3.207 \(\int \frac{\coth (x)}{\sqrt{a+b \coth ^2(x)+c \coth ^4(x)}} \, dx\)
Optimal. Leaf size=58 \[ \frac{\tanh ^{-1}\left (\frac{2 a+(b+2 c) \coth ^2(x)+b}{2 \sqrt{a+b+c} \sqrt{a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt{a+b+c}} \]
[Out]
ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b
+ c])
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Rubi [A] time = 0.117339, antiderivative size = 58, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used =
{3701, 1247, 724, 206} \[ \frac{\tanh ^{-1}\left (\frac{2 a+(b+2 c) \coth ^2(x)+b}{2 \sqrt{a+b+c} \sqrt{a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt{a+b+c}} \]
Antiderivative was successfully verified.
[In]
Int[Coth[x]/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4],x]
[Out]
ArcTanh[(2*a + b + (b + 2*c)*Coth[x]^2)/(2*Sqrt[a + b + c]*Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4])]/(2*Sqrt[a + b
+ c])
Rule 3701
Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n_.) + (c_.)*(cot[(d_.) + (e
_.)*(x_)]*(f_.))^(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*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 1247
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(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]
Rule 724
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 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rubi steps
\begin{align*} \int \frac{\coth (x)}{\sqrt{a+b \coth ^2(x)+c \coth ^4(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right ) \sqrt{a-b x^2+c x^4}} \, dx,x,-i \coth (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(1+x) \sqrt{a-b x+c x^2}} \, dx,x,-\coth ^2(x)\right )\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{4 a+4 b+4 c-x^2} \, dx,x,\frac{2 a+b+(b+2 c) \coth ^2(x)}{\sqrt{a+b \coth ^2(x)+c \coth ^4(x)}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{2 a+b+(b+2 c) \coth ^2(x)}{2 \sqrt{a+b+c} \sqrt{a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt{a+b+c}}\\ \end{align*}
Mathematica [B] time = 18.1964, size = 141, normalized size = 2.43 \[ \frac{\text{csch}^2(x) \sqrt{\cosh (4 x) (a+b+c)-4 (a-c) \cosh (2 x)+3 a-b+3 c} \tanh ^{-1}\left (\frac{\cosh (2 x) (a+b+c)-a+c}{2 \sqrt{a+b+c} \sqrt{\sinh ^4(x) (a+b+c)+(b+2 c) \sinh ^2(x)+c}}\right )}{2 \sqrt{a+b+c} \sqrt{\text{csch}^4(x) (\cosh (4 x) (a+b+c)-4 (a-c) \cosh (2 x)+3 a-b+3 c)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Coth[x]/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4],x]
[Out]
(ArcTanh[(-a + c + (a + b + c)*Cosh[2*x])/(2*Sqrt[a + b + c]*Sqrt[c + (b + 2*c)*Sinh[x]^2 + (a + b + c)*Sinh[x
]^4])]*Sqrt[3*a - b + 3*c - 4*(a - c)*Cosh[2*x] + (a + b + c)*Cosh[4*x]]*Csch[x]^2)/(2*Sqrt[a + b + c]*Sqrt[(3
*a - b + 3*c - 4*(a - c)*Cosh[2*x] + (a + b + c)*Cosh[4*x])*Csch[x]^4])
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Maple [A] time = 0.065, size = 52, normalized size = 0.9 \begin{align*}{\frac{1}{2}{\it Artanh} \left ({\frac{b \left ({\rm coth} \left (x\right ) \right ) ^{2}+2\,c \left ({\rm coth} \left (x\right ) \right ) ^{2}+2\,a+b}{2}{\frac{1}{\sqrt{a+b+c}}}{\frac{1}{\sqrt{a+b \left ({\rm coth} \left (x\right ) \right ) ^{2}+c \left ({\rm coth} \left (x\right ) \right ) ^{4}}}}} \right ){\frac{1}{\sqrt{a+b+c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(coth(x)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x)
[Out]
1/2/(a+b+c)^(1/2)*arctanh(1/2*(b*coth(x)^2+2*c*coth(x)^2+2*a+b)/(a+b+c)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2
))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left (x\right )}{\sqrt{c \coth \left (x\right )^{4} + b \coth \left (x\right )^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="maxima")
[Out]
integrate(coth(x)/sqrt(c*coth(x)^4 + b*coth(x)^2 + a), x)
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Fricas [B] time = 9.66772, size = 4811, normalized size = 82.95 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="fricas")
[Out]
[1/4*log(((a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x
)*sinh(x)^7 + (a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*sinh(x)^8 - 4*(a^2 + a*b - b*c - c^2)*cosh(x)^6 + 4*(7*(
a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^2 - a^2 - a*b + b*c + c^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2
+ 2*(a + b)*c + c^2)*cosh(x)^3 - 3*(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b + 2*(a + b)*c
+ 3*c^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^4 - 30*(a^2 + a*b - b*c - c^2)*cos
h(x)^2 + 3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)
^5 - 10*(a^2 + a*b - b*c - c^2)*cosh(x)^3 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x))*sinh(x)^3 - 4*(a^2
+ a*b - b*c - c^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^6 - 15*(a^2 + a*b - b*c -
c^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^2 - a^2 - a*b + b*c + c^2)*sinh(x)^2 + sqrt(2
)*((a + b + c)*cosh(x)^4 + 4*(a + b + c)*cosh(x)*sinh(x)^3 + (a + b + c)*sinh(x)^4 - 2*(a - c)*cosh(x)^2 + 2*(
3*(a + b + c)*cosh(x)^2 - a + c)*sinh(x)^2 + 4*((a + b + c)*cosh(x)^3 - (a - c)*cosh(x))*sinh(x) + a + b + c)*
sqrt(a + b + c)*sqrt(((a + b + c)*cosh(x)^4 + (a + b + c)*sinh(x)^4 - 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*c
osh(x)^2 - 2*a + 2*c)*sinh(x)^2 + 3*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*
cosh(x)*sinh(x)^3 + sinh(x)^4)) + a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2 + 8*((a^2 + 2*a*b + b^2 + 2*(a + b)*c
+ c^2)*cosh(x)^7 - 3*(a^2 + a*b - b*c - c^2)*cosh(x)^5 + (3*a^2 + 2*a*b + 2*(a + b)*c + 3*c^2)*cosh(x)^3 - (a^
2 + a*b - b*c - c^2)*cosh(x))*sinh(x))/(cosh(x)^4 + 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*si
nh(x)^3 + sinh(x)^4))/sqrt(a + b + c), -1/2*sqrt(-a - b - c)*arctan(sqrt(2)*((a + b + c)*cosh(x)^4 + 4*(a + b
+ c)*cosh(x)*sinh(x)^3 + (a + b + c)*sinh(x)^4 - 2*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 - a + c)*sin
h(x)^2 + 4*((a + b + c)*cosh(x)^3 - (a - c)*cosh(x))*sinh(x) + a + b + c)*sqrt(-a - b - c)*sqrt(((a + b + c)*c
osh(x)^4 + (a + b + c)*sinh(x)^4 - 4*(a - c)*cosh(x)^2 + 2*(3*(a + b + c)*cosh(x)^2 - 2*a + 2*c)*sinh(x)^2 + 3
*a - b + 3*c)/(cosh(x)^4 - 4*cosh(x)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 - 4*cosh(x)*sinh(x)^3 + sinh(x)^4))/((a
^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^8 + 8*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)*sinh(x)^7
+ (a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*sinh(x)^8 - 4*(a^2 + a*b - b*c - c^2)*cosh(x)^6 + 4*(7*(a^2 + 2*a*b
+ b^2 + 2*(a + b)*c + c^2)*cosh(x)^2 - a^2 - a*b + b*c + c^2)*sinh(x)^6 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*
c + c^2)*cosh(x)^3 - 3*(a^2 + a*b - b*c - c^2)*cosh(x))*sinh(x)^5 + 2*(3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3
*c^2)*cosh(x)^4 + 2*(35*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^4 - 30*(a^2 + a*b - b*c - c^2)*cosh(x)
^2 + 3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*sinh(x)^4 + 8*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cos
h(x)^5 - 10*(a^2 + a*b - b*c - c^2)*cosh(x)^3 + (3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*cosh(x))*sinh(x)
^3 - 4*(a^2 + a*b - b*c - c^2)*cosh(x)^2 + 4*(7*(a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^6 - 15*(a^2 +
a*b - b*c - c^2)*cosh(x)^4 + 3*(3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*cosh(x)^2 - a^2 - a*b + b*c + c^2
)*sinh(x)^2 + a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2 + 8*((a^2 + 2*a*b + b^2 + 2*(a + b)*c + c^2)*cosh(x)^7 - 3
*(a^2 + a*b - b*c - c^2)*cosh(x)^5 + (3*a^2 + 2*a*b - b^2 + 2*(3*a + b)*c + 3*c^2)*cosh(x)^3 - (a^2 + a*b - b*
c - c^2)*cosh(x))*sinh(x)))/(a + b + c)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth{\left (x \right )}}{\sqrt{a + b \coth ^{2}{\left (x \right )} + c \coth ^{4}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)/(a+b*coth(x)**2+c*coth(x)**4)**(1/2),x)
[Out]
Integral(coth(x)/sqrt(a + b*coth(x)**2 + c*coth(x)**4), x)
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="giac")
[Out]
Timed out