Integrand size = 23, antiderivative size = 78 \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {(d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+2 p),\frac {1}{2} (3-m+2 p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1-m+2 p)} \]
(d*cot(f*x+e))^m*hypergeom([1, 1/2-1/2*m+p],[3/2-1/2*m+p],-tan(f*x+e)^2)*t an(f*x+e)*(b*tan(f*x+e)^2)^p/f/(1-m+2*p)
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=-\frac {d (d \cot (e+f x))^{-1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {m}{2}+p,\frac {3}{2}-\frac {m}{2}+p,-\tan ^2(e+f x)\right ) \left (b \tan ^2(e+f x)\right )^p}{f (-1+m-2 p)} \]
-((d*(d*Cot[e + f*x])^(-1 + m)*Hypergeometric2F1[1, 1/2 - m/2 + p, 3/2 - m /2 + p, -Tan[e + f*x]^2]*(b*Tan[e + f*x]^2)^p)/(f*(-1 + m - 2*p)))
Time = 0.43 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 4141, 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.
\(\displaystyle \int \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (b \tan (e+f x)^2\right )^p (d \cot (e+f x))^mdx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int (d \cot (e+f x))^m \tan ^{2 p}(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int (d \cot (e+f x))^m \tan (e+f x)^{2 p}dx\) |
\(\Big \downarrow \) 3084 |
\(\displaystyle \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \int \tan ^{2 p-m}(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \int \tan (e+f x)^{2 p-m}dx\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \tan ^{m-2 p}(e+f x) \int \frac {\tan ^{2 p-m}(e+f x)}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m+2 p+1),\frac {1}{2} (-m+2 p+3),-\tan ^2(e+f x)\right )}{f (-m+2 p+1)}\) |
((d*Cot[e + f*x])^m*Hypergeometric2F1[1, (1 - m + 2*p)/2, (3 - m + 2*p)/2, -Tan[e + f*x]^2]*Tan[e + f*x]*(b*Tan[e + f*x]^2)^p)/(f*(1 - m + 2*p))
3.5.21.3.1 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[(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[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
\[\int \left (d \cot \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \cot {\left (e + f x \right )}\right )^{m}\, dx \]
\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
Timed out. \[ \int (d \cot (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]