\(\int \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx\) [5]

Optimal result

Integrand size = 31, antiderivative size = 571 \[ \int \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a} \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e} \] 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)*tan(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*tan(e*x+d)+c*tan(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*tan(e*x+d))/a^(1/2)/(a+b*tan(e*x+d)+c*tan(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)*tan(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*tan(e*x+d)+c*tan(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 = 0.61 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.39 \[ \int \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {-2 \sqrt {a} \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 e} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4183

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 3042

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

rule 4183

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*( 
x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(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*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n 
2, 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(cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4557 vs. \(2 (516) = 1032\).

Time = 0.67 (sec) , antiderivative size = 9127, normalized size of antiderivative = 15.98 \[ \int \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),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((-16*a*(a/4-c/4))>0)', see `assu 
me?` for m
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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