\(\int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [534]

Optimal result

Integrand size = 21, antiderivative size = 116 \[ \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}}{d} \] Input:

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

Output:

((-2*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + ArcTanh[Sqrt[a + 
 b*Tan[c + d*x]]/Sqrt[a - I*b]]/Sqrt[a - I*b] + ArcTanh[Sqrt[a + b*Tan[c + 
 d*x]]/Sqrt[a + I*b]]/Sqrt[a + I*b])/d
 

Rubi [A] (warning: unable to verify)

Time = 0.87 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {3042, 4057, 25, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4057

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4022

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4020

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 4117

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

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 4020

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

rule 4022

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

rule 4057

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

rule 4117

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
 Simp[A/f   Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; 
FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 733 vs. \(2 (90) = 180\).

Time = 0.17 (sec) , antiderivative size = 1485, normalized size of antiderivative = 12.80 \[ \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(a*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) 
+ a)/((a^2 + b^2)*d^2))*log(((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + 
 b^4)*d^4)) - a*d)*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4 
)*d^4)) + a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a)) - a*d*sqrt(((a 
^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d 
^2))*log(-((a^2 + b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a*d) 
*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 
 + b^2)*d^2)) + sqrt(b*tan(d*x + c) + a)) - a*d*sqrt(-((a^2 + b^2)*d^2*sqr 
t(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(((a^2 + 
b^2)*d^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a*d)*sqrt(-((a^2 + b^2 
)*d^2*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)) + s 
qrt(b*tan(d*x + c) + a)) + a*d*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/((a^4 + 2* 
a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2))*log(-((a^2 + b^2)*d^3*sqrt(-b 
^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a*d)*sqrt(-((a^2 + b^2)*d^2*sqrt(-b^2/ 
((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a)/((a^2 + b^2)*d^2)) + sqrt(b*tan(d*x + 
c) + a)) - 2*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt 
(a) + 2*a)/tan(d*x + c)))/(a*d), -1/2*(a*d*sqrt(((a^2 + b^2)*d^2*sqrt(-b^2 
/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2))*log(((a^2 + b^2)*d 
^3*sqrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - a*d)*sqrt(((a^2 + b^2)*d^2*s 
qrt(-b^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + a)/((a^2 + b^2)*d^2)) + sqrt(...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 1.79 (sec) , antiderivative size = 2028, normalized size of antiderivative = 17.48 \[ \int \frac {\cot (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((((((((((1/(a*d^2 - b*d^2*1i))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8 
*d^2))/d^3 - (16*(1/(a*d^2 - b*d^2*1i))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^ 
4)*(a + b*tan(c + d*x))^(1/2))/d^4))/2 + (576*a*b^8*(a + b*tan(c + d*x))^( 
1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*a*b^8)/d^3)*(1/(a*d^2 - b 
*d^2*1i))^(1/2))/2 - (96*b^8*(a + b*tan(c + d*x))^(1/2))/d^4)*(1/(a*d^2 - 
b*d^2*1i))^(1/2)*1i)/2 - ((((((((1/(a*d^2 - b*d^2*1i))^(1/2)*((32*(16*b^10 
*d^2 + 12*a^2*b^8*d^2))/d^3 + (16*(1/(a*d^2 - b*d^2*1i))^(1/2)*(16*b^10*d^ 
4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^4))/2 - (576*a*b^8*(a + 
b*tan(c + d*x))^(1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*a*b^8)/d 
^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + (96*b^8*(a + b*tan(c + d*x))^(1/2))/ 
d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2)*1i)/2)/(((((((((1/(a*d^2 - b*d^2*1i))^(1 
/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/d^3 - (16*(1/(a*d^2 - b*d^2*1i))^ 
(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/d^4))/2 + 
 (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*(1/(a*d^2 - b*d^2*1i))^(1/2)) 
/2 - (96*a*b^8)/d^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 - (96*b^8*(a + b*tan( 
c + d*x))^(1/2))/d^4)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + ((((((((1/(a*d^2 - 
 b*d^2*1i))^(1/2)*((32*(16*b^10*d^2 + 12*a^2*b^8*d^2))/d^3 + (16*(1/(a*d^2 
 - b*d^2*1i))^(1/2)*(16*b^10*d^4 + 24*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1 
/2))/d^4))/2 - (576*a*b^8*(a + b*tan(c + d*x))^(1/2))/d^2)*(1/(a*d^2 - b*d 
^2*1i))^(1/2))/2 - (96*a*b^8)/d^3)*(1/(a*d^2 - b*d^2*1i))^(1/2))/2 + (9...
 

Reduce [F]

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

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

Output:

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