\(\int \cot ^m(e+f x) \tan ^n(e+f x) \, dx\) [222]

Optimal result

Integrand size = 17, antiderivative size = 62 \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \]

[Out]

cot(f*x+e)^m*hypergeom([1, 1/2-1/2*m+1/2*n],[3/2-1/2*m+1/2*n],-tan(f*x+e)^2)*tan(f*x+e)^(1+n)/f/(1-m+n)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2684, 3557, 371} \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \tan ^{n+1}(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m+n+1),\frac {1}{2} (-m+n+3),-\tan ^2(e+f x)\right )}{f (-m+n+1)} \]

[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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2684

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 3557

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {Subst}\left (\int \frac {x^{-m+n}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\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] (verified)

Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \]

[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))

Maple [F]

\[\int \left (\cot ^{m}\left (f x +e \right )\right ) \left (\tan ^{n}\left (f x +e \right )\right )d x\]

[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)

Fricas [F]

\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \]

[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)

Sympy [F]

\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int \tan ^{n}{\left (e + f x \right )} \cot ^{m}{\left (e + f x \right )}\, dx \]

[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)

Maxima [F]

\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \]

[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)

Giac [F]

\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^m\,{\mathrm {tan}\left (e+f\,x\right )}^n \,d x \]

[In]

int(cot(e + f*x)^m*tan(e + f*x)^n,x)

[Out]

int(cot(e + f*x)^m*tan(e + f*x)^n, x)