\(\int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx\) [661]

Optimal result

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

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+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
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {-2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {-2+3 i} d}+\frac {\text {arctanh}\left (\frac {\sqrt {2+3 i} \sqrt {\tan (c+d x)}}{\sqrt {-3+2 \tan (c+d x)}}\right )}{\sqrt {2+3 i} d} \] Input:

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

Output:

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)
 

Rubi [A] (verified)

Time = 0.29 (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

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

Input:

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

Output:

(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
 

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. \(479\) vs. \(2(73)=146\).

Time = 1.02 (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 )}\)	\(480\)

Input:

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

Output:

-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)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1485 vs. \(2 (65) = 130\).

Time = 0.24 (sec) , antiderivative size = 1485, normalized size of antiderivative = 16.69 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

integrate(1/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(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...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (65) = 130\).

Time = 0.19 (sec) , antiderivative size = 470, normalized size of antiderivative = 5.28 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=\frac {\sqrt {2} {\left (\sqrt {13 \, \sqrt {13} + 26} \log \left (13689 \, {\left (\sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 2 \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} + 2 \, \sqrt {13} \sqrt {\sqrt {13} - 2} + 3 \, \sqrt {13} - 4 \, \sqrt {\sqrt {13} - 2} - 6\right )}^{2} + 54756 \, {\left (2 \, \sqrt {13} + 3 \, \sqrt {\sqrt {13} - 2} - 4\right )}^{2}\right ) - \sqrt {13 \, \sqrt {13} + 26} \log \left (13689 \, {\left (\sqrt {13} {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 2 \, {\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} - 2 \, \sqrt {13} \sqrt {\sqrt {13} - 2} + 3 \, \sqrt {13} + 4 \, \sqrt {\sqrt {13} - 2} - 6\right )}^{2} + 54756 \, {\left (2 \, \sqrt {13} - 3 \, \sqrt {\sqrt {13} - 2} - 4\right )}^{2}\right ) - \frac {6 \, \sqrt {13 \, \sqrt {13} + 26} \arctan \left (-\frac {{\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} {\left (\sqrt {13} - 2\right )} + 2 \, \sqrt {13} \sqrt {\sqrt {13} - 2} + 3 \, \sqrt {13} - 4 \, \sqrt {\sqrt {13} - 2} - 6}{2 \, {\left (2 \, \sqrt {13} + 3 \, \sqrt {\sqrt {13} - 2} - 4\right )}}\right )}{\sqrt {13} + 2} + \frac {6 \, \sqrt {13 \, \sqrt {13} + 26} \arctan \left (-\frac {{\left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} - \sqrt {2 \, \tan \left (d x + c\right ) - 3}\right )}^{2} {\left (\sqrt {13} - 2\right )} - 2 \, \sqrt {13} \sqrt {\sqrt {13} - 2} + 3 \, \sqrt {13} + 4 \, \sqrt {\sqrt {13} - 2} - 6}{2 \, {\left (2 \, \sqrt {13} - 3 \, \sqrt {\sqrt {13} - 2} - 4\right )}}\right )}{\sqrt {13} + 2}\right )}}{52 \, d} \] Input:

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

Output:

1/52*sqrt(2)*(sqrt(13*sqrt(13) + 26)*log(13689*(sqrt(13)*(sqrt(2)*sqrt(tan 
(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2 - 2*(sqrt(2)*sqrt(tan(d*x + c)) - 
 sqrt(2*tan(d*x + c) - 3))^2 + 2*sqrt(13)*sqrt(sqrt(13) - 2) + 3*sqrt(13) 
- 4*sqrt(sqrt(13) - 2) - 6)^2 + 54756*(2*sqrt(13) + 3*sqrt(sqrt(13) - 2) - 
 4)^2) - sqrt(13*sqrt(13) + 26)*log(13689*(sqrt(13)*(sqrt(2)*sqrt(tan(d*x 
+ c)) - sqrt(2*tan(d*x + c) - 3))^2 - 2*(sqrt(2)*sqrt(tan(d*x + c)) - sqrt 
(2*tan(d*x + c) - 3))^2 - 2*sqrt(13)*sqrt(sqrt(13) - 2) + 3*sqrt(13) + 4*s 
qrt(sqrt(13) - 2) - 6)^2 + 54756*(2*sqrt(13) - 3*sqrt(sqrt(13) - 2) - 4)^2 
) - 6*sqrt(13*sqrt(13) + 26)*arctan(-1/2*((sqrt(2)*sqrt(tan(d*x + c)) - sq 
rt(2*tan(d*x + c) - 3))^2*(sqrt(13) - 2) + 2*sqrt(13)*sqrt(sqrt(13) - 2) + 
 3*sqrt(13) - 4*sqrt(sqrt(13) - 2) - 6)/(2*sqrt(13) + 3*sqrt(sqrt(13) - 2) 
 - 4))/(sqrt(13) + 2) + 6*sqrt(13*sqrt(13) + 26)*arctan(-1/2*((sqrt(2)*sqr 
t(tan(d*x + c)) - sqrt(2*tan(d*x + c) - 3))^2*(sqrt(13) - 2) - 2*sqrt(13)* 
sqrt(sqrt(13) - 2) + 3*sqrt(13) + 4*sqrt(sqrt(13) - 2) - 6)/(2*sqrt(13) - 
3*sqrt(sqrt(13) - 2) - 4))/(sqrt(13) + 2))/d
 

Mupad [B] (verification not implemented)

Time = 3.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {-3+2 \tan (c+d x)}} \, dx=2\,\mathrm {atanh}\left (\frac {2\,d\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )-3}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}}\right )\,\sqrt {\frac {\frac {1}{26}-\frac {3}{52}{}\mathrm {i}}{d^2}}+2\,\mathrm {atanh}\left (\frac {2\,d\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}}\,\sqrt {2\,\mathrm {tan}\left (c+d\,x\right )-3}}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}}\right )\,\sqrt {\frac {\frac {1}{26}+\frac {3}{52}{}\mathrm {i}}{d^2}} \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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