Integrand size = 21, antiderivative size = 280 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 f}-\frac {\sqrt {1+\sqrt {2}} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\tan (e+f x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 f}+\frac {\text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )}{f}+\frac {\log \left (1+\sqrt {2}+\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{4 \sqrt {1+\sqrt {2}} f}-\frac {\log \left (1+\sqrt {2}+\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\tan (e+f x)}\right )}{4 \sqrt {1+\sqrt {2}} f}-\frac {\cot (e+f x) \sqrt {1+\tan (e+f x)}}{f} \]
arctanh((1+tan(f*x+e))^(1/2))/f+1/4*ln(1+2^(1/2)-(2+2*2^(1/2))^(1/2)*(1+ta n(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)-1/4*ln(1+2^(1/2)+(2+2*2^(1 /2))^(1/2)*(1+tan(f*x+e))^(1/2)+tan(f*x+e))/f/(1+2^(1/2))^(1/2)+1/2*arctan (((2+2*2^(1/2))^(1/2)-2*(1+tan(f*x+e))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^( 1/2))^(1/2)/f-1/2*arctan(((2+2*2^(1/2))^(1/2)+2*(1+tan(f*x+e))^(1/2))/(-2+ 2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)/f-cot(f*x+e)*(1+tan(f*x+e))^(1/2)/f
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.36 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {-2 \text {arctanh}\left (\sqrt {1+\tan (e+f x)}\right )+(1-i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1-i}}\right )+(1+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\tan (e+f x)}}{\sqrt {1+i}}\right )+2 \cot (e+f x) \sqrt {1+\tan (e+f x)}}{2 f} \]
-1/2*(-2*ArcTanh[Sqrt[1 + Tan[e + f*x]]] + (1 - I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/Sqrt[1 - I]] + (1 + I)^(3/2)*ArcTanh[Sqrt[1 + Tan[e + f*x]]/ Sqrt[1 + I]] + 2*Cot[e + f*x]*Sqrt[1 + Tan[e + f*x]])/f
Time = 1.04 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.09, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.952, Rules used = {3042, 4052, 27, 3042, 4115, 3042, 4056, 27, 3042, 3966, 484, 1407, 1142, 25, 1083, 217, 1103, 4117, 73, 220}
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.
| \(\displaystyle \int \frac {\cot ^2(e+f x)}{\sqrt {\tan (e+f x)+1}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan (e+f x)^2 \sqrt {\tan (e+f x)+1}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -\int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+2 \tan (e+f x)+1\right )}{2 \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{2} \int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+2 \tan (e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} \int \frac {\tan (e+f x)^2+2 \tan (e+f x)+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 4115 |
| \(\displaystyle -\frac {1}{2} \int \cot (e+f x) (\tan (e+f x)+1)^{3/2}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{2} \int \frac {(\tan (e+f x)+1)^{3/2}}{\tan (e+f x)}dx-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 4056 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {2}{\sqrt {\tan (e+f x)+1}}dx-\int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx-\int \frac {\cot (e+f x) \left (\tan ^2(e+f x)+1\right )}{\sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{2} \left (-2 \int \frac {1}{\sqrt {\tan (e+f x)+1}}dx-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 3966 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \int \frac {1}{\sqrt {\tan (e+f x)+1} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 484 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {4 \int \frac {1}{(\tan (e+f x)+1)^2-2 (\tan (e+f x)+1)+2}d\sqrt {\tan (e+f x)+1}}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 1407 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {4 \left (\frac {\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-\sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\int \frac {\sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {4 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\frac {1}{2} \int -\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {4 \left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \int \frac {1}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {4 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )} \int \frac {1}{-\tan (e+f x)+2 \left (1-\sqrt {2}\right )-1}d\left (2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (-\frac {4 \left (\frac {\frac {1}{2} \int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\tan (e+f x)+1}}{\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\frac {1}{2} \int \frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1}d\sqrt {\tan (e+f x)+1}+\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\tan (e+f x)^2+1}{\tan (e+f x) \sqrt {\tan (e+f x)+1}}dx-\frac {4 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\cot (e+f x)}{\sqrt {\tan (e+f x)+1}}d\tan (e+f x)}{f}-\frac {4 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (-\frac {2 \int \cot (e+f x)d\sqrt {\tan (e+f x)+1}}{f}-\frac {4 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
| \(\Big \downarrow \) 220 |
| \(\displaystyle \frac {1}{2} \left (\frac {2 \text {arctanh}\left (\sqrt {\tan (e+f x)+1}\right )}{f}-\frac {4 \left (\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}-\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \log \left (\tan (e+f x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\sqrt {\frac {1+\sqrt {2}}{\sqrt {2}-1}} \arctan \left (\frac {2 \sqrt {\tan (e+f x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \log \left (\tan (e+f x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\tan (e+f x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}\right )}{f}\right )-\frac {\sqrt {\tan (e+f x)+1} \cot (e+f x)}{f}\) |
((2*ArcTanh[Sqrt[1 + Tan[e + f*x]]])/f - (4*((Sqrt[(1 + Sqrt[2])/(-1 + Sqr t[2])]*ArcTan[(-Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt[2*( -1 + Sqrt[2])]] - Log[1 + Sqrt[2] + Tan[e + f*x] - Sqrt[2*(1 + Sqrt[2])]*S qrt[1 + Tan[e + f*x]]]/2)/(4*Sqrt[1 + Sqrt[2]]) + (Sqrt[(1 + Sqrt[2])/(-1 + Sqrt[2])]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Tan[e + f*x]])/Sqrt [2*(-1 + Sqrt[2])]] + Log[1 + Sqrt[2] + Tan[e + f*x] + Sqrt[2*(1 + Sqrt[2] )]*Sqrt[1 + Tan[e + f*x]]]/2)/(4*Sqrt[1 + Sqrt[2]])))/f)/2 - (Cot[e + f*x] *Sqrt[1 + Tan[e + f*x]])/f
3.5.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[1/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2* d Subst[Int[1/(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/ c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) Int[(r - x)/(q - r* x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(r + x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Su bst[Int[(a + x)^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c , d, n}, x] && NeQ[a^2 + b^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*Tan[e + f*x]^2, x], x], x] / ; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(3/2)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2) Int[Simp[a^2*c - b^2*c + 2*a*b*d + (2*a*b*c - a^2*d + b^2*d)*Tan[e + f*x], x]/Sqrt[a + b*Tan[e + f*x ]], x], x] + Simp[(b*c - a*d)^2/(c^2 + d^2) Int[(1 + Tan[e + f*x]^2)/(Sqr t[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e , f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[1/b^2 Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Leaf count of result is larger than twice the leaf count of optimal. \(2371\) vs. \(2(216)=432\).
Time = 142.99 (sec) , antiderivative size = 2372, normalized size of antiderivative = 8.47
1/4/f*cot(f*x+e)*(-(1+2^(1/2))^(1/2)*cos(f*x+e)*((cos(f*x+e)+sin(f*x+e))*c os(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f* x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f* x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^ 2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)*( -4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)*(-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)* (3*2^(1/2)-4)/(2*cos(f*x+e)^2-1))*(-2+2*2^(1/2))^(1/2)*2^(1/2)+3*(1+2^(1/2 ))^(1/2)*((cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x +e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2 )*arctan(1/4*((4+3*2^(1/2))*(cos(f*x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)* cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin (f*x+e)^2+1)*(3*2^(1/2)-4))^(1/2)*(-4*sin(f*x+e)*cos(f*x+e)+tan(f*x+e)+1)* (-2+2*2^(1/2))^(1/2)*(2*2^(1/2)+3)*(3*2^(1/2)-4)/(2*cos(f*x+e)^2-1))*(-2+2 *2^(1/2))^(1/2)*2^(1/2)*sin(f*x+e)+8*2^(1/2)*(1+2^(1/2))^(1/2)*((cos(f*x+e )+sin(f*x+e))*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*(cot(f*x+e)^2+cot(f*x+e)) ^(1/2)*cos(f*x+e)+2*(1+2^(1/2))^(1/2)*cos(f*x+e)*((cos(f*x+e)+sin(f*x+e))* cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e)^2*2^(1/2)-2*sin(f *x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1))^(1/2)*arctan(1/4*((4+3*2^(1/2))*(cos(f *x+e)+sin(f*x+e))*cos(f*x+e)/(2*2^(1/2)*cos(f*x+e)*sin(f*x+e)-2*sin(f*x+e) ^2*2^(1/2)-2*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)^2+1)*(3*2^(1/2)-4))^(1/...
Time = 0.26 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.40 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=-\frac {\sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right ) - \sqrt {\frac {1}{2}} f \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} + f\right )} \sqrt {-\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} + 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right ) - \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right ) + \sqrt {\frac {1}{2}} f \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} \log \left (-\sqrt {\frac {1}{2}} {\left (f^{3} \sqrt {-\frac {1}{f^{4}}} - f\right )} \sqrt {\frac {f^{2} \sqrt {-\frac {1}{f^{4}}} - 1}{f^{2}}} + \sqrt {\tan \left (f x + e\right ) + 1}\right ) \tan \left (f x + e\right ) - \log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right ) \tan \left (f x + e\right ) + \log \left (\sqrt {\tan \left (f x + e\right ) + 1} - 1\right ) \tan \left (f x + e\right ) + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}}{2 \, f \tan \left (f x + e\right )} \]
-1/2*(sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(sqrt(1/2)*(f^3*sqr t(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1)) *tan(f*x + e) - sqrt(1/2)*f*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2)*log(-sqrt(1/ 2)*(f^3*sqrt(-1/f^4) + f)*sqrt(-(f^2*sqrt(-1/f^4) + 1)/f^2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e) - sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2)*lo g(sqrt(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f^2) + sqrt (tan(f*x + e) + 1))*tan(f*x + e) + sqrt(1/2)*f*sqrt((f^2*sqrt(-1/f^4) - 1) /f^2)*log(-sqrt(1/2)*(f^3*sqrt(-1/f^4) - f)*sqrt((f^2*sqrt(-1/f^4) - 1)/f^ 2) + sqrt(tan(f*x + e) + 1))*tan(f*x + e) - log(sqrt(tan(f*x + e) + 1) + 1 )*tan(f*x + e) + log(sqrt(tan(f*x + e) + 1) - 1)*tan(f*x + e) + 2*sqrt(tan (f*x + e) + 1))/(f*tan(f*x + e))
\[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int \frac {\cot ^{2}{\left (e + f x \right )}}{\sqrt {\tan {\left (e + f x \right )} + 1}}\, dx \]
\[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\int { \frac {\cot \left (f x + e\right )^{2}}{\sqrt {\tan \left (f x + e\right ) + 1}} \,d x } \]
Time = 0.84 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.20 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\log \left (\sqrt {\tan \left (f x + e\right ) + 1} + 1\right )}{2 \, f} - \frac {\log \left ({\left | \sqrt {\tan \left (f x + e\right ) + 1} - 1 \right |}\right )}{2 \, f} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{4 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} - 2} + f \sqrt {2 \, \sqrt {2} + 2} {\left | f \right |}\right )} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {\tan \left (f x + e\right ) + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right )}{4 \, f^{3}} - \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \log \left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{8 \, f^{3}} + \frac {{\left (f^{2} \sqrt {2 \, \sqrt {2} + 2} - f \sqrt {2 \, \sqrt {2} - 2} {\left | f \right |}\right )} \log \left (-2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {\tan \left (f x + e\right ) + 1} + \sqrt {2} + \tan \left (f x + e\right ) + 1\right )}{8 \, f^{3}} - \frac {\sqrt {\tan \left (f x + e\right ) + 1}}{f \tan \left (f x + e\right )} \]
1/2*log(sqrt(tan(f*x + e) + 1) + 1)/f - 1/2*log(abs(sqrt(tan(f*x + e) + 1) - 1))/f - 1/4*(f^2*sqrt(2*sqrt(2) - 2) + f*sqrt(2*sqrt(2) + 2)*abs(f))*ar ctan(1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) + 2*sqrt(tan(f*x + e) + 1))/sq rt(-sqrt(2) + 2))/f^3 - 1/4*(f^2*sqrt(2*sqrt(2) - 2) + f*sqrt(2*sqrt(2) + 2)*abs(f))*arctan(-1/2*2^(3/4)*(2^(1/4)*sqrt(sqrt(2) + 2) - 2*sqrt(tan(f*x + e) + 1))/sqrt(-sqrt(2) + 2))/f^3 - 1/8*(f^2*sqrt(2*sqrt(2) + 2) - f*sqr t(2*sqrt(2) - 2)*abs(f))*log(2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 + 1/8*(f^2*sqrt(2*sqrt(2) + 2) - f*s qrt(2*sqrt(2) - 2)*abs(f))*log(-2^(1/4)*sqrt(sqrt(2) + 2)*sqrt(tan(f*x + e ) + 1) + sqrt(2) + tan(f*x + e) + 1)/f^3 - sqrt(tan(f*x + e) + 1)/(f*tan(f *x + e))
Time = 0.23 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.42 \[ \int \frac {\cot ^2(e+f x)}{\sqrt {1+\tan (e+f x)}} \, dx=\frac {\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}}{f-f\,\left (\mathrm {tan}\left (e+f\,x\right )+1\right )}-\frac {\mathrm {atan}\left (\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{f}-\mathrm {atan}\left (2\,f\,\sqrt {\frac {-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (2\,f\,\sqrt {\frac {-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,\sqrt {\mathrm {tan}\left (e+f\,x\right )+1}\right )\,\sqrt {\frac {-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}}{f^2}}\,2{}\mathrm {i} \]