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

3.1.8.1 Optimal result

Integrand size = 33, antiderivative size = 612 \[ \int \cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \arctan \left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 \sqrt {a} e}+\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {arctanh}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {\cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{e} \]

output

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

3.1.8.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.22 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.43 \[ \int \cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=-\frac {\frac {b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a}}-i \sqrt {a-i b-c} \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+i \sqrt {a+i b-c} \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+2 \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 e} \]

input

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

output

-1/2*((b*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*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])] + I*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*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/ 
e
 

3.1.8.3 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4183

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 7239

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

input

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

output

$Aborted
 

3.1.8.3.1 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]
 

3.1.8.4 Maple [F(-1)]

Timed out.

hanged

input

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

output

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

3.1.8.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4592 vs. \(2 (552) = 1104\).

Time = 1.05 (sec) , antiderivative size = 9221, normalized size of antiderivative = 15.07 \[ \int \cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx=\text {Too large to display} \]

input

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

output

Too large to include
 

3.1.8.6 Sympy [F]

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

input

integrate(cot(e*x+d)**2*(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)**2, x)
 

3.1.8.7 Maxima [F]

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

input

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

output

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

3.1.8.8 Giac [F]

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

input

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

output

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

3.1.8.9 Mupad [F(-1)]

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

input

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

output

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