3.46 \(\int (d \cot (e+f x))^n \sin ^2(e+f x) \, dx\)

Optimal. Leaf size=51 \[ -\frac {(d \cot (e+f x))^{n+1} \, _2F_1\left (2,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(e+f x)\right )}{d f (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \[ -\frac {(d \cot (e+f x))^{n+1} \, _2F_1\left (2,\frac {n+1}{2};\frac {n+3}{2};-\cot ^2(e+f x)\right )}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cot[e + f*x])^n*Sin[e + f*x]^2,x]

[Out]

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

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])

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])

Rubi steps

\begin {align*} \int (d \cot (e+f x))^n \sin ^2(e+f x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(-d x)^n}{\left (1+x^2\right )^2} \, dx,x,-\cot (e+f x)\right )}{f}\\ &=-\frac {(d \cot (e+f x))^{1+n} \, _2F_1\left (2,\frac {1+n}{2};\frac {3+n}{2};-\cot ^2(e+f x)\right )}{d f (1+n)}\\ \end {align*}

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Mathematica [C]  time = 3.08, size = 509, normalized size = 9.98 \[ -\frac {4 (n-3) \sin \left (\frac {1}{2} (e+f x)\right ) \sin ^2(e+f x) \cos ^3\left (\frac {1}{2} (e+f x)\right ) \left (F_1\left (\frac {1}{2}-\frac {n}{2};-n,2;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-F_1\left (\frac {1}{2}-\frac {n}{2};-n,3;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) (d \cot (e+f x))^n}{f (n-1) \left (2 (n-3) \cos ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (\frac {1}{2}-\frac {n}{2};-n,2;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 (n-3) \cos ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (\frac {1}{2}-\frac {n}{2};-n,3;\frac {3}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-2 (\cos (e+f x)-1) \left (n F_1\left (\frac {3}{2}-\frac {n}{2};1-n,2;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-n F_1\left (\frac {3}{2}-\frac {n}{2};1-n,3;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 F_1\left (\frac {3}{2}-\frac {n}{2};-n,3;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-3 F_1\left (\frac {3}{2}-\frac {n}{2};-n,4;\frac {5}{2}-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d*Cot[e + f*x])^n*Sin[e + f*x]^2,x]

[Out]

(-4*(-3 + n)*(AppellF1[1/2 - n/2, -n, 2, 3/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 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]^3*(d*Cot[e + f*x])^n*Sin[(e
+ f*x)/2]*Sin[e + f*x]^2)/(f*(-1 + n)*(2*(-3 + n)*AppellF1[1/2 - n/2, -n, 2, 3/2 - n/2, Tan[(e + f*x)/2]^2, -T
an[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2 - 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]*Cos[(e + f*x)/2]^2 - 2*(n*AppellF1[3/2 - n/2, 1 - n, 2, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Ta
n[(e + f*x)/2]^2] - n*AppellF1[3/2 - n/2, 1 - n, 3, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*Ap
pellF1[3/2 - n/2, -n, 3, 5/2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 3*AppellF1[3/2 - n/2, -n, 4, 5/
2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))

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fricas [F]  time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (\cos \left (f x + e\right )^{2} - 1\right )} \left (d \cot \left (f x + e\right )\right )^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*sin(f*x+e)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*sin(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((d*cot(f*x + e))^n*sin(f*x + e)^2, x)

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maple [F]  time = 4.19, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{n} \left (\sin ^{2}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cot(f*x+e))^n*sin(f*x+e)^2,x)

[Out]

int((d*cot(f*x+e))^n*sin(f*x+e)^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))^n*sin(f*x+e)^2,x, algorithm="maxima")

[Out]

integrate((d*cot(f*x + e))^n*sin(f*x + e)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\sin \left (e+f\,x\right )}^2\,{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^2*(d*cot(e + f*x))^n,x)

[Out]

int(sin(e + f*x)^2*(d*cot(e + f*x))^n, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot {\left (e + f x \right )}\right )^{n} \sin ^{2}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cot(f*x+e))**n*sin(f*x+e)**2,x)

[Out]

Integral((d*cot(e + f*x))**n*sin(e + f*x)**2, x)

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