\(\int \frac {\cot ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx\) [16]

Optimal result

Integrand size = 33, antiderivative size = 500 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a} e}-\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 a^{5/2} e}+\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {3 b \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 a^2 e}-\frac {\cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{2 a e} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.60 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.63 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\frac {\left (8 a^2-3 b^2+4 a c\right ) \text {arctanh}\left (\frac {2 a+b \tan (d+e x)}{2 \sqrt {a} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 a^{5/2}}-\frac {2 \text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}-\frac {2 \text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}+\frac {3 b \cot (d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a^2}-\frac {2 \cot ^2(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{a}}{4 e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 486, normalized size of antiderivative = 0.97, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4183

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(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] - ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*Tan[d + e*x])/( 
2*Sqrt[a]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(8*a^(5/2)) + (Sq 
rt[a - c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTanh[(a - c - Sqrt[a^2 + b^2 
- 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sqrt[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]*Sqrt[a 
^2 + b^2 - 2*a*c + c^2]) - (Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Ar 
cTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Tan[d + e*x])/(Sqrt[2]*Sq 
rt[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]*Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (3*b*Cot[d + e*x] 
*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(4*a^2) - (Cot[d + e*x]^2*Sq 
rt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*a))/e
 

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]
 

Maple [F(-1)]

Timed out.

hanged

Input:

int(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)
 

Output:

int(cot(e*x+d)^3/(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. 10459 vs. \(2 (441) = 882\).

Time = 2.28 (sec) , antiderivative size = 20934, normalized size of antiderivative = 41.87 \[ \int \frac {\cot ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Too large to display} \] Input:

integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm= 
"fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(cot(e*x+d)**3/(a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm= 
"maxima")
 

Output:

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

Giac [F]

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

integrate(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x, algorithm= 
"giac")
 

Output:

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

Mupad [F(-1)]

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

int(cot(d + e*x)^3/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2),x)
 

Output:

int(cot(d + e*x)^3/(a + b*tan(d + e*x) + c*tan(d + e*x)^2)^(1/2), x)
 

Reduce [F]

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

int(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)
 

Output:

int(cot(e*x+d)^3/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(1/2),x)