\(\int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\) [19]

Optimal result

Integrand size = 33, antiderivative size = 142 \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+b \cot ^2(d+e x)}{2 \sqrt {a} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a} e}-\frac {\text {arctanh}\left (\frac {2 a-b+(b-2 c) \cot ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}}\right )}{2 \sqrt {a-b+c} e} \] Output:

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

Mathematica [A] (verified)

Time = 0.98 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.39 \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\frac {\left (\frac {\text {arctanh}\left (\frac {b+2 a \tan ^2(d+e x)}{2 \sqrt {a} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {a}}-\frac {\text {arctanh}\left (\frac {b-2 c+(2 a-b) \tan ^2(d+e x)}{2 \sqrt {a-b+c} \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}\right )}{\sqrt {a-b+c}}\right ) \cot ^2(d+e x) \sqrt {c+b \tan ^2(d+e x)+a \tan ^4(d+e x)}}{2 e \sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \] Input:

Integrate[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3042, 4184, 1578, 1289, 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.

\(\displaystyle \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cot (d+e x) \sqrt {a+b \cot (d+e x)^2+c \cot (d+e x)^4}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle -\frac {\int \frac {\tan (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot (d+e x)}{e}\)

\(\Big \downarrow \) 1578

\(\displaystyle -\frac {\int \frac {\tan (d+e x)}{\left (\cot ^2(d+e x)+1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}d\cot ^2(d+e x)}{2 e}\)

\(\Big \downarrow \) 1289

\(\displaystyle -\frac {\int \left (\frac {\tan (d+e x)}{\sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}+\frac {1}{\left (-\cot ^2(d+e x)-1\right ) \sqrt {c \cot ^4(d+e x)+b \cot ^2(d+e x)+a}}\right )d\cot ^2(d+e x)}{2 e}\)

\(\Big \downarrow \) 2009

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

Input:

Int[Tan[d + e*x]/Sqrt[a + b*Cot[d + e*x]^2 + c*Cot[d + e*x]^4],x]
 

Output:

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

Defintions of rubi rules used

rule 1289

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( 
IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

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]
 

Maple [F]

\[\int \frac {\tan \left (e x +d \right )}{\sqrt {a +b \cot \left (e x +d \right )^{2}+c \cot \left (e x +d \right )^{4}}}d x\]

Input:

int(tan(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x)
 

Output:

int(tan(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (122) = 244\).

Time = 0.83 (sec) , antiderivative size = 1141, normalized size of antiderivative = 8.04 \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Too large to display} \] Input:

integrate(tan(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm= 
"fricas")
 

Output:

[1/4*((a - b + c)*sqrt(a)*log(8*a^2*tan(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 
+ b^2 + 4*a*c + 4*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(a)*sqrt((a* 
tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)) + sqrt(a - b + c)* 
a*log(((8*a^2 - 8*a*b + b^2 + 4*a*c)*tan(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4 
*(a - b)*c)*tan(e*x + d)^2 + b^2 + 4*(a - 2*b)*c + 8*c^2 - 4*((2*a - b)*ta 
n(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2)*sqrt(a - b + c)*sqrt((a*tan(e*x + 
 d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4))/(tan(e*x + d)^4 + 2*tan(e*x 
 + d)^2 + 1)))/((a^2 - a*b + a*c)*e), -1/4*(2*sqrt(-a)*(a - b + c)*arctan( 
1/2*(2*a*tan(e*x + d)^4 + b*tan(e*x + d)^2)*sqrt(-a)*sqrt((a*tan(e*x + d)^ 
4 + b*tan(e*x + d)^2 + c)/tan(e*x + d)^4)/(a^2*tan(e*x + d)^4 + a*b*tan(e* 
x + d)^2 + a*c)) - sqrt(a - b + c)*a*log(((8*a^2 - 8*a*b + b^2 + 4*a*c)*ta 
n(e*x + d)^4 + 2*(4*a*b - 3*b^2 - 4*(a - b)*c)*tan(e*x + d)^2 + b^2 + 4*(a 
 - 2*b)*c + 8*c^2 - 4*((2*a - b)*tan(e*x + d)^4 + (b - 2*c)*tan(e*x + d)^2 
)*sqrt(a - b + c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d)^2 + c)/tan(e*x + 
 d)^4))/(tan(e*x + d)^4 + 2*tan(e*x + d)^2 + 1)))/((a^2 - a*b + a*c)*e), - 
1/4*(2*a*sqrt(-a + b - c)*arctan(-1/2*((2*a - b)*tan(e*x + d)^4 + (b - 2*c 
)*tan(e*x + d)^2)*sqrt(-a + b - c)*sqrt((a*tan(e*x + d)^4 + b*tan(e*x + d) 
^2 + c)/tan(e*x + d)^4)/((a^2 - a*b + a*c)*tan(e*x + d)^4 + (a*b - b^2 + b 
*c)*tan(e*x + d)^2 + (a - b)*c + c^2)) - (a - b + c)*sqrt(a)*log(8*a^2*tan 
(e*x + d)^4 + 8*a*b*tan(e*x + d)^2 + b^2 + 4*a*c + 4*(2*a*tan(e*x + d)^...
 

Sympy [F]

\[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\tan {\left (d + e x \right )}}{\sqrt {a + b \cot ^{2}{\left (d + e x \right )} + c \cot ^{4}{\left (d + e x \right )}}}\, dx \] Input:

integrate(tan(e*x+d)/(a+b*cot(e*x+d)**2+c*cot(e*x+d)**4)**(1/2),x)
 

Output:

Integral(tan(d + e*x)/sqrt(a + b*cot(d + e*x)**2 + c*cot(d + e*x)**4), x)
 

Maxima [F]

\[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )}{\sqrt {c \cot \left (e x + d\right )^{4} + b \cot \left (e x + d\right )^{2} + a}} \,d x } \] Input:

integrate(tan(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x, algorithm= 
"maxima")
 

Output:

integrate(tan(e*x + d)/sqrt(c*cot(e*x + d)^4 + b*cot(e*x + d)^2 + a), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(tan(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(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. \[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\mathrm {tan}\left (d+e\,x\right )}{\sqrt {c\,{\mathrm {cot}\left (d+e\,x\right )}^4+b\,{\mathrm {cot}\left (d+e\,x\right )}^2+a}} \,d x \] Input:

int(tan(d + e*x)/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2),x)
 

Output:

int(tan(d + e*x)/(a + b*cot(d + e*x)^2 + c*cot(d + e*x)^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\tan (d+e x)}{\sqrt {a+b \cot ^2(d+e x)+c \cot ^4(d+e x)}} \, dx=\int \frac {\sqrt {\cot \left (e x +d \right )^{4} c +\cot \left (e x +d \right )^{2} b +a}\, \tan \left (e x +d \right )}{\cot \left (e x +d \right )^{4} c +\cot \left (e x +d \right )^{2} b +a}d x \] Input:

int(tan(e*x+d)/(a+b*cot(e*x+d)^2+c*cot(e*x+d)^4)^(1/2),x)
 

Output:

int((sqrt(cot(d + e*x)**4*c + cot(d + e*x)**2*b + a)*tan(d + e*x))/(cot(d 
+ e*x)**4*c + cot(d + e*x)**2*b + a),x)