Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {i \arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]
-I*arctan((2-3*I)^(1/2)*tan(d*x+c)^(1/2)/(3-2*tan(d*x+c))^(1/2))/d/(2-3*I) ^(1/2)+I*arctan((2+3*I)^(1/2)*tan(d*x+c)^(1/2)/(3-2*tan(d*x+c))^(1/2))/d/( 2+3*I)^(1/2)
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=\frac {i \left (\sqrt {2+3 i} \arctan \left (\frac {\sqrt {\frac {2}{13}+\frac {3 i}{13}} \sqrt {3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-2+3 i} \text {arctanh}\left (\frac {\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )\right )}{\sqrt {13} d} \]
(I*(Sqrt[2 + 3*I]*ArcTan[(Sqrt[2/13 + (3*I)/13]*Sqrt[3 - 2*Tan[c + d*x]])/ Sqrt[Tan[c + d*x]]] + Sqrt[-2 + 3*I]*ArcTanh[(Sqrt[-2/13 + (3*I)/13]*Sqrt[ 3 - 2*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)
Time = 0.27 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4058, 613, 104, 218}
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.
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 613 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {1}{\sqrt {3-2 \tan (c+d x)} \sqrt {\tan (c+d x)} (\tan (c+d x)+i)}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{\sqrt {3-2 \tan (c+d x)} (i-\tan (c+d x)) \sqrt {\tan (c+d x)}}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\int \frac {1}{\frac {(3+2 i) \tan (c+d x)}{3-2 \tan (c+d x)}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}-\int \frac {1}{i-\frac {(3-2 i) \tan (c+d x)}{3-2 \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {i \arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i}}-\frac {i \arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i}}}{d}\) |
(((-I)*ArcTan[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 - 2*Tan[c + d*x]]] )/Sqrt[2 - 3*I] + (I*ArcTan[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[3 - 2* Tan[c + d*x]]])/Sqrt[2 + 3*I])/d
3.7.68.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(e_.)*(x_)]/(Sqrt[(c_) + (d_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Sym bol] :> Simp[e/(2*b) Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] + x)), x ], x] - Simp[e/(2*b) Int[1/(Sqrt[e*x]*Sqrt[c + d*x]*(Rt[-a/b, 2] - x)), x ], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*((c + d*ff*x)^n/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(77)=154\).
Time = 3.15 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.61
| method | result | size |
| derivativedivides | \(\frac {3 \sqrt {3-2 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}-2 \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}+8 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}-34 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (-3+2 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right )}\) | \(438\) |
| default | \(\frac {3 \sqrt {3-2 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \left (\sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}-2 \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {-4+2 \sqrt {13}}+8 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}-34 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (-3+2 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right )}\) | \(438\) |
3/2/d*(3-2*tan(d*x+c))^(1/2)*(-tan(d*x+c)*(-3+2*tan(d*x+c))/(13^(1/2)-2-3* tan(d*x+c))^2)^(1/2)*(13^(1/2)-2-3*tan(d*x+c))*(13^(1/2)*(2*13^(1/2)+4)^(1 /2)*arctanh(1/6318*(-4+2*13^(1/2))^(1/2)*(4*13^(1/2)+17)*(13^(1/2)+2+3*tan (d*x+c))*(17*13^(1/2)-52)/(13^(1/2)-2-3*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)* (-3+2*tan(d*x+c))/(13^(1/2)-2-3*tan(d*x+c))^2)^(1/2))*(-4+2*13^(1/2))^(1/2 )-2*(2*13^(1/2)+4)^(1/2)*arctanh(1/6318*(-4+2*13^(1/2))^(1/2)*(4*13^(1/2)+ 17)*(13^(1/2)+2+3*tan(d*x+c))*(17*13^(1/2)-52)/(13^(1/2)-2-3*tan(d*x+c))*1 3^(1/2)/(-tan(d*x+c)*(-3+2*tan(d*x+c))/(13^(1/2)-2-3*tan(d*x+c))^2)^(1/2)) *(-4+2*13^(1/2))^(1/2)+8*arctan(6*13^(1/2)*(-tan(d*x+c)*(-3+2*tan(d*x+c))/ (13^(1/2)-2-3*tan(d*x+c))^2)^(1/2)/(26*13^(1/2)+52)^(1/2))*13^(1/2)-34*arc tan(6*13^(1/2)*(-tan(d*x+c)*(-3+2*tan(d*x+c))/(13^(1/2)-2-3*tan(d*x+c))^2) ^(1/2)/(26*13^(1/2)+52)^(1/2)))/tan(d*x+c)^(1/2)/(2*13^(1/2)+4)^(1/2)/(-3+ 2*tan(d*x+c))/(17*13^(1/2)-52)
Leaf count of result is larger than twice the leaf count of optimal. 1485 vs. \(2 (67) = 134\).
Time = 0.35 (sec) , antiderivative size = 1485, normalized size of antiderivative = 15.63 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=\text {Too large to display} \]
1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) + 2)/d^2)*log((sqrt(1/13)*(1575*d* tan(d*x + c)^2 - 212*d*tan(d*x + c) + 2*(200*d^3*tan(d*x + c)^2 - 1167*d^3 *tan(d*x + c) + 306*d^3)*sqrt(-1/d^4) - 759*d)*sqrt((3*d^2*sqrt(-1/d^4) + 2)/d^2) + 2*((204*d^2*tan(d*x + c) + 253*d^2)*sqrt(-1/d^4) - 253*tan(d*x + c) + 204)*sqrt(-2*tan(d*x + c) + 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) + 2)/d^2)*log(-(sqrt(1/13)* (1575*d*tan(d*x + c)^2 - 212*d*tan(d*x + c) + 2*(200*d^3*tan(d*x + c)^2 - 1167*d^3*tan(d*x + c) + 306*d^3)*sqrt(-1/d^4) - 759*d)*sqrt((3*d^2*sqrt(-1 /d^4) + 2)/d^2) + 2*((204*d^2*tan(d*x + c) + 253*d^2)*sqrt(-1/d^4) - 253*t an(d*x + c) + 204)*sqrt(-2*tan(d*x + c) + 3)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) + 2)/d^2)*log((sqrt (1/13)*(1575*d*tan(d*x + c)^2 - 212*d*tan(d*x + c) + 2*(200*d^3*tan(d*x + c)^2 - 1167*d^3*tan(d*x + c) + 306*d^3)*sqrt(-1/d^4) - 759*d)*sqrt((3*d^2* sqrt(-1/d^4) + 2)/d^2) - 2*((204*d^2*tan(d*x + c) + 253*d^2)*sqrt(-1/d^4) - 253*tan(d*x + c) + 204)*sqrt(-2*tan(d*x + c) + 3)*sqrt(tan(d*x + c)))/(t an(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) + 2)/d^2)*lo g(-(sqrt(1/13)*(1575*d*tan(d*x + c)^2 - 212*d*tan(d*x + c) + 2*(200*d^3*ta n(d*x + c)^2 - 1167*d^3*tan(d*x + c) + 306*d^3)*sqrt(-1/d^4) - 759*d)*sqrt ((3*d^2*sqrt(-1/d^4) + 2)/d^2) - 2*((204*d^2*tan(d*x + c) + 253*d^2)*sqrt( -1/d^4) - 253*tan(d*x + c) + 204)*sqrt(-2*tan(d*x + c) + 3)*sqrt(tan(d*...
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {3 - 2 \tan {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (67) = 134\).
Time = 0.84 (sec) , antiderivative size = 641, normalized size of antiderivative = 6.75 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=-\frac {1}{676} \, \sqrt {2} {\left (\frac {2 \, {\left (3 \, d^{2} \sqrt {169 \, \sqrt {13} - 598} - 2 \, d \sqrt {169 \, \sqrt {13} + 598} {\left | d \right |}\right )} \arctan \left (\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} + \frac {\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} - \frac {\sqrt {-2 \, \tan \left (d x + c\right ) + 3}}{\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}\right )}}{8 \, \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {2 \, {\left (3 \, d^{2} \sqrt {169 \, \sqrt {13} - 598} - 2 \, d \sqrt {169 \, \sqrt {13} + 598} {\left | d \right |}\right )} \arctan \left (-\frac {13 \, \left (\frac {4}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} - \frac {\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} + \frac {\sqrt {-2 \, \tan \left (d x + c\right ) + 3}}{\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}\right )}}{8 \, \sqrt {\frac {1}{13} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {{\left (3 \, d^{2} \sqrt {169 \, \sqrt {13} + 598} + 2 \, d \sqrt {169 \, \sqrt {13} - 598} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} - \frac {\sqrt {-2 \, \tan \left (d x + c\right ) + 3}}{\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}\right )}^{2} + 4 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} - \frac {\sqrt {-2 \, \tan \left (d x + c\right ) + 3}}{\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}\right )} + 8 \, \sqrt {\frac {1}{13}}\right )}{d^{3}} - \frac {{\left (3 \, d^{2} \sqrt {169 \, \sqrt {13} + 598} + 2 \, d \sqrt {169 \, \sqrt {13} - 598} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} - \frac {\sqrt {-2 \, \tan \left (d x + c\right ) + 3}}{\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}\right )}^{2} - 4 \, \left (\frac {4}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {1}{13} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}{\sqrt {-2 \, \tan \left (d x + c\right ) + 3}} - \frac {\sqrt {-2 \, \tan \left (d x + c\right ) + 3}}{\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {3}}\right )} + 8 \, \sqrt {\frac {1}{13}}\right )}{d^{3}}\right )} \]
-1/676*sqrt(2)*(2*(3*d^2*sqrt(169*sqrt(13) - 598) - 2*d*sqrt(169*sqrt(13) + 598)*abs(d))*arctan(13/8*(4/13)^(3/4)*(2*(4/13)^(1/4)*sqrt(-1/13*sqrt(13 ) + 1/2) + (sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))/sqrt(-2*tan(d*x + c) + 3 ) - sqrt(-2*tan(d*x + c) + 3)/(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3)))/sqrt (1/13*sqrt(13) + 1/2))/d^3 + 2*(3*d^2*sqrt(169*sqrt(13) - 598) - 2*d*sqrt( 169*sqrt(13) + 598)*abs(d))*arctan(-13/8*(4/13)^(3/4)*(2*(4/13)^(1/4)*sqrt (-1/13*sqrt(13) + 1/2) - (sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))/sqrt(-2*ta n(d*x + c) + 3) + sqrt(-2*tan(d*x + c) + 3)/(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3)))/sqrt(1/13*sqrt(13) + 1/2))/d^3 + (3*d^2*sqrt(169*sqrt(13) + 598) + 2*d*sqrt(169*sqrt(13) - 598)*abs(d))*log(((sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))/sqrt(-2*tan(d*x + c) + 3) - sqrt(-2*tan(d*x + c) + 3)/(sqrt(2)*s qrt(tan(d*x + c)) - sqrt(3)))^2 + 4*(4/13)^(1/4)*sqrt(-1/13*sqrt(13) + 1/2 )*((sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))/sqrt(-2*tan(d*x + c) + 3) - sqrt (-2*tan(d*x + c) + 3)/(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))) + 8*sqrt(1/1 3))/d^3 - (3*d^2*sqrt(169*sqrt(13) + 598) + 2*d*sqrt(169*sqrt(13) - 598)*a bs(d))*log(((sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))/sqrt(-2*tan(d*x + c) + 3) - sqrt(-2*tan(d*x + c) + 3)/(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3)))^2 - 4*(4/13)^(1/4)*sqrt(-1/13*sqrt(13) + 1/2)*((sqrt(2)*sqrt(tan(d*x + c)) - sqrt(3))/sqrt(-2*tan(d*x + c) + 3) - sqrt(-2*tan(d*x + c) + 3)/(sqrt(2)*sq rt(tan(d*x + c)) - sqrt(3))) + 8*sqrt(1/13))/d^3)
Time = 6.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {3-2 \tan (c+d x)}} \, dx=\mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (4+6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-4-6{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {3}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}-3}\right )\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {3}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (4-6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-4+6{}\mathrm {i}\right )}{2\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {3}\,\sqrt {3-2\,\mathrm {tan}\left (c+d\,x\right )}-3}\right )\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]
atan((3^(1/2)*d*tan(c + d*x)^(1/2)*((1/26 - 3i/52)/d^2)^(1/2)*(4 + 6i) - d *tan(c + d*x)^(1/2)*((1/26 - 3i/52)/d^2)^(1/2)*(3 - 2*tan(c + d*x))^(1/2)* (4 + 6i))/(2*tan(c + d*x) + 3^(1/2)*(3 - 2*tan(c + d*x))^(1/2) - 3))*((1/2 6 - 3i/52)/d^2)^(1/2)*2i - atan((3^(1/2)*d*tan(c + d*x)^(1/2)*((1/26 + 3i/ 52)/d^2)^(1/2)*(4 - 6i) - d*tan(c + d*x)^(1/2)*((1/26 + 3i/52)/d^2)^(1/2)* (3 - 2*tan(c + d*x))^(1/2)*(4 - 6i))/(2*tan(c + d*x) + 3^(1/2)*(3 - 2*tan( c + d*x))^(1/2) - 3))*((1/26 + 3i/52)/d^2)^(1/2)*2i