\(\int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx\) [8]

Optimal result

Integrand size = 31, antiderivative size = 570 \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{e}-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \cot (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e} \] Output:

1/2*(a^2+b^2+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1 
/2)))^(1/2)*arctan(1/2*(b^2+(a-c)*(a-c-(a^2-2*a*c+b^2+c^2)^(1/2))-b*(a^2-2 
*a*c+b^2+c^2)^(1/2)*cot(e*x+d))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/(a^2+b^2 
+c*(c+(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c+(a^2-2*a*c+b^2+c^2)^(1/2)))^(1/2)/ 
(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c+b^2+c^2)^(1/4)/e 
+a^(1/2)*arctanh(1/2*(2*a+b*cot(e*x+d))/a^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+ 
d)^2)^(1/2))/e-1/2*(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2* 
a*c+b^2+c^2)^(1/2)))^(1/2)*arctanh(1/2*(b^2+(a-c)*(a-c+(a^2-2*a*c+b^2+c^2) 
^(1/2))+b*(a^2-2*a*c+b^2+c^2)^(1/2)*cot(e*x+d))*2^(1/2)/(a^2-2*a*c+b^2+c^2 
)^(1/4)/(a^2+b^2+c*(c-(a^2-2*a*c+b^2+c^2)^(1/2))-a*(2*c-(a^2-2*a*c+b^2+c^2 
)^(1/2)))^(1/2)/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2))*2^(1/2)/(a^2-2*a*c+ 
b^2+c^2)^(1/4)/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.35 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.50 \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\frac {\left (i \left (\sqrt {a+i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a-i b-c} \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )\right )+2 \sqrt {a} \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )\right ) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{2 e \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}} \] Input:

Integrate[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x],x]
 

Output:

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

Rubi [F]

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.

\(\displaystyle \int \tan (d+e x) \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a+b \cot (d+e x)+c \cot (d+e x)^2}}{\cot (d+e x)}dx\)

\(\Big \downarrow \) 4184

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

Input:

Int[Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

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

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F(-1)]

Timed out.

hanged

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2423 vs. \(2 (515) = 1030\).

Time = 0.42 (sec) , antiderivative size = 4847, normalized size of antiderivative = 8.50 \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\int \sqrt {a + b \cot {\left (d + e x \right )} + c \cot ^{2}{\left (d + e x \right )}} \tan {\left (d + e x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(a + b*cot(d + e*x) + c*cot(d + e*x)**2)*tan(d + e*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume((c-b-a)*(c+b-a)>0)', see `assume 
?` for mor
 

Giac [F]

\[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\int { \sqrt {c \cot \left (e x + d\right )^{2} + b \cot \left (e x + d\right ) + a} \tan \left (e x + d\right ) \,d x } \] Input:

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

Output:

integrate(sqrt(c*cot(e*x + d)^2 + b*cot(e*x + d) + a)*tan(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\int \mathrm {tan}\left (d+e\,x\right )\,\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^2+b\,\mathrm {cot}\left (d+e\,x\right )+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x) \, dx=\int \sqrt {\cot \left (e x +d \right )^{2} c +\cot \left (e x +d \right ) b +a}\, \tan \left (e x +d \right )d x \] Input:

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

Output:

int(sqrt(cot(d + e*x)**2*c + cot(d + e*x)*b + a)*tan(d + e*x),x)