\(\int \frac {\sqrt {\cot (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx\) [863]

Optimal result

Integrand size = 25, antiderivative size = 149 \[ \int \frac {\sqrt {\cot (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {\arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a-b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {i a+b} d} \] Output:

arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^( 
1/2)*tan(d*x+c)^(1/2)/(I*a-b)^(1/2)/d+arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/ 
2)/(a+b*tan(d*x+c))^(1/2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/(I*a+b)^(1/2) 
/d
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {\cot (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {(-1)^{3/4} \left (-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d} \] Input:

Integrate[Sqrt[Cot[c + d*x]]/Sqrt[a + b*Tan[c + d*x]],x]
 

Output:

((-1)^(3/4)*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[ 
a + b*Tan[c + d*x]]]/Sqrt[-a + I*b]) - ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sq 
rt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b])*Sqrt[Cot[c + d* 
x]]*Sqrt[Tan[c + d*x]])/d
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.85, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4729, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4729

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}dx\)

\(\Big \downarrow \) 4058

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

\(\Big \downarrow \) 615

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {\arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-b+i a}}+\frac {\text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {b+i a}}\right )}{d}\)

Input:

Int[Sqrt[Cot[c + d*x]]/Sqrt[a + b*Tan[c + d*x]],x]
 

Output:

((ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt 
[I*a - b] + ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + 
d*x]]]/Sqrt[I*a + b])*Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]])/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]
 

rule 4729

Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a 
+ b*x])^m*(c*Tan[a + b*x])^m   Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownTangentIntegrandQ[u, 
x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(640\) vs. \(2(121)=242\).

Time = 3.42 (sec) , antiderivative size = 641, normalized size of antiderivative = 4.30

method	result	size
default	\(\frac {\sqrt {2}\, \sqrt {\cot \left (d x +c \right )}\, \sqrt {a +b \tan \left (d x +c \right )}\, \left (\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \ln \left (\frac {-a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+2 \sqrt {a^{2}+b^{2}}\, \left (1-\cos \left (d x +c \right )\right )+2 \sqrt {2}\, \sqrt {\frac {\left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sqrt {b +\sqrt {a^{2}+b^{2}}}\, \sin \left (d x +c \right )+2 b \left (1-\cos \left (d x +c \right )\right )+\sin \left (d x +c \right ) a}{1-\cos \left (d x +c \right )}\right )-\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \ln \left (-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+2 \sqrt {2}\, \sqrt {\frac {\left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \sqrt {b +\sqrt {a^{2}+b^{2}}}\, \sin \left (d x +c \right )-2 \sqrt {a^{2}+b^{2}}\, \left (1-\cos \left (d x +c \right )\right )-2 b \left (1-\cos \left (d x +c \right )\right )-\sin \left (d x +c \right ) a}{1-\cos \left (d x +c \right )}\right )-2 \sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \arctan \left (\frac {\left (\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\sqrt {2}\, \sqrt {\frac {\left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\right ) \sin \left (d x +c \right )}{\sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \left (1-\cos \left (d x +c \right )\right )}\right )+2 \sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \arctan \left (\frac {\left (\sqrt {b +\sqrt {a^{2}+b^{2}}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-\sqrt {2}\, \sqrt {\frac {\left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\right ) \sin \left (d x +c \right )}{\sqrt {-b +\sqrt {a^{2}+b^{2}}}\, \left (1-\cos \left (d x +c \right )\right )}\right )\right ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{4 d \sqrt {a^{2}+b^{2}}\, \sqrt {\frac {\left (a \cos \left (d x +c \right )+b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\)	\(641\)

Input:

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

Output:

1/4/d*2^(1/2)/(a^2+b^2)^(1/2)*cot(d*x+c)^(1/2)*(a+b*tan(d*x+c))^(1/2)*((b+ 
(a^2+b^2)^(1/2))^(1/2)*ln(1/(1-cos(d*x+c))*(-a*(1-cos(d*x+c))^2*csc(d*x+c) 
+2*(a^2+b^2)^(1/2)*(1-cos(d*x+c))+2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*s 
in(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)+2*b 
*(1-cos(d*x+c))+sin(d*x+c)*a))-(b+(a^2+b^2)^(1/2))^(1/2)*ln(-1/(1-cos(d*x+ 
c))*(a*(1-cos(d*x+c))^2*csc(d*x+c)+2*2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))* 
sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(b+(a^2+b^2)^(1/2))^(1/2)*sin(d*x+c)-2* 
(a^2+b^2)^(1/2)*(1-cos(d*x+c))-2*b*(1-cos(d*x+c))-sin(d*x+c)*a))-2*(-b+(a^ 
2+b^2)^(1/2))^(1/2)*arctan(1/(-b+(a^2+b^2)^(1/2))^(1/2)*((b+(a^2+b^2)^(1/2 
))^(1/2)*(-cot(d*x+c)+csc(d*x+c))+2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin 
(d*x+c)/(cos(d*x+c)+1)^2)^(1/2))/(1-cos(d*x+c))*sin(d*x+c))+2*(-b+(a^2+b^2 
)^(1/2))^(1/2)*arctan(1/(-b+(a^2+b^2)^(1/2))^(1/2)*((b+(a^2+b^2)^(1/2))^(1 
/2)*(-cot(d*x+c)+csc(d*x+c))-2^(1/2)*((a*cos(d*x+c)+b*sin(d*x+c))*sin(d*x+ 
c)/(cos(d*x+c)+1)^2)^(1/2))/(1-cos(d*x+c))*sin(d*x+c)))/((a*cos(d*x+c)+b*s 
in(d*x+c))*sin(d*x+c)/(cos(d*x+c)+1)^2)^(1/2)*(-cot(d*x+c)+csc(d*x+c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4089 vs. \(2 (117) = 234\).

Time = 0.53 (sec) , antiderivative size = 4089, normalized size of antiderivative = 27.44 \[ \int \frac {\sqrt {\cot (c+d x)}}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(cot(d*x + c))/sqrt(b*tan(d*x + c) + a), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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