Integrand size = 17, antiderivative size = 62 \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \] Output:
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)
Time = 0.05 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00 \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\frac {\cot ^m(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n),\frac {1}{2} (3-m+n),-\tan ^2(e+f x)\right ) \tan ^{1+n}(e+f x)}{f (1-m+n)} \] Input:
Integrate[Cot[e + f*x]^m*Tan[e + f*x]^n,x]
Output:
(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))
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3084, 3042, 3957, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cot (e+f x)^m \tan (e+f x)^ndx\) |
\(\Big \downarrow \) 3084 |
\(\displaystyle \tan ^m(e+f x) \cot ^m(e+f x) \int \tan ^{n-m}(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tan ^m(e+f x) \cot ^m(e+f x) \int \tan (e+f x)^{n-m}dx\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\tan ^m(e+f x) \cot ^m(e+f x) \int \frac {\tan ^{n-m}(e+f x)}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\cot ^m(e+f x) \tan ^{n+1}(e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m+n+1),\frac {1}{2} (-m+n+3),-\tan ^2(e+f x)\right )}{f (-m+n+1)}\) |
Input:
Int[Cot[e + f*x]^m*Tan[e + f*x]^n,x]
Output:
(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))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[(cot[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(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]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[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]
\[\int \cot \left (f x +e \right )^{m} \tan \left (f x +e \right )^{n}d x\]
Input:
int(cot(f*x+e)^m*tan(f*x+e)^n,x)
Output:
int(cot(f*x+e)^m*tan(f*x+e)^n,x)
\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \] Input:
integrate(cot(f*x+e)^m*tan(f*x+e)^n,x, algorithm="fricas")
Output:
integral(cot(f*x + e)^m*tan(f*x + e)^n, x)
\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int \tan ^{n}{\left (e + f x \right )} \cot ^{m}{\left (e + f x \right )}\, dx \] Input:
integrate(cot(f*x+e)**m*tan(f*x+e)**n,x)
Output:
Integral(tan(e + f*x)**n*cot(e + f*x)**m, x)
\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \] Input:
integrate(cot(f*x+e)^m*tan(f*x+e)^n,x, algorithm="maxima")
Output:
integrate(cot(f*x + e)^m*tan(f*x + e)^n, x)
\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int { \cot \left (f x + e\right )^{m} \tan \left (f x + e\right )^{n} \,d x } \] Input:
integrate(cot(f*x+e)^m*tan(f*x+e)^n,x, algorithm="giac")
Output:
integrate(cot(f*x + e)^m*tan(f*x + e)^n, x)
Timed out. \[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^m\,{\mathrm {tan}\left (e+f\,x\right )}^n \,d x \] Input:
int(cot(e + f*x)^m*tan(e + f*x)^n,x)
Output:
int(cot(e + f*x)^m*tan(e + f*x)^n, x)
\[ \int \cot ^m(e+f x) \tan ^n(e+f x) \, dx=\int \tan \left (f x +e \right )^{n} \cot \left (f x +e \right )^{m}d x \] Input:
int(cot(f*x+e)^m*tan(f*x+e)^n,x)
Output:
int(tan(e + f*x)**n*cot(e + f*x)**m,x)