Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=\frac {i \arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}-\frac {i \arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \] Output:
I*arctan((2-3*I)^(1/2)*tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2))/(2-3*I)^( 1/2)/d-I*arctan((2+3*I)^(1/2)*tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2))/(2 +3*I)^(1/2)/d
Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=-\frac {i \left (\sqrt {2+3 i} \arctan \left (\frac {\sqrt {\frac {2}{13}+\frac {3 i}{13}} \sqrt {-3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-2+3 i} \text {arctanh}\left (\frac {\sqrt {-\frac {2}{13}+\frac {3 i}{13}} \sqrt {-3-2 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )\right )}{\sqrt {13} d} \] Input:
Integrate[Sqrt[Tan[c + d*x]]/Sqrt[-3 - 2*Tan[c + d*x]],x]
Output:
((-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]*S qrt[-3 - 2*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.
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {\int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 613 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {1}{\sqrt {-2 \tan (c+d x)-3} \sqrt {\tan (c+d x)} (\tan (c+d x)+i)}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{\sqrt {-2 \tan (c+d x)-3} (i-\tan (c+d x)) \sqrt {\tan (c+d x)}}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\int \frac {1}{i-\frac {(3-2 i) \tan (c+d x)}{-2 \tan (c+d x)-3}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}-\int \frac {1}{\frac {(3+2 i) \tan (c+d x)}{-2 \tan (c+d x)-3}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {i \arctan \left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}\right )}{\sqrt {2-3 i}}-\frac {i \arctan \left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-2 \tan (c+d x)-3}}\right )}{\sqrt {2+3 i}}}{d}\) |
Input:
Int[Sqrt[Tan[c + d*x]]/Sqrt[-3 - 2*Tan[c + d*x]],x]
Output:
((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*T an[c + d*x]]])/Sqrt[2 + 3*I])/d
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. \(425\) vs. \(2(77)=154\).
Time = 0.11 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.48
| method | result | size |
| derivativedivides | \(-\frac {3 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \left (\sqrt {-4+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right )-2 \sqrt {-4+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right )+8 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}-34 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{2 d \sqrt {2 \sqrt {13}+4}\, \sqrt {-3-2 \tan \left (d x +c \right )}\, \left (17 \sqrt {13}-52\right ) \sqrt {\tan \left (d x +c \right )}}\) | \(426\) |
| default | \(-\frac {3 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \left (\sqrt {-4+2 \sqrt {13}}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right )-2 \sqrt {-4+2 \sqrt {13}}\, \sqrt {2 \sqrt {13}+4}\, \operatorname {arctanh}\left (\frac {\sqrt {-4+2 \sqrt {13}}\, \left (4 \sqrt {13}+17\right ) \left (\sqrt {13}+2-3 \tan \left (d x +c \right )\right ) \left (17 \sqrt {13}-52\right ) \sqrt {13}}{6318 \left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}\right )+8 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right ) \sqrt {13}-34 \arctan \left (\frac {6 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+52}}\right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (3+2 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-2+3 \tan \left (d x +c \right )\right )^{2}}}}{2 d \sqrt {2 \sqrt {13}+4}\, \sqrt {-3-2 \tan \left (d x +c \right )}\, \left (17 \sqrt {13}-52\right ) \sqrt {\tan \left (d x +c \right )}}\) | \(426\) |
Input:
int(tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-3/2/d*(13^(1/2)-2+3*tan(d*x+c))*((-4+2*13^(1/2))^(1/2)*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))-2*(-4+2*13^(1 /2))^(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))+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*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)))/(2*13^(1/2)+4)^(1/2)/(-3-2*tan(d*x+c))^(1/2)/(17*13^(1/2)-52 )*(-tan(d*x+c)*(3+2*tan(d*x+c))/(13^(1/2)-2+3*tan(d*x+c))^2)^(1/2)/tan(d*x +c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1485 vs. \(2 (67) = 134\).
Time = 0.20 (sec) , antiderivative size = 1485, normalized size of antiderivative = 15.63 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-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* 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((sqr t(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)*l og(-(sqrt(1/13)*(1575*d*tan(d*x + c)^2 + 212*d*tan(d*x + c) + 2*(200*d^3*t an(d*x + c)^2 + 1167*d^3*tan(d*x + c) + 306*d^3)*sqrt(-1/d^4) - 759*d)*sqr t((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...
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {- 2 \tan {\left (c + d x \right )} - 3}}\, dx \] Input:
integrate(tan(d*x+c)**(1/2)/(-3-2*tan(d*x+c))**(1/2),x)
Output:
Integral(sqrt(tan(c + d*x))/sqrt(-2*tan(c + d*x) - 3), x)
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=\int { \frac {\sqrt {\tan \left (d x + c\right )}}{\sqrt {-2 \, \tan \left (d x + c\right ) - 3}} \,d x } \] Input:
integrate(tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(tan(d*x + c))/sqrt(-2*tan(d*x + c) - 3), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (67) = 134\).
Time = 0.40 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.20 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=-\frac {\sqrt {2} {\left (-i \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} - 6 i \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 9 i\right )} \log \left ({\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{8} + 12 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{6} + 118 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{4} + 108 \, {\left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 81\right )}{27 \, d} \] Input:
integrate(tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2),x, algorithm="giac")
Output:
-1/27*sqrt(2)*(-I*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3) )^4 - 6*I*(sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^2 - 9* I)*log((sqrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^8 + 12*(s qrt(2)*sqrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^6 + 118*(sqrt(2)*s qrt(-tan(d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^4 + 108*(sqrt(2)*sqrt(-tan (d*x + c)) - sqrt(-2*tan(d*x + c) - 3))^2 + 81)/d
Time = 2.88 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.99 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=-\mathrm {atan}\left (\frac {8\,d\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{\sqrt {-2\,\mathrm {tan}\left (c+d\,x\right )-3}\,\left (\frac {2\,{\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {2}\,\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{2\,\mathrm {tan}\left (c+d\,x\right )+3}+1\right )}\right )\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {8\,d\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {6}\,1{}\mathrm {i}}{2}\right )}{\sqrt {-2\,\mathrm {tan}\left (c+d\,x\right )-3}\,\left (\frac {2\,{\left (-\sqrt {\mathrm {tan}\left (c+d\,x\right )}+\frac {\sqrt {6}\,1{}\mathrm {i}}{2}\right )}^2}{2\,\mathrm {tan}\left (c+d\,x\right )+3}+1\right )}\right )\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \] Input:
int(tan(c + d*x)^(1/2)/(- 2*tan(c + d*x) - 3)^(1/2),x)
Output:
atan((8*d*((1/26 - 3i/52)/d^2)^(1/2)*((6^(1/2)*1i)/2 - tan(c + d*x)^(1/2)) )/((- 2*tan(c + d*x) - 3)^(1/2)*((2*((6^(1/2)*1i)/2 - tan(c + d*x)^(1/2))^ 2)/(2*tan(c + d*x) + 3) + 1)))*((1/26 - 3i/52)/d^2)^(1/2)*2i - atan((8*d*( (1/26 + 3i/52)/d^2)^(1/2)*((2^(1/2)*3^(1/2)*1i)/2 - tan(c + d*x)^(1/2)))/( (- 2*tan(c + d*x) - 3)^(1/2)*((2*((2^(1/2)*3^(1/2)*1i)/2 - tan(c + d*x)^(1 /2))^2)/(2*tan(c + d*x) + 3) + 1)))*((1/26 + 3i/52)/d^2)^(1/2)*2i
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3-2 \tan (c+d x)}} \, dx=\frac {-2 \sqrt {\tan \left (d x +c \right )}\, \sqrt {-2 \tan \left (d x +c \right )-3}+4 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {-2 \tan \left (d x +c \right )-3}\, \tan \left (d x +c \right )^{2}}{2 \tan \left (d x +c \right )+3}d x \right ) d +3 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {-2 \tan \left (d x +c \right )-3}\, \tan \left (d x +c \right )}{2 \tan \left (d x +c \right )+3}d x \right ) d +3 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {-2 \tan \left (d x +c \right )-3}}{2 \tan \left (d x +c \right )^{2}+3 \tan \left (d x +c \right )}d x \right ) d}{4 d} \] Input:
int(tan(d*x+c)^(1/2)/(-3-2*tan(d*x+c))^(1/2),x)
Output:
( - 2*sqrt(tan(c + d*x))*sqrt( - 2*tan(c + d*x) - 3) + 4*int((sqrt(tan(c + d*x))*sqrt( - 2*tan(c + d*x) - 3)*tan(c + d*x)**2)/(2*tan(c + d*x) + 3),x )*d + 3*int((sqrt(tan(c + d*x))*sqrt( - 2*tan(c + d*x) - 3)*tan(c + d*x))/ (2*tan(c + d*x) + 3),x)*d + 3*int((sqrt(tan(c + d*x))*sqrt( - 2*tan(c + d* x) - 3))/(2*tan(c + d*x)**2 + 3*tan(c + d*x)),x)*d)/(4*d)