3.225 \(\int (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx\)

Optimal. Leaf size=69 \[ \frac{(a \cot (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0687408, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2604, 3476, 364} \[ \frac{(a \cot (e+f x))^m (b \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{b f (-m+n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^n,x]

[Out]

((a*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + n)/2, (3 - m + n)/2, -Tan[e + f*x]^2]*(b*Tan[e + f*x])^(1 +
n))/(b*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 (a \cot (e+f x))^m (b \tan (e+f x))^n \, dx &=\left ((a \cot (e+f x))^m (b \tan (e+f x))^m\right ) \int (b \tan (e+f x))^{-m+n} \, dx\\ &=\frac{\left (b (a \cot (e+f x))^m (b \tan (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{x^{-m+n}}{b^2+x^2} \, dx,x,b \tan (e+f x)\right )}{f}\\ &=\frac{(a \cot (e+f x))^m \, _2F_1\left (1,\frac{1}{2} (1-m+n);\frac{1}{2} (3-m+n);-\tan ^2(e+f x)\right ) (b \tan (e+f x))^{1+n}}{b f (1-m+n)}\\ \end{align*}

Mathematica [A]  time = 0.0900358, size = 67, normalized size = 0.97 \[ \frac{a (a \cot (e+f x))^{m-1} (b \tan (e+f x))^n \, _2F_1\left (1,\frac{1}{2} (-m+n+1);\frac{1}{2} (-m+n+3);-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Cot[e + f*x])^m*(b*Tan[e + f*x])^n,x]

[Out]

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

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Maple [F]  time = 0.326, size = 0, normalized size = 0. \begin{align*} \int \left ( a\cot \left ( fx+e \right ) \right ) ^{m} \left ( b\tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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