Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \]
-I*arctan((3-2*I)^(1/2)*tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2))/d/(3-2*I) ^(1/2)+I*arctan((3+2*I)^(1/2)*tan(d*x+c)^(1/2)/(2-3*tan(d*x+c))^(1/2))/d/( 3+2*I)^(1/2)
Time = 0.07 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\frac {i \left (\sqrt {3+2 i} \arctan \left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-3+2 i} \text {arctanh}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )\right )}{\sqrt {13} d} \]
(I*(Sqrt[3 + 2*I]*ArcTan[(Sqrt[3/13 + (2*I)/13]*Sqrt[2 - 3*Tan[c + d*x]])/ Sqrt[Tan[c + d*x]]] + Sqrt[-3 + 2*I]*ArcTanh[(Sqrt[-3/13 + (2*I)/13]*Sqrt[ 2 - 3*Tan[c + d*x]])/Sqrt[Tan[c + d*x]]]))/(Sqrt[13]*d)
Time = 0.26 (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.
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 613 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {1}{\sqrt {2-3 \tan (c+d x)} \sqrt {\tan (c+d x)} (\tan (c+d x)+i)}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{\sqrt {2-3 \tan (c+d x)} (i-\tan (c+d x)) \sqrt {\tan (c+d x)}}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\int \frac {1}{\frac {(2+3 i) \tan (c+d x)}{2-3 \tan (c+d x)}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}-\int \frac {1}{i-\frac {(2-3 i) \tan (c+d x)}{2-3 \tan (c+d x)}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {i \arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i}}-\frac {i \arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i}}}{d}\) |
(((-I)*ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3*Tan[c + d*x]]] )/Sqrt[3 - 2*I] + (I*ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 - 3* Tan[c + d*x]]])/Sqrt[3 + 2*I])/d
3.7.65.3.1 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 = 4.21 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.61
| method | result | size |
| derivativedivides | \(\frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {2 \sqrt {13}+6}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(438\) |
| default | \(\frac {\sqrt {2-3 \tan \left (d x +c \right )}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}\, \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \left (\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}-3 \,\operatorname {arctanh}\left (\frac {\sqrt {2 \sqrt {13}-6}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3+2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}\right ) \sqrt {2 \sqrt {13}-6}\, \sqrt {2 \sqrt {13}+6}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}-44 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (-2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3-2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right )}{2 d \sqrt {\tan \left (d x +c \right )}\, \left (-2+3 \tan \left (d x +c \right )\right ) \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\) | \(438\) |
1/2/d*(2-3*tan(d*x+c))^(1/2)*(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^(1/2)-3-2* tan(d*x+c))^2)^(1/2)*(13^(1/2)-3-2*tan(d*x+c))*(arctanh(1/208*(2*13^(1/2)- 6)^(1/2)*(3*13^(1/2)+11)*(13^(1/2)+3+2*tan(d*x+c))*(11*13^(1/2)-39)/(13^(1 /2)-3-2*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^(1/2)-3-2* tan(d*x+c))^2)^(1/2))*(2*13^(1/2)-6)^(1/2)*13^(1/2)*(2*13^(1/2)+6)^(1/2)-3 *arctanh(1/208*(2*13^(1/2)-6)^(1/2)*(3*13^(1/2)+11)*(13^(1/2)+3+2*tan(d*x+ c))*(11*13^(1/2)-39)/(13^(1/2)-3-2*tan(d*x+c))*13^(1/2)/(-tan(d*x+c)*(-2+3 *tan(d*x+c))/(13^(1/2)-3-2*tan(d*x+c))^2)^(1/2))*(2*13^(1/2)-6)^(1/2)*(2*1 3^(1/2)+6)^(1/2)+12*arctan(4*13^(1/2)*(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^( 1/2)-3-2*tan(d*x+c))^2)^(1/2)/(26*13^(1/2)+78)^(1/2))*13^(1/2)-44*arctan(4 *13^(1/2)*(-tan(d*x+c)*(-2+3*tan(d*x+c))/(13^(1/2)-3-2*tan(d*x+c))^2)^(1/2 )/(26*13^(1/2)+78)^(1/2)))/tan(d*x+c)^(1/2)/(-2+3*tan(d*x+c))/(2*13^(1/2)+ 6)^(1/2)/(11*13^(1/2)-39)
Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (67) = 134\).
Time = 0.35 (sec) , antiderivative size = 1477, normalized size of antiderivative = 15.55 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\text {Too large to display} \]
1/8*sqrt(1/13)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log((sqrt(1/13)*(155*d*t an(d*x + c)^2 + 102*d*tan(d*x + c) + (135*d^3*tan(d*x + c)^2 - 211*d^3*tan (d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2 ) + ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)*s qrt(-3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) + 1/8*s qrt(1/13)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log(-(sqrt(1/13)*(155*d*tan(d *x + c)^2 + 102*d*tan(d*x + c) + (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2) + ((33*d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)*sqrt( -3*tan(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt( 1/13)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log((sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) + (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 33*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2) - ((33* d^2*tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)*sqrt(-3*ta n(d*x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13) *sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2)*log(-(sqrt(1/13)*(155*d*tan(d*x + c)^2 + 102*d*tan(d*x + c) + (135*d^3*tan(d*x + c)^2 - 211*d^3*tan(d*x + c) + 3 3*d^3)*sqrt(-1/d^4) - 56*d)*sqrt((2*d^2*sqrt(-1/d^4) + 3)/d^2) - ((33*d^2* tan(d*x + c) + 56*d^2)*sqrt(-1/d^4) - 56*tan(d*x + c) + 33)*sqrt(-3*tan(d* x + c) + 2)*sqrt(tan(d*x + c)))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(1/13)*...
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {2 - 3 \tan {\left (c + d x \right )}}}\, dx \]
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (67) = 134\).
Time = 0.90 (sec) , antiderivative size = 641, normalized size of antiderivative = 6.75 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=-\frac {1}{2028} \, \sqrt {3} {\left (\frac {2 \, {\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} - 702} - 3 \, d \sqrt {1014 \, \sqrt {13} + 702} {\left | d \right |}\right )} \arctan \left (\frac {13 \, \left (\frac {9}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} + \frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}}{18 \, \sqrt {\frac {3}{26} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {2 \, {\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} - 702} - 3 \, d \sqrt {1014 \, \sqrt {13} + 702} {\left | d \right |}\right )} \arctan \left (-\frac {13 \, \left (\frac {9}{13}\right )^{\frac {3}{4}} {\left (2 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} - \frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} + \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}}{18 \, \sqrt {\frac {3}{26} \, \sqrt {13} + \frac {1}{2}}}\right )}{d^{3}} + \frac {{\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} + 702} + 3 \, d \sqrt {1014 \, \sqrt {13} - 702} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}^{2} + 4 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )} + 12 \, \sqrt {\frac {1}{13}}\right )}{d^{3}} - \frac {{\left (2 \, d^{2} \sqrt {1014 \, \sqrt {13} + 702} + 3 \, d \sqrt {1014 \, \sqrt {13} - 702} {\left | d \right |}\right )} \log \left ({\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )}^{2} - 4 \, \left (\frac {9}{13}\right )^{\frac {1}{4}} \sqrt {-\frac {3}{26} \, \sqrt {13} + \frac {1}{2}} {\left (\frac {\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}{\sqrt {-3 \, \tan \left (d x + c\right ) + 2}} - \frac {\sqrt {-3 \, \tan \left (d x + c\right ) + 2}}{\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {2}}\right )} + 12 \, \sqrt {\frac {1}{13}}\right )}{d^{3}}\right )} \]
-1/2028*sqrt(3)*(2*(2*d^2*sqrt(1014*sqrt(13) - 702) - 3*d*sqrt(1014*sqrt(1 3) + 702)*abs(d))*arctan(13/18*(9/13)^(3/4)*(2*(9/13)^(1/4)*sqrt(-3/26*sqr t(13) + 1/2) + (sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))/ sqrt(3/26*sqrt(13) + 1/2))/d^3 + 2*(2*d^2*sqrt(1014*sqrt(13) - 702) - 3*d* sqrt(1014*sqrt(13) + 702)*abs(d))*arctan(-13/18*(9/13)^(3/4)*(2*(9/13)^(1/ 4)*sqrt(-3/26*sqrt(13) + 1/2) - (sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqr t(-3*tan(d*x + c) + 2) + sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))/sqrt(3/26*sqrt(13) + 1/2))/d^3 + (2*d^2*sqrt(1014*sqrt(13 ) + 702) + 3*d*sqrt(1014*sqrt(13) - 702)*abs(d))*log(((sqrt(3)*sqrt(tan(d* x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2)/( sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2)))^2 + 4*(9/13)^(1/4)*sqrt(-3/26*sqrt( 13) + 1/2)*((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))) + 1 2*sqrt(1/13))/d^3 - (2*d^2*sqrt(1014*sqrt(13) + 702) + 3*d*sqrt(1014*sqrt( 13) - 702)*abs(d))*log(((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))/sqrt(-3*tan (d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2)/(sqrt(3)*sqrt(tan(d*x + c)) - s qrt(2)))^2 - 4*(9/13)^(1/4)*sqrt(-3/26*sqrt(13) + 1/2)*((sqrt(3)*sqrt(tan( d*x + c)) - sqrt(2))/sqrt(-3*tan(d*x + c) + 2) - sqrt(-3*tan(d*x + c) + 2) /(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(2))) + 12*sqrt(1/13))/d^3)
Time = 6.01 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {2-3 \tan (c+d x)}} \, dx=\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6+4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6-4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (6-4{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}\,\left (-6+4{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {2-3\,\mathrm {tan}\left (c+d\,x\right )}-2}\right )\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \]
atan((2^(1/2)*d*tan(c + d*x)^(1/2)*((3/52 - 1i/26)/d^2)^(1/2)*(6 + 4i) - d *tan(c + d*x)^(1/2)*((3/52 - 1i/26)/d^2)^(1/2)*(2 - 3*tan(c + d*x))^(1/2)* (6 + 4i))/(3*tan(c + d*x) + 2^(1/2)*(2 - 3*tan(c + d*x))^(1/2) - 2))*((3/5 2 - 1i/26)/d^2)^(1/2)*2i - atan((2^(1/2)*d*tan(c + d*x)^(1/2)*((3/52 + 1i/ 26)/d^2)^(1/2)*(6 - 4i) - d*tan(c + d*x)^(1/2)*((3/52 + 1i/26)/d^2)^(1/2)* (2 - 3*tan(c + d*x))^(1/2)*(6 - 4i))/(3*tan(c + d*x) + 2^(1/2)*(2 - 3*tan( c + d*x))^(1/2) - 2))*((3/52 + 1i/26)/d^2)^(1/2)*2i