3.1.25 \(\int \frac {\cot ^2(d+e x)}{(a+b \tan (d+e x)+c \tan ^2(d+e x))^{3/2}} \, dx\) [25]

3.1.25.1 Optimal result

Integrand size = 33, antiderivative size = 829 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=-\frac {\sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2-(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {\sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a^2-b^2-2 a c+c^2+(a-c) \sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \left (a^2+b^2-2 a c+c^2\right )^{3/2} e}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{5/2} e}+\frac {2 \cot (d+e x) \left (b^2-2 a c+b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) e \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2 \left (b^2-4 a c\right ) e} \]

output

3/2*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))/a^(5/2)/e-1/2*arctan(1/2*(b*(2*a-2*c+(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))*2^(1/2)/(2*a-2*c+( 
a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2-(a-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*a-2*c+(a^2-2*a*c 
+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2-(a-c)*(a^2-2*a*c+b^2+c^2)^(1/2)) 
^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)+1/2*arctan(1/2*(b*(2*a-2*c-(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))*2^(1/2)/(2*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-b^2-2*a*c+c^2+ 
(a-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*a-2*c-(a^2-2*a*c+b^2+c^2)^(1/2))^(1/2)*(a^2-b^2-2*a*c+c^2+(a-c)*(a^ 
2-2*a*c+b^2+c^2)^(1/2))^(1/2)/(a^2-2*a*c+b^2+c^2)^(3/2)/e*2^(1/2)-(-8*a*c+ 
3*b^2)*cot(e*x+d)*(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)/a^2/(-4*a*c+b^2)/e 
+2*cot(e*x+d)*(b^2-2*a*c+b*c*tan(e*x+d))/a/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)+ 
c*tan(e*x+d)^2)^(1/2)+2*(b*(b^2-(3*a-c)*c)+c*(b^2-2*(a-c)*c)*tan(e*x+d))/( 
b^2+(a-c)^2)/(-4*a*c+b^2)/e/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2)
 

3.1.25.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.24 (sec) , antiderivative size = 583, normalized size of antiderivative = 0.70 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\frac {\frac {2 \left (-\frac {4 \sqrt {a-i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )+\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {-2 a+i b-(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a-4 i b-4 c}-\frac {4 \sqrt {a+i b-c} \left (-\frac {1}{4} b \left (b^2-4 a c\right )-\frac {1}{4} i (a-c) \left (b^2-4 a c\right )\right ) \text {arctanh}\left (\frac {-2 a-i b-(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 a+4 i b-4 c}\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right )}-\frac {2 \cot (d+e x) \left (-b^2+2 a c-b c \tan (d+e x)\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (-b^3+b (3 a-c) c+c \left (-b^2+2 a c-2 c^2\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}-\frac {2 \left (\frac {\left (2 a b c+\frac {1}{2} b \left (-3 b^2+8 a c\right )\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{3/2}}+\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a}\right )}{a \left (b^2-4 a c\right )}}{e} \]

input

Integrate[Cot[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]
 

output

((2*((-4*Sqrt[a - I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) + (I/4)*(a - c)*(b^2 - 
4*a*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])])/(4*a - (4*I)*b - 4*c) 
- (4*Sqrt[a + I*b - c]*(-1/4*(b*(b^2 - 4*a*c)) - (I/4)*(a - c)*(b^2 - 4*a* 
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])])/(4*a + (4*I)*b - 4*c)))/(( 
b^2 + (a - c)^2)*(b^2 - 4*a*c)) - (2*Cot[d + e*x]*(-b^2 + 2*a*c - b*c*Tan[ 
d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - 
 (2*(-b^3 + b*(3*a - c)*c + c*(-b^2 + 2*a*c - 2*c^2)*Tan[d + e*x]))/((b^2 
+ (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - 
(2*(((2*a*b*c + (b*(-3*b^2 + 8*a*c))/2)*ArcTanh[(2*a + b*Tan[d + e*x])/(2* 
Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(2*a^(3/2)) + ((3*b 
^2 - 8*a*c)*Cot[d + e*x]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a 
)))/(a*(b^2 - 4*a*c)))/e
 

3.1.25.3 Rubi [A] (verified)

Time = 4.56 (sec) , antiderivative size = 815, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3042, 4183, 7276, 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.

\(\displaystyle \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\tan (d+e x)^2 \left (a+b \tan (d+e x)+c \tan (d+e x)^2\right )^{3/2}}dx\)

\(\Big \downarrow \) 4183

\(\displaystyle \frac {\int \frac {\cot ^2(d+e x)}{\left (\tan ^2(d+e x)+1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}d\tan (d+e x)}{e}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {\int \left (\frac {\cot ^2(d+e x)}{\left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}+\frac {1}{\left (-\tan ^2(d+e x)-1\right ) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}\right )d\tan (d+e x)}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {\sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \arctan \left (\frac {b \left (2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c+\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2-(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}+\frac {\sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \arctan \left (\frac {b \left (2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}\right )+\left (b^2-(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt {2 a-2 c-\sqrt {a^2-2 c a+b^2+c^2}} \sqrt {a^2-2 c a-b^2+c^2+(a-c) \sqrt {a^2-2 c a+b^2+c^2}} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \left (a^2-2 c a+b^2+c^2\right )^{3/2}}+\frac {3 b \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{2 a^{5/2}}-\frac {\left (3 b^2-8 a c\right ) \cot (d+e x) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{a^2 \left (b^2-4 a c\right )}+\frac {2 \cot (d+e x) \left (b^2+c \tan (d+e x) b-2 a c\right )}{a \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}+\frac {2 \left (b \left (b^2-(3 a-c) c\right )+c \left (b^2-2 (a-c) c\right ) \tan (d+e x)\right )}{\left (b^2+(a-c)^2\right ) \left (b^2-4 a c\right ) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}}{e}\)

input

Int[Cot[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]
 

output

(-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c 
 + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqr 
t[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*a 
*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c 
 + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c 
^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2 - 
2*a*c + c^2)^(3/2))) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sq 
rt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan 
[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c + 
Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sq 
rt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a 
^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(S 
qrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)) + (3*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])])/(2*a^(5/2)) 
+ (2*Cot[d + e*x]*(b^2 - 2*a*c + b*c*Tan[d + e*x]))/(a*(b^2 - 4*a*c)*Sqrt[ 
a + b*Tan[d + e*x] + c*Tan[d + e*x]^2]) + (2*(b*(b^2 - (3*a - c)*c) + c*(b 
^2 - 2*(a - c)*c)*Tan[d + e*x]))/((b^2 + (a - c)^2)*(b^2 - 4*a*c)*Sqrt[a + 
 b*Tan[d + e*x] + c*Tan[d + e*x]^2]) - ((3*b^2 - 8*a*c)*Cot[d + e*x]*Sqrt[ 
a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(a^2*(b^2 - 4*a*c)))/e
 

3.1.25.3.1 Defintions of rubi rules used

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 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 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]
 

3.1.25.4 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)^(3/2),x)
 

output

int(cot(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x)
 

3.1.25.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39249 vs. \(2 (764) = 1528\).

Time = 12.46 (sec) , antiderivative size = 78535, normalized size of antiderivative = 94.73 \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Too large to display} \]

input

integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm= 
"fricas")
 

output

Too large to include
 

3.1.25.6 Sympy [F]

\[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {\cot ^{2}{\left (d + e x \right )}}{\left (a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(cot(e*x+d)**2/(a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(3/2),x)
 

output

Integral(cot(d + e*x)**2/(a + b*tan(d + e*x) + c*tan(d + e*x)**2)**(3/2), 
x)
 

3.1.25.7 Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

input

integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm= 
"maxima")
 

output

Timed out
 

3.1.25.8 Giac [F(-1)]

Timed out. \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\text {Timed out} \]

input

integrate(cot(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm= 
"giac")
 

output

Timed out
 

3.1.25.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^2(d+e x)}{\left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {cot}\left (d+e\,x\right )}^2}{{\left (c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a\right )}^{3/2}} \,d x \]

input

int(cot(d + e*x)^2/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(3/2),x)
 

output

int(cot(d + e*x)^2/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(3/2), x)