\(\int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx\) [655]

Optimal result

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

arctanh((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+arctanh((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 = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {-3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}}\right )}{\sqrt {-3+2 i} d}+\frac {\text {arctanh}\left (\frac {\sqrt {3+2 i} \sqrt {\tan (c+d x)}}{\sqrt {2+3 \tan (c+d x)}}\right )}{\sqrt {3+2 i} d} \] Input:

Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[2 + 3*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]*d) + ArcTanh[(Sqrt[3 + 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[ 
c + d*x]]]/(Sqrt[3 + 2*I]*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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4058

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(ArcTanh[(Sqrt[3 - 2*I]*Sqrt[Tan[c + d*x]])/Sqrt[2 + 3*Tan[c + d*x]]]/Sqrt 
[3 - 2*I] + ArcTanh[(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. \(479\) vs. \(2(73)=146\).

Time = 20.66 (sec) , antiderivative size = 480, normalized size of antiderivative = 5.39

method	result	size
derivativedivides	\(\frac {\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}}}\, \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}}\, \arctan \left (\frac {\sqrt {-6+2 \sqrt {13}}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )-11 \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \arctan \left (\frac {\sqrt {-6+2 \sqrt {13}}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )+4 \,\operatorname {arctanh}\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 \,\operatorname {arctanh}\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 )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2+3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\)	\(480\)
default	\(\frac {\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}}}\, \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}}\, \arctan \left (\frac {\sqrt {-6+2 \sqrt {13}}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )-11 \sqrt {2 \sqrt {13}+6}\, \sqrt {-6+2 \sqrt {13}}\, \arctan \left (\frac {\sqrt {-6+2 \sqrt {13}}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (39+11 \sqrt {13}\right ) \left (2+3 \tan \left (d x +c \right )\right )}{\left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )^{2}}}\, \left (3 \sqrt {13}+11\right ) \left (\sqrt {13}+3-2 \tan \left (d x +c \right )\right ) \left (11 \sqrt {13}-39\right ) \left (\sqrt {13}-3+2 \tan \left (d x +c \right )\right )}{416 \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right )}\right )+4 \,\operatorname {arctanh}\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 \,\operatorname {arctanh}\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 )}{2 d \sqrt {\tan \left (d x +c \right )}\, \sqrt {2+3 \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {13}+6}\, \left (11 \sqrt {13}-39\right )}\)	\(480\)

Input:

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

Output:

1/2/d*(tan(d*x+c)*(2+3*tan(d*x+c))/(13^(1/2)-3+2*tan(d*x+c))^2)^(1/2)*(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)*arctan(1/416*(-6+2*13^(1/2))^(1/2)*((11*13^(1/2)-39)*tan(d*x+c)*(39+11* 
13^(1/2))*(2+3*tan(d*x+c))/(13^(1/2)-3+2*tan(d*x+c))^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))/t 
an(d*x+c)/(2+3*tan(d*x+c)))-11*(2*13^(1/2)+6)^(1/2)*(-6+2*13^(1/2))^(1/2)* 
arctan(1/416*(-6+2*13^(1/2))^(1/2)*((11*13^(1/2)-39)*tan(d*x+c)*(39+11*13^ 
(1/2))*(2+3*tan(d*x+c))/(13^(1/2)-3+2*tan(d*x+c))^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))/tan( 
d*x+c)/(2+3*tan(d*x+c)))+4*arctanh(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*ar 
ctanh(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)))/tan(d*x+c)^(1/2)/(2+3*tan(d*x+c))^(1/2)/(2 
*13^(1/2)+6)^(1/2)/(11*13^(1/2)-39)
 

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 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

integrate(1/tan(d*x+c)^(1/2)/(2+3*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*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*sqrt(1/13)*s 
qrt((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*ta 
n(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)*sqr...
 

Sympy [F]

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

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

integrate(1/tan(d*x+c)^(1/2)/(2+3*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. 663 vs. \(2 (65) = 130\).

Time = 0.22 (sec) , antiderivative size = 663, normalized size of antiderivative = 7.45 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/156*sqrt(3)*(sqrt(78*sqrt(13) + 234)*log(10816*(100*sqrt(13)*(sqrt(3)*sq 
rt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 276*(sqrt(3)*sqrt(tan(d*x 
 + c)) - sqrt(3*tan(d*x + c) + 2))^2 + 25*sqrt(13)*sqrt(150*sqrt(13) - 414 
) + 250*sqrt(13) - 69*sqrt(150*sqrt(13) - 414) - 690)^2 + 10816*(75*sqrt(1 
3)*(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 207*(sqrt(3 
)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 750*sqrt(13) - 58*sqr 
t(150*sqrt(13) - 414) + 2070)^2) - sqrt(78*sqrt(13) + 234)*log(10816*(100* 
sqrt(13)*(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 276*( 
sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 25*sqrt(13)*sqr 
t(150*sqrt(13) - 414) + 250*sqrt(13) + 69*sqrt(150*sqrt(13) - 414) - 690)^ 
2 + 10816*(75*sqrt(13)*(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 
 2))^2 - 207*(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 7 
50*sqrt(13) + 58*sqrt(150*sqrt(13) - 414) + 2070)^2) - 4*sqrt(78*sqrt(13) 
+ 234)*(arctan(3/4) + arctan(((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x 
 + c) + 2))^2*(1725*sqrt(13) - 6443) + 363*sqrt(13)*sqrt(150*sqrt(13) - 41 
4) - 3450*sqrt(13) - 1271*sqrt(150*sqrt(13) - 414) + 12886)/(91*sqrt(13)*s 
qrt(150*sqrt(13) - 414) + 10350*sqrt(13) - 453*sqrt(150*sqrt(13) - 414) - 
38658)))/(sqrt(13) + 3) + 4*sqrt(78*sqrt(13) + 234)*(arctan(3/4) + arctan( 
-((sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2*(1725*sqrt(13) 
 - 6443) - 363*sqrt(13)*sqrt(150*sqrt(13) - 414) - 3450*sqrt(13) + 1271...
 

Mupad [B] (verification not implemented)

Time = 3.43 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.33 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx=-\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (4-6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}\,\left (-4+6{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}+2}\right )\,\sqrt {\frac {\frac {3}{52}-\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {\sqrt {2}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\left (4+6{}\mathrm {i}\right )+d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}\,\left (-4-6{}\mathrm {i}\right )}{3\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {2}\,\sqrt {3\,\mathrm {tan}\left (c+d\,x\right )+2}+2}\right )\,\sqrt {\frac {\frac {3}{52}+\frac {1}{26}{}\mathrm {i}}{d^2}}\,2{}\mathrm {i} \] Input:

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

Output:

atan((2^(1/2)*d*tan(c + d*x)^(1/2)*((3/52 + 1i/26)/d^2)^(1/2)*(4 + 6i) - d 
*tan(c + d*x)^(1/2)*((3/52 + 1i/26)/d^2)^(1/2)*(3*tan(c + d*x) + 2)^(1/2)* 
(4 + 6i))/(3*tan(c + d*x) - 2^(1/2)*(3*tan(c + d*x) + 2)^(1/2) + 2))*((3/5 
2 + 1i/26)/d^2)^(1/2)*2i - atan((2^(1/2)*d*tan(c + d*x)^(1/2)*((3/52 - 1i/ 
26)/d^2)^(1/2)*(4 - 6i) - d*tan(c + d*x)^(1/2)*((3/52 - 1i/26)/d^2)^(1/2)* 
(3*tan(c + d*x) + 2)^(1/2)*(4 - 6i))/(3*tan(c + d*x) - 2^(1/2)*(3*tan(c + 
d*x) + 2)^(1/2) + 2))*((3/52 - 1i/26)/d^2)^(1/2)*2i
 

Reduce [F]

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

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