\(\int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx\) [1140]

Optimal result

Integrand size = 32, antiderivative size = 121 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {2} \sqrt {a} f}+\frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \] Output:

-1/2*I*(c-I*d)^(1/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/(c-I*d 
)^(1/2)/(a+I*a*tan(f*x+e))^(1/2))*2^(1/2)/a^(1/2)/f+I*(c+d*tan(f*x+e))^(1/ 
2)/f/(a+I*a*tan(f*x+e))^(1/2)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {i \left (\frac {\sqrt {2} \sqrt {-a (c-i d)} \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{a}+\frac {2 \sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{2 f} \] Input:

Integrate[Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + I*a*Tan[e + f*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3042, 4029, 3042, 4027, 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.

\(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}dx\)

\(\Big \downarrow \) 4029

\(\displaystyle \frac {(c-i d) \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c-i d) \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a}+\frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 4027

\(\displaystyle \frac {i \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {i a (c-i d) \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}\)

\(\Big \downarrow \) 221

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

Input:

Int[Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + I*a*Tan[e + f*x]],x]
 

Output:

((-I)*Sqrt[c - I*d]*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sq 
rt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*Sqrt[a]*f) + (I*Sqrt[c 
+ d*Tan[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])
 

Defintions of rubi rules used

rule 221

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]
 

rule 3042

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

rule 4027

Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) 
 + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f)   Subst[Int[1/(a*c - b*d - 2* 
a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; 
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N 
eQ[c^2 + d^2, 0]
 

rule 4029

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e 
 + f*x])^n/(2*b*f*m)), x] - Simp[(a*c - b*d)/(2*b^2)   Int[(a + b*Tan[e + f 
*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, 
f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Eq 
Q[m + n, 0] && LeQ[m, -2^(-1)]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 876 vs. \(2 (95 ) = 190\).

Time = 0.99 (sec) , antiderivative size = 877, normalized size of antiderivative = 7.25

method	result	size
derivativedivides	\(-\frac {\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (-i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \tan \left (f x +e \right )^{2}+2 i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \tan \left (f x +e \right )-\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \tan \left (f x +e \right )^{2}+i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d -2 \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \tan \left (f x +e \right )+\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c -4 i c \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \tan \left (f x +e \right )-4 i \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, d +4 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, d \tan \left (f x +e \right )-4 c \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\right )}{4 f a \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \left (i c -d \right ) \left (-\tan \left (f x +e \right )+i\right )^{2}}\)	\(877\)
default	\(-\frac {\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (-i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \tan \left (f x +e \right )^{2}+2 i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \tan \left (f x +e \right )-\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c \tan \left (f x +e \right )^{2}+i \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d -2 \ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, d \tan \left (f x +e \right )+\ln \left (\frac {3 a c +i a \tan \left (f x +e \right ) c -i a d +3 a d \tan \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}}{i+\tan \left (f x +e \right )}\right ) \sqrt {2}\, \sqrt {-a \left (i d -c \right )}\, c -4 i c \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \tan \left (f x +e \right )-4 i \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, d +4 \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, d \tan \left (f x +e \right )-4 c \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\right )}{4 f a \sqrt {a \left (c +d \tan \left (f x +e \right )\right ) \left (1+i \tan \left (f x +e \right )\right )}\, \left (i c -d \right ) \left (-\tan \left (f x +e \right )+i\right )^{2}}\)	\(877\)

Input:

int((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

-1/4/f*(c+d*tan(f*x+e))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a*(-I*ln((3*a*c+I 
*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+ 
d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c) 
)^(1/2)*d*tan(f*x+e)^2+2*I*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+ 
e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2 
))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1/2)*c*tan(f*x+e)-ln((3*a*c+I*a*t 
an(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*ta 
n(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1 
/2)*c*tan(f*x+e)^2+I*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2 
^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+ 
tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1/2)*d-2*ln((3*a*c+I*a*tan(f*x+e)*c-I*a 
*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I* 
tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2)*(-a*(I*d-c))^(1/2)*d*tan(f*x+e 
)+ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c)) 
^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(I+tan(f*x+e)))*2^(1/2 
)*(-a*(I*d-c))^(1/2)*c-4*I*c*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*t 
an(f*x+e)-4*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d+4*(a*(c+d*tan( 
f*x+e))*(1+I*tan(f*x+e)))^(1/2)*d*tan(f*x+e)-4*c*(a*(c+d*tan(f*x+e))*(1+I* 
tan(f*x+e)))^(1/2))/(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)/(I*c-d)/(- 
tan(f*x+e)+I)^2
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (89) = 178\).

Time = 0.13 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.91 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {{\left (\sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} \log \left (i \, \sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - \sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} \log \left (-i \, \sqrt {2} a f \sqrt {-\frac {c - i \, d}{a f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}\right ) - 2 \, \sqrt {2} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (2 i \, f x + 2 i \, e\right )} - i\right )}\right )} e^{\left (-i \, f x - i \, e\right )}}{4 \, a f} \] Input:

integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(1/2),x, algorithm="fr 
icas")
 

Output:

1/4*(sqrt(2)*a*f*sqrt(-(c - I*d)/(a*f^2))*e^(I*f*x + I*e)*log(I*sqrt(2)*a* 
f*sqrt(-(c - I*d)/(a*f^2))*e^(I*f*x + I*e) + sqrt(2)*sqrt(((c - I*d)*e^(2* 
I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 
2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 1)) - sqrt(2)*a*f*sqrt(-(c - I*d)/(a*f 
^2))*e^(I*f*x + I*e)*log(-I*sqrt(2)*a*f*sqrt(-(c - I*d)/(a*f^2))*e^(I*f*x 
+ I*e) + sqrt(2)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f* 
x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*(e^(2*I*f*x + 2*I*e) + 
1)) - 2*sqrt(2)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x 
 + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*(-I*e^(2*I*f*x + 2*I*e) 
- I))*e^(-I*f*x - I*e)/(a*f)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\sqrt {c + d \tan {\left (e + f x \right )}}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )}}\, dx \] Input:

integrate((c+d*tan(f*x+e))**(1/2)/(a+I*a*tan(f*x+e))**(1/2),x)
 

Output:

Integral(sqrt(c + d*tan(e + f*x))/sqrt(I*a*(tan(e + f*x) - I)), x)
 

Maxima [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Timed out} \] Input:

integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(1/2),x, algorithm="ma 
xima")
 

Output:

Timed out
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(1/2),x, algorithm="gi 
ac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeError: Bad Argument TypeError: Bad Argument TypeRecursive ass 
umption s
 

Mupad [B] (verification not implemented)

Time = 16.80 (sec) , antiderivative size = 1724, normalized size of antiderivative = 14.25 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Too large to display} \] Input:

int((c + d*tan(e + f*x))^(1/2)/(a + a*tan(e + f*x)*1i)^(1/2),x)
 

Output:

(2*(c + d*1i)*((a + a*tan(e + f*x)*1i)^(1/2) - a^(1/2)))/(d*f*((c + d*tan( 
e + f*x))^(1/2) - c^(1/2))*((a*1i)/d + ((a + a*tan(e + f*x)*1i)^(1/2) - a^ 
(1/2))^2/((c + d*tan(e + f*x))^(1/2) - c^(1/2))^2 - (a^(1/2)*c^(1/2)*((a + 
 a*tan(e + f*x)*1i)^(1/2) - a^(1/2))*2i)/(d*((c + d*tan(e + f*x))^(1/2) - 
c^(1/2))))) - (2^(1/2)*atan(((2^(1/2)*(d*1i - c)^(1/2)*(4*d^7*(4*a^(3/2)*c 
^(3/2)*f - a^(3/2)*c^(1/2)*d*f*4i) + (16*d^7*((a + a*tan(e + f*x)*1i)^(1/2 
) - a^(1/2))*(a*d^2*f - a*c^2*f + a*c*d*f*2i))/((c + d*tan(e + f*x))^(1/2) 
 - c^(1/2)) - (4*d^8*(a^(1/2)*c^(3/2)*f*4i + 4*a^(1/2)*c^(1/2)*d*f)*((a + 
a*tan(e + f*x)*1i)^(1/2) - a^(1/2))^2)/((c + d*tan(e + f*x))^(1/2) - c^(1/ 
2))^2 - (2^(1/2)*(d*1i - c)^(1/2)*(4*d^7*(a^2*c*f^2*4i - 4*a^2*d*f^2) - (1 
6*d^7*(a^(3/2)*c^(3/2)*f^2*2i + 6*a^(3/2)*c^(1/2)*d*f^2)*((a + a*tan(e + f 
*x)*1i)^(1/2) - a^(1/2)))/((c + d*tan(e + f*x))^(1/2) - c^(1/2)) + (4*d^8* 
(20*a*c*f^2 - a*d*f^2*12i)*((a + a*tan(e + f*x)*1i)^(1/2) - a^(1/2))^2)/(( 
c + d*tan(e + f*x))^(1/2) - c^(1/2))^2))/(4*a^(1/2)*f))*1i)/(4*a^(1/2)*f) 
+ (2^(1/2)*(d*1i - c)^(1/2)*(4*d^7*(4*a^(3/2)*c^(3/2)*f - a^(3/2)*c^(1/2)* 
d*f*4i) + (16*d^7*((a + a*tan(e + f*x)*1i)^(1/2) - a^(1/2))*(a*d^2*f - a*c 
^2*f + a*c*d*f*2i))/((c + d*tan(e + f*x))^(1/2) - c^(1/2)) - (4*d^8*(a^(1/ 
2)*c^(3/2)*f*4i + 4*a^(1/2)*c^(1/2)*d*f)*((a + a*tan(e + f*x)*1i)^(1/2) - 
a^(1/2))^2)/((c + d*tan(e + f*x))^(1/2) - c^(1/2))^2 + (2^(1/2)*(d*1i - c) 
^(1/2)*(4*d^7*(a^2*c*f^2*4i - 4*a^2*d*f^2) - (16*d^7*(a^(3/2)*c^(3/2)*f...
 

Reduce [F]

\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {too large to display} \] Input:

int((c+d*tan(f*x+e))^(1/2)/(a+I*a*tan(f*x+e))^(1/2),x)
 

Output:

(2*sqrt(a)*(3*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + f* 
x)*c**3*d*i + 7*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e + 
f*x)*c**2*d**2 - 5*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e 
 + f*x)*c*d**3*i - sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*tan(e 
 + f*x)*d**4 - sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*c**4*i - 
5*sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*c**3*d + 7*sqrt(tan(e 
+ f*x)*i + 1)*sqrt(tan(e + f*x)*d + c)*c**2*d**2*i + 3*sqrt(tan(e + f*x)*i 
 + 1)*sqrt(tan(e + f*x)*d + c)*c*d**3 + 6*int((sqrt(tan(e + f*x)*i + 1)*sq 
rt(tan(e + f*x)*d + c)*tan(e + f*x))/(3*tan(e + f*x)**3*c**4*d*i + 8*tan(e 
 + f*x)**3*c**3*d**2 - 14*tan(e + f*x)**3*c**2*d**3*i - 25*tan(e + f*x)**3 
*d**5*i + 3*tan(e + f*x)**2*c**5*i + 8*tan(e + f*x)**2*c**4*d - 14*tan(e + 
 f*x)**2*c**3*d**2*i - 25*tan(e + f*x)**2*c*d**4*i + 3*tan(e + f*x)*c**4*d 
*i + 8*tan(e + f*x)*c**3*d**2 - 14*tan(e + f*x)*c**2*d**3*i - 25*tan(e + f 
*x)*d**5*i + 3*c**5*i + 8*c**4*d - 14*c**3*d**2*i - 25*c*d**4*i),x)*tan(e 
+ f*x)**2*c**9*f - 34*int((sqrt(tan(e + f*x)*i + 1)*sqrt(tan(e + f*x)*d + 
c)*tan(e + f*x))/(3*tan(e + f*x)**3*c**4*d*i + 8*tan(e + f*x)**3*c**3*d**2 
 - 14*tan(e + f*x)**3*c**2*d**3*i - 25*tan(e + f*x)**3*d**5*i + 3*tan(e + 
f*x)**2*c**5*i + 8*tan(e + f*x)**2*c**4*d - 14*tan(e + f*x)**2*c**3*d**2*i 
 - 25*tan(e + f*x)**2*c*d**4*i + 3*tan(e + f*x)*c**4*d*i + 8*tan(e + f*x)* 
c**3*d**2 - 14*tan(e + f*x)*c**2*d**3*i - 25*tan(e + f*x)*d**5*i + 3*c*...