2*n],[3/2+1/2*n],cos(f*x+ 
e)^2)*sin(f*x+e)^3*(sin(f*x+e)^2)^(-1+1/2*n)/d/f/(1+n)
 

Mathematica [C] (warning: unable to verify)

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

Time = 1.98 (sec) , antiderivative size = 477, normalized size of antiderivative = 6.04 \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=-\frac {4 (-4+n) \left (\operatorname {AppellF1}\left (1-\frac {n}{2},-n,3,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-\operatorname {AppellF1}\left (1-\frac {n}{2},-n,4,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^3(e+f x)}{f (-2+n) \left (2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,3,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 (-4+n) \operatorname {AppellF1}\left (1-\frac {n}{2},-n,4,2-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \cos ^2\left (\frac {1}{2} (e+f x)\right )-2 \left (n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,3,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n \operatorname {AppellF1}\left (2-\frac {n}{2},1-n,4,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+3 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,4,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-4 \operatorname {AppellF1}\left (2-\frac {n}{2},-n,5,3-\frac {n}{2},\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (-1+\cos (e+f x))\right )} \] Input:

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

Output:

(-4*(-4 + n)*(AppellF1[1 - n/2, -n, 3, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[( 
e + f*x)/2]^2] - AppellF1[1 - n/2, -n, 4, 2 - n/2, Tan[(e + f*x)/2]^2, -Ta 
n[(e + f*x)/2]^2])*Cos[(e + f*x)/2]^3*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]* 
Sin[e + f*x]^3)/(f*(-2 + n)*(2*(-4 + n)*AppellF1[1 - n/2, -n, 3, 2 - n/2, 
Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 2*(-4 + n)*A 
ppellF1[1 - n/2, -n, 4, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]* 
Cos[(e + f*x)/2]^2 - 2*(n*AppellF1[2 - n/2, 1 - n, 3, 3 - n/2, Tan[(e + f* 
x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[2 - n/2, 1 - n, 4, 3 - n/2, Tan 
[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 3*AppellF1[2 - n/2, -n, 4, 3 - n/2 
, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 4*AppellF1[2 - n/2, -n, 5, 3 
- n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))
 

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.

\(\displaystyle \int \sin ^3(e+f x) (d \cot (e+f x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (-d \tan \left (e+f x-\frac {\pi }{2}\right )\right )^n}{\sec \left (e+f x-\frac {\pi }{2}\right )^3}dx\)

\(\Big \downarrow \) 3097

\(\displaystyle -\frac {\sin ^3(e+f x) \sin ^2(e+f x)^{\frac {n-2}{2}} (d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (\frac {n-2}{2},\frac {n+1}{2},\frac {n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)}\)

Input:

Int[(d*Cot[e + f*x])^n*Sin[e + f*x]^3,x]
 

Output:

-(((d*Cot[e + f*x])^(1 + n)*Hypergeometric2F1[(-2 + n)/2, (1 + n)/2, (3 + 
n)/2, Cos[e + f*x]^2]*Sin[e + f*x]^3*(Sin[e + f*x]^2)^((-2 + n)/2))/(d*f*( 
1 + n)))
 

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]
 

Maple [F]

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

Input:

int((d*cot(f*x+e))^n*sin(f*x+e)^3,x)
 

Output:

int((d*cot(f*x+e))^n*sin(f*x+e)^3,x)
 

Fricas [F]

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

integrate((d*cot(f*x+e))^n*sin(f*x+e)^3,x, algorithm="fricas")
 

Output:

integral(-(cos(f*x + e)^2 - 1)*(d*cot(f*x + e))^n*sin(f*x + e), x)
 

Sympy [F(-1)]

Timed out. \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\text {Timed out} \] Input:

integrate((d*cot(f*x+e))**n*sin(f*x+e)**3,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((d*cot(f*x+e))^n*sin(f*x+e)^3,x, algorithm="maxima")
 

Output:

integrate((d*cot(f*x + e))^n*sin(f*x + e)^3, x)
 

Giac [F]

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

integrate((d*cot(f*x+e))^n*sin(f*x+e)^3,x, algorithm="giac")
 

Output:

integrate((d*cot(f*x + e))^n*sin(f*x + e)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d \cot (e+f x))^n \sin ^3(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^3\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \] Input:

int(sin(e + f*x)^3*(d*cot(e + f*x))^n,x)
 

Output:

int(sin(e + f*x)^3*(d*cot(e + f*x))^n, x)
 

Reduce [F]

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

int((d*cot(f*x+e))^n*sin(f*x+e)^3,x)
 

Output:

d**n*int(cot(e + f*x)**n*sin(e + f*x)**3,x)