3.361 \(\int \cot ^6(e+f x) (b \sec (e+f x))^m \, dx\)

Optimal. Leaf size=63 \[ -\frac {\cot ^5(e+f x) \cos ^2(e+f x)^{\frac {m-5}{2}} (b \sec (e+f x))^m \, _2F_1\left (-\frac {5}{2},\frac {m-5}{2};-\frac {3}{2};\sin ^2(e+f x)\right )}{5 f} \]

[Out]

-1/5*(cos(f*x+e)^2)^(-5/2+1/2*m)*cot(f*x+e)^5*hypergeom([-5/2, -5/2+1/2*m],[-3/2],sin(f*x+e)^2)*(b*sec(f*x+e))
^m/f

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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} \[ -\frac {\cot ^5(e+f x) \cos ^2(e+f x)^{\frac {m-5}{2}} (b \sec (e+f x))^m \, _2F_1\left (-\frac {5}{2},\frac {m-5}{2};-\frac {3}{2};\sin ^2(e+f x)\right )}{5 f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^6*(b*Sec[e + f*x])^m,x]

[Out]

-((Cos[e + f*x]^2)^((-5 + m)/2)*Cot[e + f*x]^5*Hypergeometric2F1[-5/2, (-5 + m)/2, -3/2, Sin[e + f*x]^2]*(b*Se
c[e + f*x])^m)/(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 ^6(e+f x) (b \sec (e+f x))^m \, dx &=-\frac {\cos ^2(e+f x)^{\frac {1}{2} (-5+m)} \cot ^5(e+f x) \, _2F_1\left (-\frac {5}{2},\frac {1}{2} (-5+m);-\frac {3}{2};\sin ^2(e+f x)\right ) (b \sec (e+f x))^m}{5 f}\\ \end {align*}

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Mathematica [F]  time = 0.60, size = 0, normalized size = 0.00 \[ \int \cot ^6(e+f x) (b \sec (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cot[e + f*x]^6*(b*Sec[e + f*x])^m,x]

[Out]

Integrate[Cot[e + f*x]^6*(b*Sec[e + f*x])^m, x]

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6*(b*sec(f*x+e))^m,x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e))^m*cot(f*x + e)^6, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6*(b*sec(f*x+e))^m,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e))^m*cot(f*x + e)^6, x)

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \left (\cot ^{6}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^6*(b*sec(f*x+e))^m,x)

[Out]

int(cot(f*x+e)^6*(b*sec(f*x+e))^m,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{6}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^6*(b*sec(f*x+e))^m,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e))^m*cot(f*x + e)^6, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^6\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(e + f*x)^6*(b/cos(e + f*x))^m,x)

[Out]

int(cot(e + f*x)^6*(b/cos(e + f*x))^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{6}{\left (e + f x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**6*(b*sec(f*x+e))**m,x)

[Out]

Integral((b*sec(e + f*x))**m*cot(e + f*x)**6, x)

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