Integrand size = 25, antiderivative size = 89 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {\text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]
arctanh((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)+arctanh((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 = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {-2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {-2+3 i} d}+\frac {\text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \]
ArcTan[(Sqrt[-2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/(Sqr t[-2 + 3*I]*d) + ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Ta n[c + d*x]]]/(Sqrt[2 + 3*I]*d)
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 4058, 615, 2009}
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.
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2 \tan (c+d x)-3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2 \tan (c+d x)-3}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2 \tan (c+d x)-3} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {\int \left (\frac {i}{2 (i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {2 \tan (c+d x)-3}}+\frac {i}{2 \sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {2 \tan (c+d x)-3}}\right )d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)-3}}\right )}{\sqrt {2-3 i}}+\frac {\text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)-3}}\right )}{\sqrt {2+3 i}}}{d}\) |
(ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/Sqr t[2 - 3*I] + ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x]]]/Sqrt[2 + 3*I])/d
3.7.61.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
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. \(479\) vs. \(2(73)=146\).
Time = 4.30 (sec) , antiderivative size = 480, normalized size of antiderivative = 5.39
| method | result | size |
| derivativedivides | \(-\frac {\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 (4 \sqrt {-4+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {-4+2 \sqrt {13}}\, \sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (52+17 \sqrt {13}\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 (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}\right )-17 \sqrt {-4+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {-4+2 \sqrt {13}}\, \sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (52+17 \sqrt {13}\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 (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}\right )+18 \,\operatorname {arctanh}\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}-36 \,\operatorname {arctanh}\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 {-3+2 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (17 \sqrt {13}-52\right )}\) | \(480\) |
| default | \(-\frac {\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 (4 \sqrt {-4+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {-4+2 \sqrt {13}}\, \sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (52+17 \sqrt {13}\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 (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}\right )-17 \sqrt {-4+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+4}\, \arctan \left (\frac {\sqrt {-4+2 \sqrt {13}}\, \sqrt {\frac {\left (17 \sqrt {13}-52\right ) \tan \left (d x +c \right ) \left (52+17 \sqrt {13}\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 (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2+3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \left (\sqrt {13}-2-3 \tan \left (d x +c \right )\right )}{56862 \tan \left (d x +c \right ) \left (-3+2 \tan \left (d x +c \right )\right )}\right )+18 \,\operatorname {arctanh}\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}-36 \,\operatorname {arctanh}\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 {-3+2 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+4}\, \left (17 \sqrt {13}-52\right )}\) | \(480\) |
-1/2/d*(tan(d*x+c)*(-3+2*tan(d*x+c))/(13^(1/2)-2-3*tan(d*x+c))^2)^(1/2)*(1 3^(1/2)-2-3*tan(d*x+c))*(4*(-4+2*13^(1/2))^(1/2)*13^(1/2)*(2*13^(1/2)+4)^( 1/2)*arctan(1/56862*(-4+2*13^(1/2))^(1/2)*((17*13^(1/2)-52)*tan(d*x+c)*(52 +17*13^(1/2))*(-3+2*tan(d*x+c))/(13^(1/2)-2-3*tan(d*x+c))^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))/tan(d*x+c)/(-3+2*tan(d*x+c)))-17*(-4+2*13^(1/2))^(1/2)*(2*13^(1/2)+4)^ (1/2)*arctan(1/56862*(-4+2*13^(1/2))^(1/2)*((17*13^(1/2)-52)*tan(d*x+c)*(5 2+17*13^(1/2))*(-3+2*tan(d*x+c))/(13^(1/2)-2-3*tan(d*x+c))^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))/tan(d*x+c)/(-3+2*tan(d*x+c)))+18*arctanh(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)-36*arctanh(6*13^(1/2)*(tan(d*x+c)*(-3+2*tan(d*x+c))/(13^(1/2)-2-3*t an(d*x+c))^2)^(1/2)/(26*13^(1/2)+52)^(1/2)))/tan(d*x+c)^(1/2)/(-3+2*tan(d* x+c))^(1/2)/(2*13^(1/2)+4)^(1/2)/(17*13^(1/2)-52)
Leaf count of result is larger than twice the leaf count of optimal. 1485 vs. \(2 (65) = 130\).
Time = 0.35 (sec) , antiderivative size = 1485, normalized size of antiderivative = 16.69 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\text {Too large to display} \]
-1/8*sqrt(1/13)*sqrt((3*d^2*sqrt(-1/d^4) + 2)/d^2)*log(1/2*(sqrt(1/13)*(40 0*d*tan(d*x + c)^2 - 2334*d*tan(d*x + c) - (1575*d^3*tan(d*x + c)^2 - 212* d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*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(-1/2*(sqrt( 1/13)*(400*d*tan(d*x + c)^2 - 2334*d*tan(d*x + c) - (1575*d^3*tan(d*x + c) ^2 - 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*d)*sqrt((3*d^2*sqr t(-1/d^4) + 2)/d^2) + 2*((204*d^2*tan(d*x + c) + 253*d^2)*sqrt(-1/d^4) - 2 53*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(1/ 2*(sqrt(1/13)*(400*d*tan(d*x + c)^2 - 2334*d*tan(d*x + c) - (1575*d^3*tan( d*x + c)^2 - 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*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(-1/2*(sqrt(1/13)*(400*d*tan(d*x + c)^2 - 2334*d*tan(d*x + c) - (1575 *d^3*tan(d*x + c)^2 - 212*d^3*tan(d*x + c) - 759*d^3)*sqrt(-1/d^4) + 612*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(t...
\[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {2 \tan {\left (c + d x \right )} - 3} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Exception generated. \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\text {Exception raised: RuntimeError} \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 479 vs. \(2 (65) = 130\).
Time = 0.43 (sec) , antiderivative size = 479, normalized size of antiderivative = 5.38 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=-\frac {\sqrt {2} {\left (\sqrt {\sqrt {13} - 2} {\left (\frac {9 i - 6}{\sqrt {13} - 2} - 2 i - 3\right )} \log \left (\left (915 i + 1098\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + \left (2370 i + 2844\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 366 \, \sqrt {13} \sqrt {61 \, \sqrt {13} + 158} - \left (1647 i - 6954\right ) \, \sqrt {13} - \left (918 i - 948\right ) \, \sqrt {61 \, \sqrt {13} + 158} - 4266 i + 18012\right ) - \sqrt {\sqrt {13} - 2} {\left (\frac {9 i - 6}{\sqrt {13} - 2} - 2 i - 3\right )} \log \left (\left (915 i + 1098\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + \left (2370 i + 2844\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 366 \, \sqrt {13} \sqrt {61 \, \sqrt {13} + 158} - \left (1647 i - 6954\right ) \, \sqrt {13} + \left (918 i - 948\right ) \, \sqrt {61 \, \sqrt {13} + 158} - 4266 i + 18012\right ) - \sqrt {\sqrt {13} + 2} {\left (\frac {6 i - 9}{\sqrt {13} + 2} - 3 i - 2\right )} \log \left (\left (90 i + 45\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - \left (108 i + 54\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 90 \, \sqrt {13} \sqrt {5 \, \sqrt {13} - 6} + \left (450 i - 225\right ) \, \sqrt {13} - \left (306 i + 108\right ) \, \sqrt {5 \, \sqrt {13} - 6} - 540 i + 270\right ) + \sqrt {\sqrt {13} + 2} {\left (\frac {6 i - 9}{\sqrt {13} + 2} - 3 i - 2\right )} \log \left (\left (90 i + 45\right ) \, \sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - \left (108 i + 54\right ) \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 90 \, \sqrt {13} \sqrt {5 \, \sqrt {13} - 6} + \left (450 i - 225\right ) \, \sqrt {13} + \left (306 i + 108\right ) \, \sqrt {5 \, \sqrt {13} - 6} - 540 i + 270\right )\right )}}{52 \, d} \]
-1/52*sqrt(2)*(sqrt(sqrt(13) - 2)*((9*I - 6)/(sqrt(13) - 2) - 2*I - 3)*log ((915*I + 1098)*sqrt(13)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 + (2370*I + 2844)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 + 366*sqrt(13)*sqrt(61*sqrt(13) + 158) - (1647*I - 6954)*sqrt(1 3) - (918*I - 948)*sqrt(61*sqrt(13) + 158) - 4266*I + 18012) - sqrt(sqrt(1 3) - 2)*((9*I - 6)/(sqrt(13) - 2) - 2*I - 3)*log((915*I + 1098)*sqrt(13)*( sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 + (2370*I + 2844) *(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 - 366*sqrt(13)* sqrt(61*sqrt(13) + 158) - (1647*I - 6954)*sqrt(13) + (918*I - 948)*sqrt(61 *sqrt(13) + 158) - 4266*I + 18012) - sqrt(sqrt(13) + 2)*((6*I - 9)/(sqrt(1 3) + 2) - 3*I - 2)*log((90*I + 45)*sqrt(13)*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 - (108*I + 54)*(sqrt(2)*sqrt(tan(d*x + c)) - s qrt(2*tan(d*x + c) - 3))^2 + 90*sqrt(13)*sqrt(5*sqrt(13) - 6) + (450*I - 2 25)*sqrt(13) - (306*I + 108)*sqrt(5*sqrt(13) - 6) - 540*I + 270) + sqrt(sq rt(13) + 2)*((6*I - 9)/(sqrt(13) + 2) - 3*I - 2)*log((90*I + 45)*sqrt(13)* (sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 - (108*I + 54)*( sqrt(2)*sqrt(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 - 90*sqrt(13)*sqr t(5*sqrt(13) - 6) + (450*I - 225)*sqrt(13) + (306*I + 108)*sqrt(5*sqrt(13) - 6) - 540*I + 270))/d
Time = 7.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=2\,\mathrm {atanh}\left (\frac {2\,d\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )-3}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}}\right )\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}+2\,\mathrm {atanh}\left (\frac {2\,d\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )-3}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}}\right )\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}} \]