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

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

3.7.55.2 Mathematica [A] (verified)

Time = 0.19 (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)
 

3.7.55.3 Rubi [A] (verified)

Time = 0.31 (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
 

3.7.55.3.1 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]
 

3.7.55.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(479\) vs. \(2(73)=146\).

Time = 5.40 (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 {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\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 {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\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 {2 \sqrt {13}-6}\, \sqrt {13}\, \sqrt {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\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 {2 \sqrt {13}+6}\, \arctan \left (\frac {\sqrt {2 \sqrt {13}-6}\, \sqrt {\frac {\left (11 \sqrt {13}-39\right ) \tan \left (d x +c \right ) \left (2+3 \tan \left (d x +c \right )\right ) \left (39+11 \sqrt {13}\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*(2*13^(1/2)-6)^(1/2)*13^(1/2)*(2*13^(1/2)+6)^(1/2 
)*arctan(1/416*(2*13^(1/2)-6)^(1/2)*((11*13^(1/2)-39)*tan(d*x+c)*(2+3*tan( 
d*x+c))*(39+11*13^(1/2))/(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)))-11*(2*13^(1/2)-6)^(1/2)*(2*13^(1/2)+6)^(1/2)*arc 
tan(1/416*(2*13^(1/2)-6)^(1/2)*((11*13^(1/2)-39)*tan(d*x+c)*(2+3*tan(d*x+c 
))*(39+11*13^(1/2))/(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*arctan 
h(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)
 

3.7.55.5 Fricas [B] (verification not implemented)

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

Time = 0.35 (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...
 

3.7.55.6 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)
 

3.7.55.7 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.
 

3.7.55.8 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. 489 vs. \(2 (65) = 130\).

Time = 0.45 (sec) , antiderivative size = 489, normalized size of antiderivative = 5.49 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {2+3 \tan (c+d x)}} \, dx=-\frac {\sqrt {3} {\left (\left (3 i + 2\right ) \, \sqrt {6 \, \sqrt {13} - 18} {\left (-\frac {2 i}{\sqrt {13} - 3} + 1\right )} \log \left (\left (120 i + 40\right ) \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + \left (432 i + 144\right ) \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + 80 \, \sqrt {13} \sqrt {15 \, \sqrt {13} + 54} - 800 \, \sqrt {13} - \left (16 i - 288\right ) \, \sqrt {15 \, \sqrt {13} + 54} - 2880\right ) - \left (3 i + 2\right ) \, \sqrt {6 \, \sqrt {13} - 18} {\left (-\frac {2 i}{\sqrt {13} - 3} + 1\right )} \log \left (\left (120 i + 40\right ) \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + \left (432 i + 144\right ) \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 80 \, \sqrt {13} \sqrt {15 \, \sqrt {13} + 54} - 800 \, \sqrt {13} + \left (16 i - 288\right ) \, \sqrt {15 \, \sqrt {13} + 54} - 2880\right ) + \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )} \log \left (8 \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 24 \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} + 8 \, \sqrt {13} \sqrt {6 \, \sqrt {13} - 18} - \left (48 i + 16\right ) \, \sqrt {13} + \left (16 i - 24\right ) \, \sqrt {6 \, \sqrt {13} - 18} + 144 i + 48\right ) - \left (2 i + 3\right ) \, \sqrt {6 \, \sqrt {13} + 18} {\left (-\frac {2 i}{\sqrt {13} + 3} + 1\right )} \log \left (8 \, \sqrt {13} {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 24 \, {\left (\sqrt {3} \sqrt {\tan \left (d x + c\right )} - \sqrt {3 \, \tan \left (d x + c\right ) + 2}\right )}^{2} - 8 \, \sqrt {13} \sqrt {6 \, \sqrt {13} - 18} - \left (48 i + 16\right ) \, \sqrt {13} - \left (16 i - 24\right ) \, \sqrt {6 \, \sqrt {13} - 18} + 144 i + 48\right )\right )}}{156 \, d} \]

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)*((3*I + 2)*sqrt(6*sqrt(13) - 18)*(-2*I/(sqrt(13) - 3) + 1)* 
log((120*I + 40)*sqrt(13)*(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c 
) + 2))^2 + (432*I + 144)*(sqrt(3)*sqrt(tan(d*x + c)) - sqrt(3*tan(d*x + c 
) + 2))^2 + 80*sqrt(13)*sqrt(15*sqrt(13) + 54) - 800*sqrt(13) - (16*I - 28 
8)*sqrt(15*sqrt(13) + 54) - 2880) - (3*I + 2)*sqrt(6*sqrt(13) - 18)*(-2*I/ 
(sqrt(13) - 3) + 1)*log((120*I + 40)*sqrt(13)*(sqrt(3)*sqrt(tan(d*x + c)) 
- sqrt(3*tan(d*x + c) + 2))^2 + (432*I + 144)*(sqrt(3)*sqrt(tan(d*x + c)) 
- sqrt(3*tan(d*x + c) + 2))^2 - 80*sqrt(13)*sqrt(15*sqrt(13) + 54) - 800*s 
qrt(13) + (16*I - 288)*sqrt(15*sqrt(13) + 54) - 2880) + (2*I + 3)*sqrt(6*s 
qrt(13) + 18)*(-2*I/(sqrt(13) + 3) + 1)*log(8*sqrt(13)*(sqrt(3)*sqrt(tan(d 
*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 24*(sqrt(3)*sqrt(tan(d*x + c)) - 
sqrt(3*tan(d*x + c) + 2))^2 + 8*sqrt(13)*sqrt(6*sqrt(13) - 18) - (48*I + 1 
6)*sqrt(13) + (16*I - 24)*sqrt(6*sqrt(13) - 18) + 144*I + 48) - (2*I + 3)* 
sqrt(6*sqrt(13) + 18)*(-2*I/(sqrt(13) + 3) + 1)*log(8*sqrt(13)*(sqrt(3)*sq 
rt(tan(d*x + c)) - sqrt(3*tan(d*x + c) + 2))^2 - 24*(sqrt(3)*sqrt(tan(d*x 
+ c)) - sqrt(3*tan(d*x + c) + 2))^2 - 8*sqrt(13)*sqrt(6*sqrt(13) - 18) - ( 
48*I + 16)*sqrt(13) - (16*I - 24)*sqrt(6*sqrt(13) - 18) + 144*I + 48))/d
 

3.7.55.9 Mupad [B] (verification not implemented)

Time = 6.86 (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