3.48 \(\int (d \cot (e+f x))^n \csc ^3(e+f x) \, dx\)

Optimal. Leaf size=79 \[ -\frac {\csc ^3(e+f x) \sin ^2(e+f x)^{\frac {n+4}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+4}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 79, normalized size of antiderivative = 1.00, 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} \[ -\frac {\csc ^3(e+f x) \sin ^2(e+f x)^{\frac {n+4}{2}} (d \cot (e+f x))^{n+1} \, _2F_1\left (\frac {n+1}{2},\frac {n+4}{2};\frac {n+3}{2};\cos ^2(e+f x)\right )}{d f (n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((d*Cot[e + f*x])^(1 + n)*Csc[e + f*x]^3*Hypergeometric2F1[(1 + n)/2, (4 + n)/2, (3 + n)/2, Cos[e + f*x]^2]*
(Sin[e + f*x]^2)^((4 + 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 \csc ^3(e+f x) \, dx &=-\frac {(d \cot (e+f x))^{1+n} \csc ^3(e+f x) \, _2F_1\left (\frac {1+n}{2},\frac {4+n}{2};\frac {3+n}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {4+n}{2}}}{d f (1+n)}\\ \end {align*}

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Mathematica [C]  time = 14.08, size = 784, normalized size = 9.92 \[ \frac {8 (n-4) \sin ^4\left (\frac {1}{2} (e+f x)\right ) \cos ^6\left (\frac {1}{2} (e+f x)\right ) \csc ^2(e+f x) (d \cot (e+f x))^n \left (n F_1\left (1-\frac {n}{2};-n,1;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-2) \cot ^2\left (\frac {1}{2} (e+f x)\right ) \, _2F_1\left (-n,-\frac {n}{2};1-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )}{f (n-2) n \left (-8 n \sin ^4\left (\frac {1}{2} (e+f x)\right ) F_1\left (2-\frac {n}{2};1-n,1;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 )-8 \sin ^4\left (\frac {1}{2} (e+f x)\right ) F_1\left (2-\frac {n}{2};-n,2;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-4) \left (4 \cos ^4\left (\frac {1}{2} (e+f x)\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^n-\sin ^2(e+f x) F_1\left (1-\frac {n}{2};-n,1;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 )}+\frac {(n-4) \sin ^2\left (\frac {1}{2} (e+f x)\right ) F_1\left (1-\frac {n}{2};-n,1;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 ) (d \cot (e+f x))^n}{f (4-2 n) \left (2 \tan ^2\left (\frac {1}{2} (e+f x)\right ) \left (n F_1\left (2-\frac {n}{2};1-n,1;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 )+F_1\left (2-\frac {n}{2};-n,2;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 )+(n-4) F_1\left (1-\frac {n}{2};-n,1;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 )}-\frac {\cot ^2\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \, _2F_1\left (-\frac {n}{2}-1,-n;-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n}}{f (4 n+8)}+\frac {\tan ^2\left (\frac {1}{2} (e+f x)\right ) (d \cot (e+f x))^n \, _2F_1\left (1-\frac {n}{2},-n;2-\frac {n}{2};\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos (e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )^{-n}}{f (8-4 n)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((Cot[(e + f*x)/2]^2*(d*Cot[e + f*x])^n*Hypergeometric2F1[-1 - n/2, -n, -1/2*n, Tan[(e + f*x)/2]^2])/(f*(8 +
4*n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n)) + (8*(-4 + n)*Cos[(e + f*x)/2]^6*(d*Cot[e + f*x])^n*Csc[e + f*x]^2*
(n*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (-2 + n)*Cot[(e + f*x)/2]^2*Hy
pergeometric2F1[-n, -1/2*n, 1 - n/2, Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^4)/(f*(-2 + n)*n*(-8*n*AppellF1[2 -
n/2, 1 - n, 1, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^4 - 8*AppellF1[2 - n/2, -n,
2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^4 + (-4 + n)*(4*Cos[(e + f*x)/2]^4*(Cos
[e + f*x]*Sec[(e + f*x)/2]^2)^n - AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*S
in[e + f*x]^2))) + ((d*Cot[e + f*x])^n*Hypergeometric2F1[1 - n/2, -n, 2 - n/2, Tan[(e + f*x)/2]^2]*Tan[(e + f*
x)/2]^2)/(f*(8 - 4*n)*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) + ((-4 + n)*AppellF1[1 - n/2, -n, 1, 2 - n/2, Tan[(
e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(d*Cot[e + f*x])^n*Sin[(e + f*x)/2]^2)/(f*(4 - 2*n)*((-4 + n)*AppellF1[1 -
n/2, -n, 1, 2 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*(n*AppellF1[2 - n/2, 1 - n, 1, 3 - n/2, Tan
[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[2 - n/2, -n, 2, 3 - n/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2
]^2])*Tan[(e + f*x)/2]^2))

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*cot(f*x + e))^n*csc(f*x + e)^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.64, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x +e \right )\right )^{n} \left (\csc ^{3}\left (f x +e \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot \left (f x + e\right )\right )^{n} \csc \left (f x + e\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^n}{{\sin \left (e+f\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cot {\left (e + f x \right )}\right )^{n} \csc ^{3}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d*cot(e + f*x))**n*csc(e + f*x)**3, x)

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