\(\int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx\) [47]

Optimal result

Integrand size = 19, antiderivative size = 51 \[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=-\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (3,\frac {1+n}{2},\frac {3+n}{2},-\cot ^2(e+f x)\right )}{d f (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 51, 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 = {2687, 371} \[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=-\frac {(d \cot (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (3,\frac {n+1}{2},\frac {n+3}{2},-\cot ^2(e+f x)\right )}{d f (n+1)} \]

[In]

Int[(d*Cot[e + f*x])^n*Sin[e + f*x]^4,x]

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(-d x)^n}{\left (1+x^2\right )^3} \, dx,x,-\cot (e+f x)\right )}{f} \\ & = -\frac {(d \cot (e+f x))^{1+n} \operatorname {Hypergeometric2F1}\left (3,\frac {1+n}{2},\frac {3+n}{2},-\cot ^2(e+f x)\right )}{d f (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.16 \[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=\frac {(d \cot (e+f x))^n \operatorname {Hypergeometric2F1}\left (3,\frac {5}{2}-\frac {n}{2},\frac {7}{2}-\frac {n}{2},-\tan ^2(e+f x)\right ) \tan ^5(e+f x)}{f (5-n)} \]

[In]

Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x]^4,x]

[Out]

((d*Cot[e + f*x])^n*Hypergeometric2F1[3, 5/2 - n/2, 7/2 - n/2, -Tan[e + f*x]^2]*Tan[e + f*x]^5)/(f*(5 - n))

Maple [F]

\[\int \left (d \cot \left (f x +e \right )\right )^{n} \sin \left (f x +e \right )^{4}d x\]

[In]

int((d*cot(f*x+e))^n*sin(f*x+e)^4,x)

[Out]

int((d*cot(f*x+e))^n*sin(f*x+e)^4,x)

Fricas [F]

\[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4} \,d x } \]

[In]

integrate((d*cot(f*x+e))^n*sin(f*x+e)^4,x, algorithm="fricas")

[Out]

integral((cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1)*(d*cot(f*x + e))^n, x)

Sympy [F]

\[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{n} \sin ^{4}{\left (e + f x \right )}\, dx \]

[In]

integrate((d*cot(f*x+e))**n*sin(f*x+e)**4,x)

[Out]

Integral((d*cot(e + f*x))**n*sin(e + f*x)**4, x)

Maxima [F]

\[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4} \,d x } \]

[In]

integrate((d*cot(f*x+e))^n*sin(f*x+e)^4,x, algorithm="maxima")

[Out]

integrate((d*cot(f*x + e))^n*sin(f*x + e)^4, x)

Giac [F]

\[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4} \,d x } \]

[In]

integrate((d*cot(f*x+e))^n*sin(f*x+e)^4,x, algorithm="giac")

[Out]

integrate((d*cot(f*x + e))^n*sin(f*x + e)^4, x)

Mupad [F(-1)]

Timed out. \[ \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx=\int {\sin \left (e+f\,x\right )}^4\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]

[In]

int(sin(e + f*x)^4*(d*cot(e + f*x))^n,x)

[Out]

int(sin(e + f*x)^4*(d*cot(e + f*x))^n, x)