3.47 \(\int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx\)
Optimal. Leaf size=51 \[ -\frac{(d \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (3,\frac{n+1}{2},\frac{n+3}{2},-\cot ^2(e+f x)\right )}{d f (n+1)} \]
[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)))
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Rubi [A] time = 0.0479054, antiderivative size = 51, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used =
{2607, 364} \[ -\frac{(d \cot (e+f x))^{n+1} \, _2F_1\left (3,\frac{n+1}{2};\frac{n+3}{2};-\cot ^2(e+f x)\right )}{d f (n+1)} \]
Antiderivative was successfully verified.
[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 2607
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])
Rule 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rubi steps
\begin{align*} \int (d \cot (e+f x))^n \sin ^4(e+f x) \, dx &=\frac{\operatorname{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} \, _2F_1\left (3,\frac{1+n}{2};\frac{3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)}\\ \end{align*}
Mathematica [C] time = 7.33255, size = 1099, normalized size = 21.55 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x]^4,x]
[Out]
(2*(-3 + n)*(AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*AppellF1[1/2 -
n/2, -n, 4, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[1/2 - n/2, -n, 5, 3/2 - n/2, Tan[(
e + f*x)/2]^2, -Tan[(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]^4)/(f
*(-1 + n)*(-3*AppellF1[3/2 - n/2, -n, 4, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 8*AppellF1[3/2
- n/2, -n, 5, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 5*AppellF1[3/2 - n/2, -n, 6, 5/2 - n/2, Ta
n[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 3*AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e +
f*x)/2]^2]*Cos[(e + f*x)/2]^2 - n*AppellF1[1/2 - n/2, -n, 3, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]
^2]*Cos[(e + f*x)/2]^2 - 6*AppellF1[1/2 - n/2, -n, 4, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[
(e + f*x)/2]^2 + 2*n*AppellF1[1/2 - n/2, -n, 4, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f
*x)/2]^2 + 3*AppellF1[1/2 - n/2, -n, 5, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2
- n*AppellF1[1/2 - n/2, -n, 5, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 + n*App
ellF1[3/2 - n/2, 1 - n, 3, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + n*AppellF
1[3/2 - n/2, 1 - n, 5, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(-1 + Cos[e + f*x]) + 3*AppellF1[3/
2 - n/2, -n, 4, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] - 8*AppellF1[3/2 - n/2, -n, 5
, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 5*AppellF1[3/2 - n/2, -n, 6, 5/2 - n/2, T
an[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x] + 4*n*AppellF1[3/2 - n/2, 1 - n, 4, 5/2 - n/2, Tan[(e + f
*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^2))
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Maple [F] time = 1.15, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( \sin \left ( fx+e \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\right )} \left (d \cot \left (f x + e\right )\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*cot(f*x+e))**n*sin(f*x+e)**4,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)