\(\int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx\) [5]

Optimal result

Integrand size = 33, antiderivative size = 501 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {a} e}+\frac {\left (3 b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b \cot (d+e x)}{2 \sqrt {a} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{8 a^{5/2} e}-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \cot (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {3 b \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan (d+e x)}{4 a^2 e}+\frac {\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)} \tan ^2(d+e x)}{2 a e} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.75 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.84 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=-\frac {\cot (d+e x) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)} \left (\sqrt {a-i b-c} \sqrt {a+i b-c} \left (8 a^2-3 b^2+4 a c\right ) \text {arctanh}\left (\frac {b+2 a \tan (d+e x)}{2 \sqrt {a} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+2 \sqrt {a} \left (2 i a^2 \sqrt {a-i b-c} \arctan \left (\frac {i b-2 c+(2 i a-b) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a+i b-c} \left (2 i a^2 \arctan \left (\frac {i b+2 c+(2 i a+b) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}}\right )+\sqrt {a-i b-c} (3 b-2 a \tan (d+e x)) \sqrt {c+b \tan (d+e x)+a \tan ^2(d+e x)}\right )\right )\right )}{8 a^{5/2} \sqrt {a-i b-c} \sqrt {a+i b-c} e \sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \] Input:

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

Output:

-1/8*(Cot[d + e*x]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2]*(Sqrt[a - I 
*b - c]*Sqrt[a + I*b - c]*(8*a^2 - 3*b^2 + 4*a*c)*ArcTanh[(b + 2*a*Tan[d + 
 e*x])/(2*Sqrt[a]*Sqrt[c + b*Tan[d + e*x] + a*Tan[d + e*x]^2])] + 2*Sqrt[a 
]*((2*I)*a^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])] + 
Sqrt[a + I*b - c]*((2*I)*a^2*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])] + Sq 
rt[a - I*b - c]*(3*b - 2*a*Tan[d + e*x])*Sqrt[c + b*Tan[d + e*x] + a*Tan[d 
 + e*x]^2]))))/(a^(5/2)*Sqrt[a - I*b - c]*Sqrt[a + I*b - c]*e*Sqrt[a + b*C 
ot[d + e*x] + c*Cot[d + e*x]^2])
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 487, 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, 4184, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4184

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((ArcTanh[(2*a + b*Cot[d + e*x])/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*C 
ot[d + e*x]^2])]/Sqrt[a] - ((3*b^2 - 4*a*c)*ArcTanh[(2*a + b*Cot[d + e*x]) 
/(2*Sqrt[a]*Sqrt[a + b*Cot[d + e*x] + c*Cot[d + e*x]^2])])/(8*a^(5/2)) + ( 
Sqrt[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*Cot[d + e*x])/(Sqrt[2]*Sqrt[a - c - Sqrt[a^2 + b^2 - 
2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Cot[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]]* 
ArcTanh[(a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2] + b*Cot[d + e*x])/(Sqrt[2]* 
Sqrt[a - c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Cot[d + e*x] + c*Co 
t[d + e*x]^2])])/(Sqrt[2]*Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (3*b*Sqrt[a + b 
*Cot[d + e*x] + c*Cot[d + e*x]^2]*Tan[d + e*x])/(4*a^2) - (Sqrt[a + b*Cot[ 
d + e*x] + c*Cot[d + e*x]^2]*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 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 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(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(e*x+d)^2)^(1/2),x)
 

Output:

int(tan(e*x+d)^3/(a+b*cot(e*x+d)+c*cot(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. 5247 vs. \(2 (442) = 884\).

Time = 1.28 (sec) , antiderivative size = 10495, normalized size of antiderivative = 20.95 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \cot (d+e x)+c \cot ^2(d+e x)}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument 
Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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