\(\int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx\) [203]

Optimal result

Integrand size = 21, antiderivative size = 106 \[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \tanh ^2(x)}{2 \sqrt {a} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a}}+\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]

[Out]

-1/2*arctanh(1/2*(2*a+b*tanh(x)^2)/a^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/a^(1/2)+1/2*arctanh(1/2*(2*a+b+(
b+2*c)*tanh(x)^2)/(a+b+c)^(1/2)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2))/(a+b+c)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3781, 1265, 974, 738, 212} \[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {2 a+(b+2 c) \tanh ^2(x)+b}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}}-\frac {\text {arctanh}\left (\frac {2 a+b \tanh ^2(x)}{2 \sqrt {a} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a}} \]

[In]

Int[Coth[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 974

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] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] &&
ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[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] &&
IntegerQ[(m - 1)/2]

Rule 3781

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[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[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \sqrt {a-b x^2+c x^4}} \, dx,x,i \tanh (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x) \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{(-1-x) \sqrt {a-b x+c x^2}}+\frac {1}{x \sqrt {a-b x+c x^2}}\right ) \, dx,x,-\tanh ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(-1-x) \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a-b x+c x^2}} \, dx,x,-\tanh ^2(x)\right ) \\ & = -\text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \tanh ^2(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )-\text {Subst}\left (\int \frac {1}{4 a+4 b+4 c-x^2} \, dx,x,\frac {-2 a-b+(-b-2 c) \tanh ^2(x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right ) \\ & = -\frac {\text {arctanh}\left (\frac {2 a+b \tanh ^2(x)}{2 \sqrt {a} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a}}-\frac {\text {arctanh}\left (\frac {-2 a-b+(-b-2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03 \[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {2 a+b \tanh ^2(x)}{2 \sqrt {a} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a}}-\frac {\text {arctanh}\left (\frac {-2 a-b-(b+2 c) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]

[In]

Integrate[Coth[x]/Sqrt[a + b*Tanh[x]^2 + c*Tanh[x]^4],x]

[Out]

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

Maple [F]

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

[In]

int(coth(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)

[Out]

int(coth(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1520 vs. \(2 (86) = 172\).

Time = 0.92 (sec) , antiderivative size = 6705, normalized size of antiderivative = 63.25 \[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\coth {\left (x \right )}}{\sqrt {a + b \tanh ^{2}{\left (x \right )} + c \tanh ^{4}{\left (x \right )}}}\, dx \]

[In]

integrate(coth(x)/(a+b*tanh(x)**2+c*tanh(x)**4)**(1/2),x)

[Out]

Integral(coth(x)/sqrt(a + b*tanh(x)**2 + c*tanh(x)**4), x)

Maxima [F]

\[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int { \frac {\coth \left (x\right )}{\sqrt {c \tanh \left (x\right )^{4} + b \tanh \left (x\right )^{2} + a}} \,d x } \]

[In]

integrate(coth(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(coth(x)/sqrt(c*tanh(x)^4 + b*tanh(x)^2 + a), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\text {Timed out} \]

[In]

integrate(coth(x)/(a+b*tanh(x)^2+c*tanh(x)^4)^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth (x)}{\sqrt {a+b \tanh ^2(x)+c \tanh ^4(x)}} \, dx=\int \frac {\mathrm {coth}\left (x\right )}{\sqrt {c\,{\mathrm {tanh}\left (x\right )}^4+b\,{\mathrm {tanh}\left (x\right )}^2+a}} \,d x \]

[In]

int(coth(x)/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2),x)

[Out]

int(coth(x)/(a + b*tanh(x)^2 + c*tanh(x)^4)^(1/2), x)