3.222 \(\int \cot ^m(e+f x) \tan ^n(e+f x) \, dx\)

Optimal. Leaf size=62 \[ \frac{\cot ^m(e+f x) \tan ^{n+1}(e+f x) \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]

[Out]

(Cot[e + f*x]^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/(f*(
1 - m + n))

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Rubi [A]  time = 0.0582763, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2604, 3476, 364} \[ \frac{\cot ^m(e+f x) \tan ^{n+1}(e+f x) \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^m*Tan[e + f*x]^n,x]

[Out]

(Cot[e + f*x]^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/(f*(
1 - m + n))

Rule 2604

Int[(cot[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(a*Cot[e + f*
x])^m*(b*Tan[e + f*x])^m, Int[(b*Tan[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[m
] &&  !IntegerQ[n]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx &=\left (\cot ^m(e+f x) \tan ^m(e+f x)\right ) \int \tan ^{-m+n}(e+f x) \, dx\\ &=\frac{\left (\cot ^m(e+f x) \tan ^m(e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^{-m+n}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\cot ^m(e+f x) \, _2F_1\left (1,\frac{1}{2} (1-m+n);\frac{1}{2} (3-m+n);-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)}\\ \end{align*}

Mathematica [A]  time = 0.0743462, size = 62, normalized size = 1. \[ \frac{\cot ^m(e+f x) \tan ^{n+1}(e+f x) \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^m*Tan[e + f*x]^n,x]

[Out]

(Cot[e + f*x]^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x]^(1 + n))/(f*(
1 - m + n))

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Maple [F]  time = 0.444, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( fx+e \right ) \right ) ^{m} \left ( \tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^m*tan(f*x+e)^n,x)

[Out]

int(cot(f*x+e)^m*tan(f*x+e)^n,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^m*tan(f*x+e)^n,x, algorithm="maxima")

[Out]

integrate(cot(f*x + e)^m*tan(f*x + e)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^m*tan(f*x+e)^n,x, algorithm="fricas")

[Out]

integral(cot(f*x + e)^m*tan(f*x + e)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**m*tan(f*x+e)**n,x)

[Out]

Integral(tan(e + f*x)**n*cot(e + f*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^m*tan(f*x+e)^n,x, algorithm="giac")

[Out]

integrate(cot(f*x + e)^m*tan(f*x + e)^n, x)