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

Optimal. Leaf size=79 \[ -\frac{\sin ^3(e+f x) \sin ^2(e+f x)^{\frac{n-2}{2}} (d \cot (e+f x))^{n+1} \text{Hypergeometric2F1}\left (\frac{n-2}{2},\frac{n+1}{2},\frac{n+3}{2},\cos ^2(e+f x)\right )}{d f (n+1)} \]

[Out]

-(((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)))

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Rubi [A]  time = 0.0424696, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2617} \[ -\frac{\sin ^3(e+f x) \sin ^2(e+f x)^{\frac{n-2}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac{n-2}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((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)))

Rule 2617

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)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2,
(n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps

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

Mathematica [C]  time = 2.30768, size = 477, normalized size = 6.04 \[ -\frac{4 (n-4) \sin \left (\frac{1}{2} (e+f x)\right ) \sin ^3(e+f x) \cos ^3\left (\frac{1}{2} (e+f x)\right ) \left (F_1\left (1-\frac{n}{2};-n,3;2-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-F_1\left (1-\frac{n}{2};-n,4;2-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right ) (d \cot (e+f x))^n}{f (n-2) \left (2 (n-4) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (1-\frac{n}{2};-n,3;2-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 (n-4) \cos ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (1-\frac{n}{2};-n,4;2-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 (\cos (e+f x)-1) \left (n F_1\left (2-\frac{n}{2};1-n,3;3-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-n F_1\left (2-\frac{n}{2};1-n,4;3-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+3 F_1\left (2-\frac{n}{2};-n,4;3-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-4 F_1\left (2-\frac{n}{2};-n,5;3-\frac{n}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-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, -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]^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)*AppellF1[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, -Ta
n[(e + f*x)/2]^2])*(-1 + Cos[e + f*x])))

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Maple [F]  time = 1.074, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cot \left ( fx+e \right ) \right ) ^{n} \left ( \sin \left ( fx+e \right ) \right ) ^{3}\, 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)^3,x)

[Out]

int((d*cot(f*x+e))^n*sin(f*x+e)^3,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 )^{3}\,{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)^3,x, algorithm="maxima")

[Out]

integrate((d*cot(f*x + e))^n*sin(f*x + e)^3, 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 )^{2} - 1\right )} \left (d \cot \left (f x + e\right )\right )^{n} \sin \left (f x + e\right ), 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)^3,x, algorithm="fricas")

[Out]

integral(-(cos(f*x + e)^2 - 1)*(d*cot(f*x + e))^n*sin(f*x + e), 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)**3,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 )^{3}\,{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)^3,x, algorithm="giac")

[Out]

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