\(\int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx\) [658]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}}\right )}{\sqrt {3-2 i} d}+\frac {\arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-2-3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \] Output:

arctan((3-2*I)^(1/2)*tan(d*x+c)^(1/2)/(-2-3*tan(d*x+c))^(1/2))/(3-2*I)^(1/ 
2)/d+arctan((3+2*I)^(1/2)*tan(d*x+c)^(1/2)/(-2-3*tan(d*x+c))^(1/2))/(3+2*I 
)^(1/2)/d
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\frac {-\sqrt {3+2 i} \arctan \left (\frac {\sqrt {\frac {3}{13}+\frac {2 i}{13}} \sqrt {-2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )+\sqrt {-3+2 i} \text {arctanh}\left (\frac {\sqrt {-\frac {3}{13}+\frac {2 i}{13}} \sqrt {-2-3 \tan (c+d x)}}{\sqrt {\tan (c+d x)}}\right )}{\sqrt {13} d} \] Input:

Integrate[1/(Sqrt[-2 - 3*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]
 

Output:

(-(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)
 

Rubi [A] (verified)

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.

\(\displaystyle \int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)}}dx\)

\(\Big \downarrow \) 4058

\(\displaystyle \frac {\int \frac {1}{\sqrt {-3 \tan (c+d x)-2} \sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 615

\(\displaystyle \frac {\int \left (\frac {i}{2 \sqrt {-3 \tan (c+d x)-2} (i-\tan (c+d x)) \sqrt {\tan (c+d x)}}+\frac {i}{2 \sqrt {-3 \tan (c+d x)-2} (\tan (c+d x)+i) \sqrt {\tan (c+d x)}}\right )d\tan (c+d x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {3-2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3-2 i}}+\frac {\arctan \left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {-3 \tan (c+d x)-2}}\right )}{\sqrt {3+2 i}}}{d}\)

Input:

Int[1/(Sqrt[-2 - 3*Tan[c + d*x]]*Sqrt[Tan[c + d*x]]),x]
 

Output:

(ArcTan[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + d*x]]]/Sqrt 
[3 - 2*I] + ArcTan[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[-2 - 3*Tan[c + 
d*x]]]/Sqrt[3 + 2*I])/d
 

Defintions of rubi rules used

rule 615

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4058

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(426\) vs. \(2(73)=146\).

Time = 0.58 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.80

method	result	size
derivativedivides	\(\frac {\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (3 \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-11 \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-4 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{2 d \sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right ) \sqrt {\tan \left (d x +c \right )}}\)	\(427\)
default	\(\frac {\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \left (3 \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-11 \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \operatorname {arctanh}\left (\frac {\sqrt {-6+2 \sqrt {13}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \sqrt {13}}{208 \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}\right )-4 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right ) \sqrt {13}+12 \arctan \left (\frac {4 \sqrt {13}\, \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{\sqrt {26 \sqrt {13}+78}}\right )\right ) \sqrt {-\frac {\tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}}{2 d \sqrt {-2-3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right ) \sqrt {\tan \left (d x +c \right )}}\)	\(427\)

Input:

int(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/2/d*(13^(1/2)-3+2*tan(d*x+c))*(3*13^(1/2)*(2*13^(1/2)+6)^(1/2)*(-6+2*13^ 
(1/2))^(1/2)*arctanh(1/208*(-6+2*13^(1/2))^(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))-11*(2*13^(1/2 
)+6)^(1/2)*(-6+2*13^(1/2))^(1/2)*arctanh(1/208*(-6+2*13^(1/2))^(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))-4*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)+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)))/(-2-3*tan(d*x+c))^(1/2)/(2*13^(1/2)+6)^(1/2)/(11*13^(1/2)-39 
)*(-tan(d*x+c)*(2+3*tan(d*x+c))/(13^(1/2)-3+2*tan(d*x+c))^2)^(1/2)/tan(d*x 
+c)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (65) = 130\).

Time = 0.16 (sec) , antiderivative size = 1477, normalized size of antiderivative = 16.60 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="fricas" 
)
 

Output:

1/8*sqrt(1/13)*sqrt(-(2*d^2*sqrt(-1/d^4) + 3)/d^2)*log((sqrt(1/13)*(135*d* 
tan(d*x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*ta 
n(d*x + c) - 56*d^3)*sqrt(-1/d^4) + 33*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)*(135*d*ta 
n(d*x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*tan( 
d*x + c) - 56*d^3)*sqrt(-1/d^4) + 33*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)*(135*d*tan(d 
*x + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*tan(d*x 
 + c) - 56*d^3)*sqrt(-1/d^4) + 33*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)*(135*d*tan(d*x 
 + c)^2 + 211*d*tan(d*x + c) - (155*d^3*tan(d*x + c)^2 - 102*d^3*tan(d*x + 
 c) - 56*d^3)*sqrt(-1/d^4) + 33*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*sqr...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {- 3 \tan {\left (c + d x \right )} - 2} \sqrt {\tan {\left (c + d x \right )}}}\, dx \] Input:

integrate(1/(-2-3*tan(d*x+c))**(1/2)/tan(d*x+c)**(1/2),x)
 

Output:

Integral(1/(sqrt(-3*tan(c + d*x) - 2)*sqrt(tan(c + d*x))), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="maxima" 
)
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (65) = 130\).

Time = 0.20 (sec) , antiderivative size = 269, normalized size of antiderivative = 3.02 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + 3 \, {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} - \left (6 i - 2\right ) \, \sqrt {13} - 18 i + 6}{\sqrt {13} \sqrt {6 \, \sqrt {13} + 18} + \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18}}\right )}{d \sqrt {6 \, \sqrt {13} + 18} {\left (\frac {2 i}{\sqrt {13} + 3} + 1\right )}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {13} {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + 3 \, {\left (\sqrt {3} \sqrt {-\tan \left (d x + c\right )} - \sqrt {-3 \, \tan \left (d x + c\right ) - 2}\right )}^{2} + \left (6 i + 2\right ) \, \sqrt {13} + 18 i + 6}{\sqrt {13} \sqrt {6 \, \sqrt {13} + 18} - \left (2 i - 3\right ) \, \sqrt {6 \, \sqrt {13} + 18}}\right )}{d \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )}} \] Input:

integrate(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="giac")
 

Output:

-2*sqrt(3)*arctan((sqrt(13)*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(-3*tan(d*x 
 + c) - 2))^2 + 3*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(-3*tan(d*x + c) - 2) 
)^2 - (6*I - 2)*sqrt(13) - 18*I + 6)/(sqrt(13)*sqrt(6*sqrt(13) + 18) + (2* 
I + 3)*sqrt(6*sqrt(13) + 18)))/(d*sqrt(6*sqrt(13) + 18)*(2*I/(sqrt(13) + 3 
) + 1)) + 2*sqrt(3)*arctan((sqrt(13)*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(- 
3*tan(d*x + c) - 2))^2 + 3*(sqrt(3)*sqrt(-tan(d*x + c)) - sqrt(-3*tan(d*x 
+ c) - 2))^2 + (6*I + 2)*sqrt(13) + 18*I + 6)/(sqrt(13)*sqrt(6*sqrt(13) + 
18) - (2*I - 3)*sqrt(6*sqrt(13) + 18)))/(d*sqrt(6*sqrt(13) + 18)*(-2*I/(sq 
rt(13) + 3) + 1))
 

Mupad [B] (verification not implemented)

Time = 3.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-2\,\mathrm {atanh}\left (\frac {4\,d\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}+6\,d\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}}\right )\,\sqrt {\frac {-\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}-2\,\mathrm {atanh}\left (\frac {4\,d\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}+6\,d\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-3\,\mathrm {tan}\left (c+d\,x\right )-2}}\right )\,\sqrt {\frac {-\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}} \] Input:

int(1/(tan(c + d*x)^(1/2)*(- 3*tan(c + d*x) - 2)^(1/2)),x)
 

Output:

- 2*atanh((4*d*((- 3/52 - 1i/26)/d^2)^(1/2) + 6*d*tan(c + d*x)*((- 3/52 - 
1i/26)/d^2)^(1/2))/(tan(c + d*x)^(1/2)*(- 3*tan(c + d*x) - 2)^(1/2)))*((- 
3/52 - 1i/26)/d^2)^(1/2) - 2*atanh((4*d*((- 3/52 + 1i/26)/d^2)^(1/2) + 6*d 
*tan(c + d*x)*((- 3/52 + 1i/26)/d^2)^(1/2))/(tan(c + d*x)^(1/2)*(- 3*tan(c 
 + d*x) - 2)^(1/2)))*((- 3/52 + 1i/26)/d^2)^(1/2)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {-2-3 \tan (c+d x)} \sqrt {\tan (c+d x)}} \, dx=-\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {-3 \tan \left (d x +c \right )-2}}{3 \tan \left (d x +c \right )^{2}+2 \tan \left (d x +c \right )}d x \right ) \] Input:

int(1/(-2-3*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x)
 

Output:

 - int((sqrt(tan(c + d*x))*sqrt( - 3*tan(c + d*x) - 2))/(3*tan(c + d*x)**2 
 + 2*tan(c + d*x)),x)