\(\int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx\) [579]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx=\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-\tan (c+d x)}}{1-\tan (c+d x)}\right )}{\sqrt {2} d} \] Output:

-1/2*(a-b)*arctan(-1+2^(1/2)*(-tan(d*x+c))^(1/2))*2^(1/2)/d-1/2*(a-b)*arct 
an(1+2^(1/2)*(-tan(d*x+c))^(1/2))*2^(1/2)/d-1/2*(a+b)*arctanh(2^(1/2)*(-ta 
n(d*x+c))^(1/2)/(1-tan(d*x+c)))*2^(1/2)/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx=\frac {\sqrt [4]{-1} \left ((a-i b) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(a+i b) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right ) \tan ^{\frac {3}{2}}(c+d x)}{d (-\tan (c+d x))^{3/2}} \] Input:

Integrate[(a + b*Tan[c + d*x])/Sqrt[-Tan[c + d*x]],x]
 

Output:

((-1)^(1/4)*((a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (a + I*b)*A 
rcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])*Tan[c + d*x]^(3/2))/(d*(-Tan[c + d* 
x])^(3/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.34, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 4017, 25, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}}dx\)

\(\Big \downarrow \) 4017

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1482

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

\(\displaystyle \frac {2 \left (-\frac {1}{2} (a+b) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {-\tan (c+d x)}}{-\tan (c+d x)-\sqrt {2} \sqrt {-\tan (c+d x)}+1}d\sqrt {-\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{-\tan (c+d x)+\sqrt {2} \sqrt {-\tan (c+d x)}+1}d\sqrt {-\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \left (-\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {-\tan (c+d x)}}{-\tan (c+d x)-\sqrt {2} \sqrt {-\tan (c+d x)}+1}d\sqrt {-\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{-\tan (c+d x)+\sqrt {2} \sqrt {-\tan (c+d x)}+1}d\sqrt {-\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (-\frac {1}{2} (a+b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {-\tan (c+d x)}}{-\tan (c+d x)-\sqrt {2} \sqrt {-\tan (c+d x)}+1}d\sqrt {-\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {-\tan (c+d x)}+1}{-\tan (c+d x)+\sqrt {2} \sqrt {-\tan (c+d x)}+1}d\sqrt {-\tan (c+d x)}\right )-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {-\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a+b) \left (\frac {\log \left (-\tan (c+d x)+\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (-\tan (c+d x)-\sqrt {2} \sqrt {-\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}\)

Input:

Int[(a + b*Tan[c + d*x])/Sqrt[-Tan[c + d*x]],x]
 

Output:

(2*(-1/2*((a - b)*(-(ArcTan[1 - Sqrt[2]*Sqrt[-Tan[c + d*x]]]/Sqrt[2]) + Ar 
cTan[1 + Sqrt[2]*Sqrt[-Tan[c + d*x]]]/Sqrt[2])) - ((a + b)*(-1/2*Log[1 - S 
qrt[2]*Sqrt[-Tan[c + d*x]] - Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[ 
-Tan[c + d*x]] - Tan[c + d*x]]/(2*Sqrt[2])))/2))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 1482

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
a*c, 2]}, Simp[(d*q + a*e)/(2*a*c)   Int[(q + c*x^2)/(a + c*x^4), x], x] + 
Simp[(d*q - a*e)/(2*a*c)   Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a 
, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- 
a)*c]
 

rule 3042

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

rule 4017

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ 
)]], x_Symbol] :> Simp[2/f   Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq 
rt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] & 
& NeQ[c^2 + d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(201\) vs. \(2(99)=198\).

Time = 0.21 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.73

method	result	size
derivativedivides	\(\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {-\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}{-\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {-\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}{-\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}}{d}\)	\(202\)
default	\(\frac {-\frac {a \sqrt {2}\, \left (\ln \left (\frac {-\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}{-\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {-\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}{-\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4}}{d}\)	\(202\)
parts	\(-\frac {a \sqrt {2}\, \left (\ln \left (\frac {-\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}{-\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4 d}+\frac {b \sqrt {2}\, \left (\ln \left (\frac {-\tan \left (d x +c \right )-\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}{-\tan \left (d x +c \right )+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}+1}\right )+2 \arctan \left (1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )+2 \arctan \left (-1+\sqrt {2}\, \sqrt {-\tan \left (d x +c \right )}\right )\right )}{4 d}\)	\(204\)

Input:

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

Output:

1/d*(-1/4*a*2^(1/2)*(ln((-tan(d*x+c)+2^(1/2)*(-tan(d*x+c))^(1/2)+1)/(-tan( 
d*x+c)-2^(1/2)*(-tan(d*x+c))^(1/2)+1))+2*arctan(1+2^(1/2)*(-tan(d*x+c))^(1 
/2))+2*arctan(-1+2^(1/2)*(-tan(d*x+c))^(1/2)))+1/4*b*2^(1/2)*(ln((-tan(d*x 
+c)-2^(1/2)*(-tan(d*x+c))^(1/2)+1)/(-tan(d*x+c)+2^(1/2)*(-tan(d*x+c))^(1/2 
)+1))+2*arctan(1+2^(1/2)*(-tan(d*x+c))^(1/2))+2*arctan(-1+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. 362 vs. \(2 (98) = 196\).

Time = 0.08 (sec) , antiderivative size = 362, normalized size of antiderivative = 3.09 \[ \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx=\sqrt {\frac {1}{2}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} d \sqrt {-\tan \left (d x + c\right )} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} + d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) + \sqrt {\frac {1}{2}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} \arctan \left (-\frac {2 \, \sqrt {\frac {1}{2}} {\left (a + b\right )} d \sqrt {-\tan \left (d x + c\right )} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}} - d^{2} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \sqrt {\frac {a^{2} - 2 \, a b + b^{2}}{d^{2}}}}{a^{2} - b^{2}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \log \left (2 \, \sqrt {\frac {1}{2}} d \sqrt {-\tan \left (d x + c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} - {\left (a + b\right )} \tan \left (d x + c\right ) + a + b\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} \log \left (-2 \, \sqrt {\frac {1}{2}} d \sqrt {-\tan \left (d x + c\right )} \sqrt {\frac {a^{2} + 2 \, a b + b^{2}}{d^{2}}} - {\left (a + b\right )} \tan \left (d x + c\right ) + a + b\right ) \] Input:

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

Output:

sqrt(1/2)*sqrt((a^2 - 2*a*b + b^2)/d^2)*arctan(-(2*sqrt(1/2)*(a + b)*d*sqr 
t(-tan(d*x + c))*sqrt((a^2 - 2*a*b + b^2)/d^2) + d^2*sqrt((a^2 + 2*a*b + b 
^2)/d^2)*sqrt((a^2 - 2*a*b + b^2)/d^2))/(a^2 - b^2)) + sqrt(1/2)*sqrt((a^2 
 - 2*a*b + b^2)/d^2)*arctan(-(2*sqrt(1/2)*(a + b)*d*sqrt(-tan(d*x + c))*sq 
rt((a^2 - 2*a*b + b^2)/d^2) - d^2*sqrt((a^2 + 2*a*b + b^2)/d^2)*sqrt((a^2 
- 2*a*b + b^2)/d^2))/(a^2 - b^2)) - 1/2*sqrt(1/2)*sqrt((a^2 + 2*a*b + b^2) 
/d^2)*log(2*sqrt(1/2)*d*sqrt(-tan(d*x + c))*sqrt((a^2 + 2*a*b + b^2)/d^2) 
- (a + b)*tan(d*x + c) + a + b) + 1/2*sqrt(1/2)*sqrt((a^2 + 2*a*b + b^2)/d 
^2)*log(-2*sqrt(1/2)*d*sqrt(-tan(d*x + c))*sqrt((a^2 + 2*a*b + b^2)/d^2) - 
 (a + b)*tan(d*x + c) + a + b)
 

Sympy [F]

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

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

Output:

Integral((a + b*tan(c + d*x))/sqrt(-tan(c + d*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx=-\frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {-\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {-\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {-\tan \left (d x + c\right )} - \tan \left (d x + c\right ) + 1\right )}{4 \, d} \] Input:

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

Output:

-1/4*(2*sqrt(2)*(a - b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(-tan(d*x + c) 
))) + 2*sqrt(2)*(a - b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(-tan(d*x + c 
)))) + sqrt(2)*(a + b)*log(sqrt(2)*sqrt(-tan(d*x + c)) - tan(d*x + c) + 1) 
 - sqrt(2)*(a + b)*log(-sqrt(2)*sqrt(-tan(d*x + c)) - tan(d*x + c) + 1))/d
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 1.72 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \tan (c+d x)}{\sqrt {-\tan (c+d x)}} \, dx=\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )}{d}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {-\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d} \] Input:

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

Output:

((-1)^(1/4)*a*atan((-1)^(1/4)*(-tan(c + d*x))^(1/2))*1i)/d + ((-1)^(1/4)*a 
*atanh((-1)^(1/4)*(-tan(c + d*x))^(1/2))*1i)/d + ((-1)^(1/4)*b*atan((-1)^( 
1/4)*(-tan(c + d*x))^(1/2)))/d - ((-1)^(1/4)*b*atanh((-1)^(1/4)*(-tan(c + 
d*x))^(1/2)))/d
 

Reduce [F]

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

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

Output:

 - i*(int(sqrt(tan(c + d*x))/tan(c + d*x),x)*a + int(sqrt(tan(c + d*x)),x) 
*b)