Integrand size = 25, antiderivative size = 95 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2-3 i} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \] Output:
-I*arctanh((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*arctanh((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 = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}} \, dx=-\frac {i \arctan \left (\frac {\sqrt {-2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {-2+3 i} d}+\frac {i \text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \] Input:
Integrate[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]*d) + (I*ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt [-3 + 2*Tan[c + d*x]]])/(Sqrt[2 + 3*I]*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, 221}
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 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {1}{\sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {2 \tan (c+d x)-3}}d\tan (c+d x)-\frac {1}{2} \int \frac {1}{(i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {2 \tan (c+d x)-3}}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\int \frac {1}{i-\frac {(3+2 i) \tan (c+d x)}{2 \tan (c+d x)-3}}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)-3}}-\int \frac {1}{\frac {(3-2 i) \tan (c+d x)}{2 \tan (c+d x)-3}+i}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)-3}}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {i \text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)-3}}\right )}{\sqrt {2+3 i}}-\frac {i \text {arctanh}\left (\frac {\sqrt {2-3 i} \sqrt {\tan (c+d x)}}{\sqrt {2 \tan (c+d x)-3}}\right )}{\sqrt {2-3 i}}}{d}\) |
Input:
Int[Sqrt[Tan[c + d*x]]/Sqrt[-3 + 2*Tan[c + d*x]],x]
Output:
(((-I)*ArcTanh[(Sqrt[2 - 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[c + d*x] ]])/Sqrt[2 - 3*I] + (I*ArcTanh[(Sqrt[2 + 3*I]*Sqrt[Tan[c + d*x]])/Sqrt[-3 + 2*Tan[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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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. \(478\) vs. \(2(77)=154\).
Time = 0.17 (sec) , antiderivative size = 479, normalized size of antiderivative = 5.04
| method | result | size |
| derivativedivides | \(-\frac {3 \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 {-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 )-2 \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 )-8 \,\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}+34 \,\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 )}\) | \(479\) |
| default | \(-\frac {3 \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 {-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 )-2 \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 )-8 \,\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}+34 \,\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 )}\) | \(479\) |
Input:
int(tan(d*x+c)^(1/2)/(-3+2*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-3/2/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+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+1 7*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)))-2*(-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)*(52+1 7*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)))-8*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)+34*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)))/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. \(2 (67) = 134\).
Time = 0.16 (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(1/1 3)*(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*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(( 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(t...
\[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}} \, dx=\int \frac {\sqrt {\tan {\left (c + d x \right )}}}{\sqrt {2 \tan {\left (c + d x \right )} - 3}}\, dx \] 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)
Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (67) = 134\).
Time = 0.66 (sec) , antiderivative size = 498, normalized size of antiderivative = 5.24 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:
integrate(tan(d*x+c)^(1/2)/(-3+2*tan(d*x+c))^(1/2),x, algorithm="giac")
Output:
1/676*sqrt(2)*(3*(2*sqrt(169*sqrt(13) + 598)*arctan(13/4*(4/13)^(3/4)*((4/ 13)^(1/4)*sqrt(1/13*sqrt(13) + 1/2) + 1)/sqrt(-1/13*sqrt(13) + 1/2)) + 2*s qrt(169*sqrt(13) + 598)*arctan(-13/4*(4/13)^(3/4)*((4/13)^(1/4)*sqrt(1/13* sqrt(13) + 1/2) - 1)/sqrt(-1/13*sqrt(13) + 1/2)) + sqrt(169*sqrt(13) - 598 )*log(2*(4/13)^(1/4)*sqrt(1/13*sqrt(13) + 1/2) + 2*sqrt(1/13) + 1) - sqrt( 169*sqrt(13) - 598)*log(-2*(4/13)^(1/4)*sqrt(1/13*sqrt(13) + 1/2) + 2*sqrt (1/13) + 1))/d - 2*(3*d^2*sqrt(169*sqrt(13) + 598) + 2*d*sqrt(169*sqrt(13) - 598)*abs(d))*arctan(13/4*(4/13)^(3/4)*((4/13)^(1/4)*sqrt(1/13*sqrt(13) + 1/2) + sqrt(3/(2*tan(d*x + c) - 3) + 1))/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(-13/4*(4/13)^(3/4)*((4/13)^(1/4)*sqrt(1/13*sqrt(13) + 1/2) - sqrt (3/(2*tan(d*x + c) - 3) + 1))/sqrt(-1/13*sqrt(13) + 1/2))/d^3 - (3*d^2*sqr t(169*sqrt(13) - 598) - 2*d*sqrt(169*sqrt(13) + 598)*abs(d))*log(2*(4/13)^ (1/4)*sqrt(1/13*sqrt(13) + 1/2)*sqrt(3/(2*tan(d*x + c) - 3) + 1) + 2*sqrt( 1/13) + 3/(2*tan(d*x + c) - 3) + 1)/d^3 + (3*d^2*sqrt(169*sqrt(13) - 598) - 2*d*sqrt(169*sqrt(13) + 598)*abs(d))*log(-2*(4/13)^(1/4)*sqrt(1/13*sqrt( 13) + 1/2)*sqrt(3/(2*tan(d*x + c) - 3) + 1) + 2*sqrt(1/13) + 3/(2*tan(d*x + c) - 3) + 1)/d^3)
Time = 2.88 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.01 \[ \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}} \, dx=-\mathrm {atan}\left (\frac {8\,d\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (\frac {\sqrt {2}\,\sqrt {3}}{2}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )-3}\,\left (\frac {2\,{\left (\frac {\sqrt {2}\,\sqrt {3}}{2}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}^2}{2\,\mathrm {tan}\left (c+d\,x\right )-3}+1\right )}\right )\,\sqrt {\frac {-\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {8\,d\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\left (\frac {\sqrt {2}\,\sqrt {3}}{2}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )-3}\,\left (\frac {2\,{\left (\frac {\sqrt {2}\,\sqrt {3}}{2}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}^2}{2\,\mathrm {tan}\left (c+d\,x\right )-3}+1\right )}\right )\,\sqrt {\frac {-\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \] 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)*((2^(1/2)*3^(1/2))/2 - tan(c + d*x) ^(1/2)))/((2*tan(c + d*x) - 3)^(1/2)*((2*((2^(1/2)*3^(1/2))/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))/2 - tan(c + d*x )^(1/2)))/((2*tan(c + d*x) - 3)^(1/2)*((2*((2^(1/2)*3^(1/2))/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*i nt((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)