3.382 \(\int \cot ^4(e+f x) (b \csc (e+f x))^m \, dx\)

Optimal. Leaf size=63 \[ -\frac {\cot ^5(e+f x) \sin ^2(e+f x)^{\frac {m+5}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {7}{2};\cos ^2(e+f x)\right )}{5 f} \]

[Out]

-1/5*cot(f*x+e)^5*(b*csc(f*x+e))^m*hypergeom([5/2, 5/2+1/2*m],[7/2],cos(f*x+e)^2)*(sin(f*x+e)^2)^(5/2+1/2*m)/f

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 63, 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} \[ -\frac {\cot ^5(e+f x) \sin ^2(e+f x)^{\frac {m+5}{2}} (b \csc (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {m+5}{2};\frac {7}{2};\cos ^2(e+f x)\right )}{5 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^4*(b*Csc[e + f*x])^m,x]

[Out]

-(Cot[e + f*x]^5*(b*Csc[e + f*x])^m*Hypergeometric2F1[5/2, (5 + m)/2, 7/2, Cos[e + f*x]^2]*(Sin[e + f*x]^2)^((
5 + m)/2))/(5*f)

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 \cot ^4(e+f x) (b \csc (e+f x))^m \, dx &=-\frac {\cot ^5(e+f x) (b \csc (e+f x))^m \, _2F_1\left (\frac {5}{2},\frac {5+m}{2};\frac {7}{2};\cos ^2(e+f x)\right ) \sin ^2(e+f x)^{\frac {5+m}{2}}}{5 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.26, size = 106, normalized size = 1.68 \[ -\frac {\cot (e+f x) \sin ^2(e+f x)^{\frac {m+1}{2}} (b \csc (e+f x))^m \left (\, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {3}{2};\cos ^2(e+f x)\right )-2 \, _2F_1\left (\frac {1}{2},\frac {m+3}{2};\frac {3}{2};\cos ^2(e+f x)\right )+\, _2F_1\left (\frac {1}{2},\frac {m+5}{2};\frac {3}{2};\cos ^2(e+f x)\right )\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^4*(b*Csc[e + f*x])^m,x]

[Out]

-((Cot[e + f*x]*(b*Csc[e + f*x])^m*(Hypergeometric2F1[1/2, (1 + m)/2, 3/2, Cos[e + f*x]^2] - 2*Hypergeometric2
F1[1/2, (3 + m)/2, 3/2, Cos[e + f*x]^2] + Hypergeometric2F1[1/2, (5 + m)/2, 3/2, Cos[e + f*x]^2])*(Sin[e + f*x
]^2)^((1 + m)/2))/f)

________________________________________________________________________________________

fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(b*csc(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*csc(f*x + e))^m*cot(f*x + e)^4, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(b*csc(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e)^4, x)

________________________________________________________________________________________

maple [F]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{4}\left (f x +e \right )\right ) \left (b \csc \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^4*(b*csc(f*x+e))^m,x)

[Out]

int(cot(f*x+e)^4*(b*csc(f*x+e))^m,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^4*(b*csc(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*csc(f*x + e))^m*cot(f*x + e)^4, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (\frac {b}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^4*(b/sin(e + f*x))^m,x)

[Out]

int(cot(e + f*x)^4*(b/sin(e + f*x))^m, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \csc {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**4*(b*csc(f*x+e))**m,x)

[Out]

Integral((b*csc(e + f*x))**m*cot(e + f*x)**4, x)

________________________________________________________________________________________