3.360 \(\int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx\)

Optimal. Leaf size=63 \[ -\frac{\cot ^3(e+f x) \cos ^2(e+f x)^{\frac{m-3}{2}} (b \sec (e+f x))^m \, _2F_1\left (-\frac{3}{2},\frac{m-3}{2};-\frac{1}{2};\sin ^2(e+f x)\right )}{3 f} \]

[Out]

-((Cos[e + f*x]^2)^((-3 + m)/2)*Cot[e + f*x]^3*Hypergeometric2F1[-3/2, (-3 + m)/2, -1/2, Sin[e + f*x]^2]*(b*Se
c[e + f*x])^m)/(3*f)

________________________________________________________________________________________

Rubi [A]  time = 0.0373571, antiderivative size = 63, 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} \[ -\frac{\cot ^3(e+f x) \cos ^2(e+f x)^{\frac{m-3}{2}} (b \sec (e+f x))^m \, _2F_1\left (-\frac{3}{2},\frac{m-3}{2};-\frac{1}{2};\sin ^2(e+f x)\right )}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [C]  time = 25.1862, size = 6671, normalized size = 105.89 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [F]  time = 0.179, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{4} \left ( b\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec{\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \sec \left (f x + e\right )\right )^{m} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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