Integrand size = 23, antiderivative size = 63 \[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=-\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n p),\frac {1}{2} (1+n p),-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)} \]
-cot(f*x+e)*hypergeom([1, 1/2*n*p-1/2],[1/2*n*p+1/2],-tan(f*x+e)^2)*(b*(c* tan(f*x+e))^n)^p/f/(-n*p+1)
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1+n p),\frac {1}{2} (1+n p),-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (-1+n p)} \]
(Cot[e + f*x]*Hypergeometric2F1[1, (-1 + n*p)/2, (1 + n*p)/2, -Tan[e + f*x ]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(-1 + n*p))
Time = 0.37 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4142, 3042, 2030, 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.
\(\displaystyle \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (b (c \tan (e+f x))^n\right )^p}{\tan (e+f x)^2}dx\) |
\(\Big \downarrow \) 4142 |
\(\displaystyle (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int \cot ^2(e+f x) (c \tan (e+f x))^{n p}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int \frac {(c \tan (e+f x))^{n p}}{\tan (e+f x)^2}dx\) |
\(\Big \downarrow \) 2030 |
\(\displaystyle c^2 (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int (c \tan (e+f x))^{n p-2}dx\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {c^3 (c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p \int \frac {(c \tan (e+f x))^{n p-2}}{\tan ^2(e+f x) c^2+c^2}d(c \tan (e+f x))}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\frac {\cot (e+f x) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (n p-1),\frac {1}{2} (n p+1),-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (1-n p)}\) |
-((Cot[e + f*x]*Hypergeometric2F1[1, (-1 + n*p)/2, (1 + n*p)/2, -Tan[e + f *x]^2]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 - n*p)))
3.5.13.3.1 Defintions of rubi rules used
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[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> S imp[b^IntPart[p]*((b*(c*Tan[e + f*x])^n)^FracPart[p]/(c*Tan[e + f*x])^(n*Fr acPart[p])) Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; FreeQ[{ b, c, e, f, n, p}, x] && !IntegerQ[p] && !IntegerQ[n] && (EqQ[u, 1] || Ma tchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
\[\int \cot \left (f x +e \right )^{2} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}d x\]
\[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cot \left (f x + e\right )^{2} \,d x } \]
\[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=\int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \cot ^{2}{\left (e + f x \right )}\, dx \]
\[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cot \left (f x + e\right )^{2} \,d x } \]
\[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \cot \left (f x + e\right )^{2} \,d x } \]
Timed out. \[ \int \cot ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]