Integrand size = 25, antiderivative size = 278 \[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}-\frac {\sqrt {d} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}+\frac {\sqrt {d} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}-\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}+\frac {\sqrt {d} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )} \]
-1/2*arctan((d*tan(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/a^2/f-1/4*arctan(1-2^(1/ 2)*(d*tan(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/a^2/f*2^(1/2)+1/4*arctan(1+2^(1/2 )*(d*tan(f*x+e))^(1/2)/d^(1/2))*d^(1/2)/a^2/f*2^(1/2)-1/8*ln(d^(1/2)-2^(1/ 2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))*d^(1/2)/a^2/f*2^(1/2)+1/8*ln(d ^(1/2)+2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))*d^(1/2)/a^2/f*2^(1 /2)-1/2*(d*tan(f*x+e))^(1/2)/f/(a^2+a^2*tan(f*x+e))
Time = 2.90 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=\frac {-4 d^{3/2} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-2 \sqrt {2} d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )+2 \sqrt {2} d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-\sqrt {2} d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )+\sqrt {2} d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )-4 d \sqrt {d \tan (e+f x)}+\frac {4 (d \tan (e+f x))^{3/2}}{1+\tan (e+f x)}}{8 a^2 d f} \]
(-4*d^(3/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]] - 2*Sqrt[2]*d^(3/2)*ArcTa n[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] + 2*Sqrt[2]*d^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] - Sqrt[2]*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*Tan[e + f*x]]] + Sqrt[2]*d^(3/2)*Lo g[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]] - 4*d*Sqr t[d*Tan[e + f*x]] + (4*(d*Tan[e + f*x])^(3/2))/(1 + Tan[e + f*x]))/(8*a^2* d*f)
Time = 1.13 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.92, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.840, Rules used = {3042, 4051, 27, 3042, 4136, 27, 3042, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103, 4117, 27, 73, 216}
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.
\(\displaystyle \int \frac {\sqrt {d \tan (e+f x)}}{(a \tan (e+f x)+a)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {d \tan (e+f x)}}{(a \tan (e+f x)+a)^2}dx\) |
\(\Big \downarrow \) 4051 |
\(\displaystyle -\frac {\int -\frac {-a d \tan ^2(e+f x)+2 a d \tan (e+f x)+a d}{2 \sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{2 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {-a d \tan ^2(e+f x)+2 a d \tan (e+f x)+a d}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {-a d \tan (e+f x)^2+2 a d \tan (e+f x)+a d}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 4136 |
\(\displaystyle \frac {\frac {\int \frac {4 a^2 d}{\sqrt {d \tan (e+f x)}}dx}{2 a^2}-a d \int \frac {\tan ^2(e+f x)+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 d \int \frac {1}{\sqrt {d \tan (e+f x)}}dx-a d \int \frac {\tan ^2(e+f x)+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 d \int \frac {1}{\sqrt {d \tan (e+f x)}}dx-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\frac {2 d^2 \int \frac {1}{\sqrt {d \tan (e+f x)} \left (\tan ^2(e+f x) d^2+d^2\right )}d(d \tan (e+f x))}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\frac {4 d^2 \int \frac {1}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {\int \frac {d^2 \tan ^2(e+f x)+d}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-a d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {a d \int \frac {1}{a \sqrt {d \tan (e+f x)} (\tan (e+f x)+1)}d\tan (e+f x)}{f}}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {d \int \frac {1}{\sqrt {d \tan (e+f x)} (\tan (e+f x)+1)}d\tan (e+f x)}{f}}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 \int \frac {1}{\tan (e+f x)+1}d\sqrt {d \tan (e+f x)}}{f}}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\frac {4 d^2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}-\frac {2 \sqrt {d} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}}{4 a^2}-\frac {\sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )}\) |
((-2*Sqrt[d]*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/f + (4*d^2*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*S qrt[d]*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d]))/(2*d) + (-1/2*Log[d - Sqrt[2]*d^(3 /2)*Tan[e + f*x] + d^2*Tan[e + f*x]^2]/(Sqrt[2]*Sqrt[d]) + Log[d + Sqrt[2] *d^(3/2)*Tan[e + f*x] + d^2*Tan[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]))/(2*d)))/f )/(4*a^2) - Sqrt[d*Tan[e + f*x]]/(2*f*(a^2 + a^2*Tan[e + f*x]))
3.4.66.3.1 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^n/(f*(m + 1)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(a^2 + b^2 )) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[a*c *(m + 1) - b*d*n - (b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(m + n + 1)*Tan[e + f*x]^2, 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] && LtQ[m, -1] && GtQ[n, 0] && Int egerQ[2*m]
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]
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^ n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Simp[ (A*b^2 - a*b*B + a^2*C)/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*((1 + Tan[ e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] & & !GtQ[n, 0] && !LeQ[n, -1]
Time = 0.73 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(\frac {2 d^{3} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d^{3}}-\frac {\frac {\sqrt {d \tan \left (f x +e \right )}}{2 d \left (d \tan \left (f x +e \right )+d \right )}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}}{2 d}\right )}{f \,a^{2}}\) | \(200\) |
default | \(\frac {2 d^{3} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d^{3}}-\frac {\frac {\sqrt {d \tan \left (f x +e \right )}}{2 d \left (d \tan \left (f x +e \right )+d \right )}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}}{2 d}\right )}{f \,a^{2}}\) | \(200\) |
2/f/a^2*d^3*(1/16/d^3*(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d^2)^(1/4)*(d *tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f *x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x +e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))-1/2/d *(1/2*(d*tan(f*x+e))^(1/2)/d/(d*tan(f*x+e)+d)+1/2/d^(3/2)*arctan((d*tan(f* x+e))^(1/2)/d^(1/2))))
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.37 \[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=\left [\frac {\sqrt {-d} {\left (\tan \left (f x + e\right ) + 1\right )} \log \left (\frac {d \tan \left (f x + e\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} - d}{\tan \left (f x + e\right ) + 1}\right ) + {\left (a^{2} f \tan \left (f x + e\right ) + a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) - {\left (-i \, a^{2} f \tan \left (f x + e\right ) - i \, a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) - {\left (i \, a^{2} f \tan \left (f x + e\right ) + i \, a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) - {\left (a^{2} f \tan \left (f x + e\right ) + a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (-a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )}}{4 \, {\left (a^{2} f \tan \left (f x + e\right ) + a^{2} f\right )}}, -\frac {2 \, \sqrt {d} {\left (\tan \left (f x + e\right ) + 1\right )} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) - {\left (a^{2} f \tan \left (f x + e\right ) + a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) + {\left (-i \, a^{2} f \tan \left (f x + e\right ) - i \, a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) + {\left (i \, a^{2} f \tan \left (f x + e\right ) + i \, a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) + {\left (a^{2} f \tan \left (f x + e\right ) + a^{2} f\right )} \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} \log \left (-a^{2} f \left (-\frac {d^{2}}{a^{8} f^{4}}\right )^{\frac {1}{4}} + \sqrt {d \tan \left (f x + e\right )}\right ) + 2 \, \sqrt {d \tan \left (f x + e\right )}}{4 \, {\left (a^{2} f \tan \left (f x + e\right ) + a^{2} f\right )}}\right ] \]
[1/4*(sqrt(-d)*(tan(f*x + e) + 1)*log((d*tan(f*x + e) - 2*sqrt(d*tan(f*x + e))*sqrt(-d) - d)/(tan(f*x + e) + 1)) + (a^2*f*tan(f*x + e) + a^2*f)*(-d^ 2/(a^8*f^4))^(1/4)*log(a^2*f*(-d^2/(a^8*f^4))^(1/4) + sqrt(d*tan(f*x + e)) ) - (-I*a^2*f*tan(f*x + e) - I*a^2*f)*(-d^2/(a^8*f^4))^(1/4)*log(I*a^2*f*( -d^2/(a^8*f^4))^(1/4) + sqrt(d*tan(f*x + e))) - (I*a^2*f*tan(f*x + e) + I* a^2*f)*(-d^2/(a^8*f^4))^(1/4)*log(-I*a^2*f*(-d^2/(a^8*f^4))^(1/4) + sqrt(d *tan(f*x + e))) - (a^2*f*tan(f*x + e) + a^2*f)*(-d^2/(a^8*f^4))^(1/4)*log( -a^2*f*(-d^2/(a^8*f^4))^(1/4) + sqrt(d*tan(f*x + e))) - 2*sqrt(d*tan(f*x + e)))/(a^2*f*tan(f*x + e) + a^2*f), -1/4*(2*sqrt(d)*(tan(f*x + e) + 1)*arc tan(sqrt(d*tan(f*x + e))/sqrt(d)) - (a^2*f*tan(f*x + e) + a^2*f)*(-d^2/(a^ 8*f^4))^(1/4)*log(a^2*f*(-d^2/(a^8*f^4))^(1/4) + sqrt(d*tan(f*x + e))) + ( -I*a^2*f*tan(f*x + e) - I*a^2*f)*(-d^2/(a^8*f^4))^(1/4)*log(I*a^2*f*(-d^2/ (a^8*f^4))^(1/4) + sqrt(d*tan(f*x + e))) + (I*a^2*f*tan(f*x + e) + I*a^2*f )*(-d^2/(a^8*f^4))^(1/4)*log(-I*a^2*f*(-d^2/(a^8*f^4))^(1/4) + sqrt(d*tan( f*x + e))) + (a^2*f*tan(f*x + e) + a^2*f)*(-d^2/(a^8*f^4))^(1/4)*log(-a^2* f*(-d^2/(a^8*f^4))^(1/4) + sqrt(d*tan(f*x + e))) + 2*sqrt(d*tan(f*x + e))) /(a^2*f*tan(f*x + e) + a^2*f)]
\[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=\frac {\int \frac {\sqrt {d \tan {\left (e + f x \right )}}}{\tan ^{2}{\left (e + f x \right )} + 2 \tan {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]
Time = 0.35 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=-\frac {\frac {4 \, \sqrt {d \tan \left (f x + e\right )} d^{2}}{a^{2} d \tan \left (f x + e\right ) + a^{2} d} + \frac {4 \, d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{2}} - \frac {2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right ) + 2 \, \sqrt {2} d^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right ) + \sqrt {2} d^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right ) - \sqrt {2} d^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{a^{2}}}{8 \, d f} \]
-1/8*(4*sqrt(d*tan(f*x + e))*d^2/(a^2*d*tan(f*x + e) + a^2*d) + 4*d^(3/2)* arctan(sqrt(d*tan(f*x + e))/sqrt(d))/a^2 - (2*sqrt(2)*d^(3/2)*arctan(1/2*s qrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(f*x + e)))/sqrt(d)) + 2*sqrt(2)*d^( 3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d )) + sqrt(2)*d^(3/2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqr t(d) + d) - sqrt(2)*d^(3/2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d))/a^2)/(d*f)
\[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{{\left (a \tan \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Time = 5.73 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {d \tan (e+f x)}}{(a+a \tan (e+f x))^2} \, dx=-\frac {\mathrm {atan}\left (\frac {4\,d^{12}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^2}{a^8\,f^4}\right )}^{1/4}}{\frac {4\,d^{13}}{a^2\,f}-4\,a^2\,d^{12}\,f\,\sqrt {-\frac {d^2}{a^8\,f^4}}}+\frac {4\,d^{11}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^2}{a^8\,f^4}\right )}^{3/4}}{\frac {4\,d^{13}}{a^6\,f^3}-\frac {4\,d^{12}\,\sqrt {-\frac {d^2}{a^8\,f^4}}}{a^2\,f}}\right )\,{\left (-\frac {d^2}{a^8\,f^4}\right )}^{1/4}}{2}-\mathrm {atan}\left (\frac {d^{12}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^2}{256\,a^8\,f^4}\right )}^{1/4}\,16{}\mathrm {i}}{\frac {4\,d^{13}}{a^2\,f}+64\,a^2\,d^{12}\,f\,\sqrt {-\frac {d^2}{256\,a^8\,f^4}}}-\frac {d^{11}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^2}{256\,a^8\,f^4}\right )}^{3/4}\,256{}\mathrm {i}}{\frac {4\,d^{13}}{a^6\,f^3}+\frac {64\,d^{12}\,\sqrt {-\frac {d^2}{256\,a^8\,f^4}}}{a^2\,f}}\right )\,{\left (-\frac {d^2}{256\,a^8\,f^4}\right )}^{1/4}\,2{}\mathrm {i}-\frac {d\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\left (a^2\,d\,f+a^2\,d\,f\,\mathrm {tan}\left (e+f\,x\right )\right )}+\frac {\sqrt {-d}\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{\sqrt {-d}}\right )\,1{}\mathrm {i}}{2\,a^2\,f} \]
((-d)^(1/2)*atan(((d*tan(e + f*x))^(1/2)*1i)/(-d)^(1/2))*1i)/(2*a^2*f) - a tan((d^12*(d*tan(e + f*x))^(1/2)*(-d^2/(256*a^8*f^4))^(1/4)*16i)/((4*d^13) /(a^2*f) + 64*a^2*d^12*f*(-d^2/(256*a^8*f^4))^(1/2)) - (d^11*(d*tan(e + f* x))^(1/2)*(-d^2/(256*a^8*f^4))^(3/4)*256i)/((4*d^13)/(a^6*f^3) + (64*d^12* (-d^2/(256*a^8*f^4))^(1/2))/(a^2*f)))*(-d^2/(256*a^8*f^4))^(1/4)*2i - (d*( d*tan(e + f*x))^(1/2))/(2*(a^2*d*f + a^2*d*f*tan(e + f*x))) - (atan((4*d^1 2*(d*tan(e + f*x))^(1/2)*(-d^2/(a^8*f^4))^(1/4))/((4*d^13)/(a^2*f) - 4*a^2 *d^12*f*(-d^2/(a^8*f^4))^(1/2)) + (4*d^11*(d*tan(e + f*x))^(1/2)*(-d^2/(a^ 8*f^4))^(3/4))/((4*d^13)/(a^6*f^3) - (4*d^12*(-d^2/(a^8*f^4))^(1/2))/(a^2* f)))*(-d^2/(a^8*f^4))^(1/4))/2