Integrand size = 11, antiderivative size = 54 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=x-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}-\frac {\tanh (x)}{4 \left (2-\tanh ^2(x)\right )} \]
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)
Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\frac {76 x+48 x \cosh (2 x)-7 \sqrt {2} \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right ) (3+\cosh (2 x))^2+4 x \cosh (4 x)-2 \sinh (2 x)-3 \sinh (4 x)}{8 (3+\cosh (2 x))^2} \]
(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)
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 4889, 372, 27, 440, 25, 397, 219}
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.
\(\displaystyle \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (-\cot (i x)^2-\csc (i x)^2\right )^3}dx\) |
\(\Big \downarrow \) 4889 |
\(\displaystyle \int \frac {\tanh ^6(x)}{\left (1-\tanh ^2(x)\right ) \left (2-\tanh ^2(x)\right )^3}d\tanh (x)\) |
\(\Big \downarrow \) 372 |
\(\displaystyle \frac {1}{4} \int \frac {2 \tanh ^2(x) \left (3-\tanh ^2(x)\right )}{\left (1-\tanh ^2(x)\right ) \left (2-\tanh ^2(x)\right )^2}d\tanh (x)-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {\tanh ^2(x) \left (3-\tanh ^2(x)\right )}{\left (1-\tanh ^2(x)\right ) \left (2-\tanh ^2(x)\right )^2}d\tanh (x)-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}\) |
\(\Big \downarrow \) 440 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int -\frac {3 \tanh ^2(x)+1}{\left (1-\tanh ^2(x)\right ) \left (2-\tanh ^2(x)\right )}d\tanh (x)-\frac {\tanh (x)}{2 \left (2-\tanh ^2(x)\right )}\right )-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {3 \tanh ^2(x)+1}{\left (1-\tanh ^2(x)\right ) \left (2-\tanh ^2(x)\right )}d\tanh (x)-\frac {\tanh (x)}{2 \left (2-\tanh ^2(x)\right )}\right )-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}\) |
\(\Big \downarrow \) 397 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (4 \int \frac {1}{1-\tanh ^2(x)}d\tanh (x)-7 \int \frac {1}{2-\tanh ^2(x)}d\tanh (x)\right )-\frac {\tanh (x)}{2 \left (2-\tanh ^2(x)\right )}\right )-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (4 \text {arctanh}(\tanh (x))-\frac {7 \text {arctanh}\left (\frac {\tanh (x)}{\sqrt {2}}\right )}{\sqrt {2}}\right )-\frac {\tanh (x)}{2 \left (2-\tanh ^2(x)\right )}\right )-\frac {\tanh ^3(x)}{2 \left (2-\tanh ^2(x)\right )^2}\) |
-1/2*Tanh[x]^3/(2 - Tanh[x]^2)^2 + ((4*ArcTanh[Tanh[x]] - (7*ArcTanh[Tanh[ x]/Sqrt[2]])/Sqrt[2])/2 - Tanh[x]/(2*(2 - Tanh[x]^2)))/2
3.9.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
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 | \(\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )+\frac {-\frac {\tanh \left (\frac {x}{2}\right )^{7}}{4}-\frac {5 \tanh \left (\frac {x}{2}\right )^{5}}{4}-\frac {5 \tanh \left (\frac {x}{2}\right )^{3}}{4}-\frac {\tanh \left (\frac {x}{2}\right )}{4}}{\left (\tanh \left (\frac {x}{2}\right )^{4}+1\right )^{2}}-\frac {7 \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}+\frac {7 \sqrt {2}\, \left (\ln \left (\frac {\tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}{\tanh \left (\frac {x}{2}\right )^{2}+\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1}\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}+1\right )+2 \arctan \left (\tanh \left (\frac {x}{2}\right ) \sqrt {2}-1\right )\right )}{32}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )\) | \(199\) |
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 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\text {Too large to display} \]
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*(14 0*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 + 11 4*(4*x + 3)*cosh(x)^2 + 24*x + 19)*sinh(x)^2 + 7*(sqrt(2)*cosh(x)^8 + 8*sq rt(2)*cosh(x)*sinh(x)^7 + sqrt(2)*sinh(x)^8 + 4*(7*sqrt(2)*cosh(x)^2 + 3*s qrt(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*sqrt(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 + 1 2*sqrt(2)*cosh(x)^2 + 8*(sqrt(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)/(cosh(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))*sin h(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 + 12*cosh(x)^6 + 8*(7*cosh(x)^3 + 9*cosh(x))*sinh(...
\[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\int \frac {1}{\left (\coth ^{2}{\left (x \right )} + \operatorname {csch}^{2}{\left (x \right )}\right )^{3}}\, dx \]
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 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (-2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (-2 \, x\right )} + 3}\right ) + x - \frac {19 \, e^{\left (-2 \, x\right )} + 57 \, e^{\left (-4 \, x\right )} + 17 \, e^{\left (-6 \, x\right )} + 3}{2 \, {\left (12 \, e^{\left (-2 \, x\right )} + 38 \, e^{\left (-4 \, x\right )} + 12 \, e^{\left (-6 \, x\right )} + e^{\left (-8 \, x\right )} + 1\right )}} \]
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)
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.33 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=-\frac {7}{16} \, \sqrt {2} \log \left (-\frac {2 \, \sqrt {2} - e^{\left (2 \, x\right )} - 3}{2 \, \sqrt {2} + e^{\left (2 \, x\right )} + 3}\right ) + x + \frac {17 \, e^{\left (6 \, x\right )} + 57 \, e^{\left (4 \, x\right )} + 19 \, e^{\left (2 \, x\right )} + 3}{2 \, {\left (e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} + 1\right )}^{2}} \]
-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) + 19*e^(2*x) + 3)/(e^(4*x) + 6*e^(2*x) + 1)^2
Time = 2.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.07 \[ \int \frac {1}{\left (\coth ^2(x)+\text {csch}^2(x)\right )^3} \, dx=x+\frac {136\,{\mathrm {e}}^{2\,x}+24}{12\,{\mathrm {e}}^{2\,x}+38\,{\mathrm {e}}^{4\,x}+12\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}-\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}-\frac {7\,\sqrt {2}\,\ln \left (7\,{\mathrm {e}}^{2\,x}+\frac {7\,\sqrt {2}\,\left (12\,{\mathrm {e}}^{2\,x}+4\right )}{16}\right )}{16}+\frac {\frac {17\,{\mathrm {e}}^{2\,x}}{2}-\frac {45}{2}}{6\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1} \]