3.4.60 \(\int \cot ^4(e+f x) (b \sec (e+f x))^m \, dx\) [360]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {2697} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*((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*Sec[e + f*x])^m)/f

Rule 2697

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)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], 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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
time = 26.70, size = 6532, normalized size = 103.68 \begin {gather*} \text {Result too large to show} \end {gather*}

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

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (\cot ^{4}\left (f x +e \right )\right ) \left (b \sec \left (f x +e \right )\right )^{m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (b \sec {\left (e + f x \right )}\right )^{m} \cot ^{4}{\left (e + f x \right )}\, dx \end {gather*}

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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