\(\int \frac {1}{(\coth ^2(x)+\text {csch}^2(x))^3} \, dx\) [822]

Optimal result

Integrand size = 11, antiderivative size = 54 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=x-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )} \]

[Out]

x-7/8*arctanh(1/2*2^(1/2)*tanh(x))*2^(1/2)-1/2*tanh(x)^3/(2-tanh(x)^2)^2-1/4*tanh(x)/(2-tanh(x)^2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {481, 592, 536, 212} \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}+x-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2} \]

[In]

Int[(Coth[x]^2 + Csch[x]^2)^(-3),x]

[Out]

x - (7*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - Tanh[x]^3/(2*(2 - Tanh[x]^2)^2) - Tanh[x]/(4*(2 - Tanh[x]^2))

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 481

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-a)*e^(
2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Dist[e^
(2*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1)
+ (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0]
&& IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 536

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 592

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c -
a*d)*(p + 1))), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^6}{\left (1-x^2\right ) \left (2-x^2\right )^3} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}+\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (6-2 x^2\right )}{\left (1-x^2\right ) \left (2-x^2\right )^2} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {1}{8} \text {Subst}\left (\int \frac {-2-6 x^2}{\left (1-x^2\right ) \left (2-x^2\right )} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )}-\frac {7}{4} \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\tanh (x)\right )+\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = x-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\frac {76 x+48 x \cosh (2 x)-7 \sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) (3+\cosh (2 x))^2+4 x \cosh (4 x)-2 \sinh (2 x)-3 \sinh (4 x)}{8 (3+\cosh (2 x))^2} \]

[In]

Integrate[(Coth[x]^2 + Csch[x]^2)^(-3),x]

[Out]

(76*x + 48*x*Cosh[2*x] - 7*Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]*(3 + Cosh[2*x])^2 + 4*x*Cosh[4*x] - 2*Sinh[2*x] -
3*Sinh[4*x])/(8*(3 + Cosh[2*x])^2)

Maple [A] (verified)

Time = 37.53 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.04

method	result	size
parallelrisch	\(0\)	\(2\)
risch	\(x +\frac {17 \,{\mathrm e}^{6 x}+57 \,{\mathrm e}^{4 x}+19 \,{\mathrm e}^{2 x}+3}{2 \left ({\mathrm e}^{4 x}+6 \,{\mathrm e}^{2 x}+1\right )^{2}}+\frac {7 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}+3+2 \sqrt {2}\right )}{16}-\frac {7 \sqrt {2}\, \ln \left ({\mathrm e}^{2 x}-2 \sqrt {2}+3\right )}{16}\)	\(73\)
default	\(\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )+\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{7}}{4}-\frac {5 \tanh \left (\frac {x}{2}\right )^{5}}{4}-\frac {5 \tanh \left (\frac {x}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {x}{2}\right )}{4}}{\left (\tanh \left (\frac {x}{2}\right )^{4}+1\right )^{2}}-\frac {7 \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\)	\(199\)

[In]

int(1/(coth(x)^2+csch(x)^2)^3,x,method=_RETURNVERBOSE)

[Out]

0

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (41) = 82\).

Time = 0.28 (sec) , antiderivative size = 715, normalized size of antiderivative = 13.24 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate(1/(coth(x)^2+csch(x)^2)^3,x, algorithm="fricas")

[Out]

1/16*(16*x*cosh(x)^8 + 128*x*cosh(x)*sinh(x)^7 + 16*x*sinh(x)^8 + 8*(24*x + 17)*cosh(x)^6 + 8*(56*x*cosh(x)^2
+ 24*x + 17)*sinh(x)^6 + 16*(56*x*cosh(x)^3 + 3*(24*x + 17)*cosh(x))*sinh(x)^5 + 152*(4*x + 3)*cosh(x)^4 + 8*(
140*x*cosh(x)^4 + 15*(24*x + 17)*cosh(x)^2 + 76*x + 57)*sinh(x)^4 + 32*(28*x*cosh(x)^5 + 5*(24*x + 17)*cosh(x)
^3 + 19*(4*x + 3)*cosh(x))*sinh(x)^3 + 8*(24*x + 19)*cosh(x)^2 + 8*(56*x*cosh(x)^6 + 15*(24*x + 17)*cosh(x)^4
+ 114*(4*x + 3)*cosh(x)^2 + 24*x + 19)*sinh(x)^2 + 7*(sqrt(2)*cosh(x)^8 + 8*sqrt(2)*cosh(x)*sinh(x)^7 + sqrt(2
)*sinh(x)^8 + 4*(7*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^6 + 12*sqrt(2)*cosh(x)^6 + 8*(7*sqrt(2)*cosh(x)^3 +
9*sqrt(2)*cosh(x))*sinh(x)^5 + 2*(35*sqrt(2)*cosh(x)^4 + 90*sqrt(2)*cosh(x)^2 + 19*sqrt(2))*sinh(x)^4 + 38*sqr
t(2)*cosh(x)^4 + 8*(7*sqrt(2)*cosh(x)^5 + 30*sqrt(2)*cosh(x)^3 + 19*sqrt(2)*cosh(x))*sinh(x)^3 + 4*(7*sqrt(2)*
cosh(x)^6 + 45*sqrt(2)*cosh(x)^4 + 57*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^2 + 12*sqrt(2)*cosh(x)^2 + 8*(sqr
t(2)*cosh(x)^7 + 9*sqrt(2)*cosh(x)^5 + 19*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*log((3*(2*
sqrt(2) + 3)*cosh(x)^2 - 4*(3*sqrt(2) + 4)*cosh(x)*sinh(x) + 3*(2*sqrt(2) + 3)*sinh(x)^2 + 2*sqrt(2) + 3)/(cos
h(x)^2 + sinh(x)^2 + 3)) + 16*(8*x*cosh(x)^7 + 3*(24*x + 17)*cosh(x)^5 + 38*(4*x + 3)*cosh(x)^3 + (24*x + 19)*
cosh(x))*sinh(x) + 16*x + 24)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 + 3)*sinh(x)^6 + 1
2*cosh(x)^6 + 8*(7*cosh(x)^3 + 9*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 + 90*cosh(x)^2 + 19)*sinh(x)^4 + 38*cosh
(x)^4 + 8*(7*cosh(x)^5 + 30*cosh(x)^3 + 19*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 + 45*cosh(x)^4 + 57*cosh(x)^2 +
3)*sinh(x)^2 + 12*cosh(x)^2 + 8*(cosh(x)^7 + 9*cosh(x)^5 + 19*cosh(x)^3 + 3*cosh(x))*sinh(x) + 1)

Sympy [F]

\[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\int \frac {1}{\left (\coth ^{2}{\left (x \right )} + \operatorname {csch}^{2}{\left (x \right )}\right )^{3}}\, dx \]

[In]

integrate(1/(coth(x)**2+csch(x)**2)**3,x)

[Out]

Integral((coth(x)**2 + csch(x)**2)**(-3), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (41) = 82\).

Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.56 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x - \frac {19 \, e^{\left (-2 \, x\right )} + 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} + 3}{2 \, {\left (12 \, e^{\left (-2 \, x\right )} + 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]

[In]

integrate(1/(coth(x)^2+csch(x)^2)^3,x, algorithm="maxima")

[Out]

7/16*sqrt(2)*log(-(2*sqrt(2) - e^(-2*x) - 3)/(2*sqrt(2) + e^(-2*x) + 3)) + x - 1/2*(19*e^(-2*x) + 57*e^(-4*x)
+ 17*e^(-6*x) + 3)/(12*e^(-2*x) + 38*e^(-4*x) + 12*e^(-6*x) + e^(-8*x) + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=-\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x + \frac {17 \, e^{\left (6 \, x\right )} + 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} + 3}{2 \, {\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \]

[In]

integrate(1/(coth(x)^2+csch(x)^2)^3,x, algorithm="giac")

[Out]

-7/16*sqrt(2)*log(-(2*sqrt(2) - e^(2*x) - 3)/(2*sqrt(2) + e^(2*x) + 3)) + x + 1/2*(17*e^(6*x) + 57*e^(4*x) + 1
9*e^(2*x) + 3)/(e^(4*x) + 6*e^(2*x) + 1)^2

Mupad [B] (verification not implemented)

Time = 2.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=x+\frac {136\,{\mathrm {e}}^{2\,x}+24}{12\,{\mathrm {e}}^{2\,x}+38\,{\mathrm {e}}^{4\,x}+12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}-\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}-\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}+\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}+\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{2}-\frac {45}{2}}{6\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]

[In]

int(1/(coth(x)^2 + 1/sinh(x)^2)^3,x)

[Out]

x + (136*exp(2*x) + 24)/(12*exp(2*x) + 38*exp(4*x) + 12*exp(6*x) + exp(8*x) + 1) + (7*2^(1/2)*log(7*exp(2*x) -
(7*2^(1/2)*(12*exp(2*x) + 4))/16))/16 - (7*2^(1/2)*log(7*exp(2*x) + (7*2^(1/2)*(12*exp(2*x) + 4))/16))/16 + (
(17*exp(2*x))/2 - 45/2)/(6*exp(2*x) + exp(4*x) + 1)