Integrand size = 23, antiderivative size = 98 \[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\cos ^2(e+f x)^{\frac {1}{2}+p} (d \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1-m+2 p),\frac {1}{2} (3-m+2 p),\sin ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1-m+2 p)} \]
(cos(f*x+e)^2)^(1/2+p)*(d*csc(f*x+e))^m*hypergeom([1/2+p, 1/2-1/2*m+p],[3/ 2-1/2*m+p],sin(f*x+e)^2)*tan(f*x+e)*(b*tan(f*x+e)^2)^p/f/(1-m+2*p)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 3.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.05 \[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=-\frac {d (-3+m-2 p) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {m}{2}+p,2 p,1-m,\frac {3}{2}-\frac {m}{2}+p,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) (d \csc (e+f x))^{-1+m} \left (b \tan ^2(e+f x)\right )^p}{f (-1+m-2 p) \left ((-3+m-2 p) \operatorname {AppellF1}\left (\frac {1}{2}-\frac {m}{2}+p,2 p,1-m,\frac {3}{2}-\frac {m}{2}+p,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )+2 \left (-\left ((-1+m) \operatorname {AppellF1}\left (\frac {3}{2}-\frac {m}{2}+p,2 p,2-m,\frac {5}{2}-\frac {m}{2}+p,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )-2 p \operatorname {AppellF1}\left (\frac {3}{2}-\frac {m}{2}+p,1+2 p,1-m,\frac {5}{2}-\frac {m}{2}+p,\tan ^2\left (\frac {1}{2} (e+f x)\right ),-\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right )\right )} \]
-((d*(-3 + m - 2*p)*AppellF1[1/2 - m/2 + p, 2*p, 1 - m, 3/2 - m/2 + p, Tan [(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(d*Csc[e + f*x])^(-1 + m)*(b*Tan[e + f*x]^2)^p)/(f*(-1 + m - 2*p)*((-3 + m - 2*p)*AppellF1[1/2 - m/2 + p, 2*p, 1 - m, 3/2 - m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*(-((-1 + m)*AppellF1[3/2 - m/2 + p, 2*p, 2 - m, 5/2 - m/2 + p, Tan[(e + f*x)/2]^ 2, -Tan[(e + f*x)/2]^2]) - 2*p*AppellF1[3/2 - m/2 + p, 1 + 2*p, 1 - m, 5/2 - m/2 + p, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) )
Time = 0.60 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4141, 3042, 3098, 3042, 3082, 3042, 3057}
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.
| \(\displaystyle \int \left (b \tan ^2(e+f x)\right )^p (d \csc (e+f x))^m \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (b \tan (e+f x)^2\right )^p (d \csc (e+f x))^mdx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int (d \csc (e+f x))^m \tan ^{2 p}(e+f x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \int (d \csc (e+f x))^m \tan (e+f x)^{2 p}dx\) |
| \(\Big \downarrow \) 3098 |
| \(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \left (\frac {\sin (e+f x)}{d}\right )^m (d \csc (e+f x))^m \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} \tan ^{2 p}(e+f x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p \left (\frac {\sin (e+f x)}{d}\right )^m (d \csc (e+f x))^m \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} \tan (e+f x)^{2 p}dx\) |
| \(\Big \downarrow \) 3082 |
| \(\displaystyle \frac {\sin (e+f x) \cos ^{2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p (d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^{m-2 p-1} \int \cos ^{-2 p}(e+f x) \left (\frac {\sin (e+f x)}{d}\right )^{2 p-m}dx}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sin (e+f x) \cos ^{2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p (d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^{m-2 p-1} \int \cos (e+f x)^{-2 p} \left (\frac {\sin (e+f x)}{d}\right )^{2 p-m}dx}{d}\) |
| \(\Big \downarrow \) 3057 |
| \(\displaystyle \frac {\tan (e+f x) \cos ^2(e+f x)^{p+\frac {1}{2}} \left (b \tan ^2(e+f x)\right )^p (d \csc (e+f x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (2 p+1),\frac {1}{2} (-m+2 p+1),\frac {1}{2} (-m+2 p+3),\sin ^2(e+f x)\right )}{f (-m+2 p+1)}\) |
((Cos[e + f*x]^2)^(1/2 + p)*(d*Csc[e + f*x])^m*Hypergeometric2F1[(1 + 2*p) /2, (1 - m + 2*p)/2, (3 - m + 2*p)/2, Sin[e + f*x]^2]*Tan[e + f*x]*(b*Tan[ e + f*x]^2)^p)/(f*(1 - m + 2*p))
3.5.96.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a*Cos[e + f*x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b* (a*Sin[e + f*x])^(n + 1))) Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x ], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[n]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(a*Csc[e + f*x])^FracPart[m]*(Sin[e + f*x]/a)^FracPar t[m] Int[(b*Tan[e + f*x])^n/(Sin[e + f*x]/a)^m, x], x] /; FreeQ[{a, b, e, f, m, n}, x] && !IntegerQ[m] && !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]])
\[\int \left (d \csc \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \csc {\left (e + f x \right )}\right )^{m}\, dx \]
\[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \csc \left (f x + e\right )\right )^{m} \,d x } \]
Timed out. \[ \int (d \csc (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^m \,d x \]