Integrand size = 13, antiderivative size = 86 \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=-\frac {1}{x \left (1-e^{2 a} x^4\right )}+\frac {2 e^{2 a} x^3}{1-e^{2 a} x^4}-\frac {1}{2} e^{a/2} \arctan \left (e^{a/2} x\right )+\frac {1}{2} e^{a/2} \text {arctanh}\left (e^{a/2} x\right ) \]
-1/x/(1-exp(2*a)*x^4)+2*exp(2*a)*x^3/(1-exp(2*a)*x^4)-1/2*exp(1/2*a)*arcta n(exp(1/2*a)*x)+1/2*exp(1/2*a)*arctanh(exp(1/2*a)*x)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 3.35 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.78 \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=\frac {e^{-2 a} \left (-343-1163 e^{2 a} x^4-241 e^{4 a} x^8+3 e^{6 a} x^{12}+\left (343+632 e^{2 a} x^4+362 e^{4 a} x^8-56 e^{6 a} x^{12}-e^{8 a} x^{16}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},e^{2 a} x^4\right )\right )}{384 x^5}+\frac {16}{231} e^{2 a} x^3 \left (1+e^{2 a} x^4\right )^2 \, _4F_3\left (\frac {3}{4},2,2,2;1,1,\frac {15}{4};e^{2 a} x^4\right ) \]
(-343 - 1163*E^(2*a)*x^4 - 241*E^(4*a)*x^8 + 3*E^(6*a)*x^12 + (343 + 632*E ^(2*a)*x^4 + 362*E^(4*a)*x^8 - 56*E^(6*a)*x^12 - E^(8*a)*x^16)*Hypergeomet ric2F1[3/4, 1, 7/4, E^(2*a)*x^4])/(384*E^(2*a)*x^5) + (16*E^(2*a)*x^3*(1 + E^(2*a)*x^4)^2*HypergeometricPFQ[{3/4, 2, 2, 2}, {1, 1, 15/4}, E^(2*a)*x^ 4])/231
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6072, 962, 957, 827, 216, 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.
\(\displaystyle \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6072 |
\(\displaystyle \int \frac {\left (-e^{2 a} x^4-1\right )^2}{x^2 \left (1-e^{2 a} x^4\right )^2}dx\) |
\(\Big \downarrow \) 962 |
\(\displaystyle \int \frac {x^2 \left (e^{4 a} x^4+7 e^{2 a}\right )}{\left (1-e^{2 a} x^4\right )^2}dx-\frac {1}{x \left (1-e^{2 a} x^4\right )}\) |
\(\Big \downarrow \) 957 |
\(\displaystyle e^{2 a} \int \frac {x^2}{1-e^{2 a} x^4}dx-\frac {1}{x \left (1-e^{2 a} x^4\right )}+\frac {2 e^{2 a} x^3}{1-e^{2 a} x^4}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle e^{2 a} \left (\frac {1}{2} e^{-a} \int \frac {1}{1-e^a x^2}dx-\frac {1}{2} e^{-a} \int \frac {1}{e^a x^2+1}dx\right )-\frac {1}{x \left (1-e^{2 a} x^4\right )}+\frac {2 e^{2 a} x^3}{1-e^{2 a} x^4}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle e^{2 a} \left (\frac {1}{2} e^{-a} \int \frac {1}{1-e^a x^2}dx-\frac {1}{2} e^{-3 a/2} \arctan \left (e^{a/2} x\right )\right )-\frac {1}{x \left (1-e^{2 a} x^4\right )}+\frac {2 e^{2 a} x^3}{1-e^{2 a} x^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle e^{2 a} \left (\frac {1}{2} e^{-3 a/2} \text {arctanh}\left (e^{a/2} x\right )-\frac {1}{2} e^{-3 a/2} \arctan \left (e^{a/2} x\right )\right )-\frac {1}{x \left (1-e^{2 a} x^4\right )}+\frac {2 e^{2 a} x^3}{1-e^{2 a} x^4}\) |
-(1/(x*(1 - E^(2*a)*x^4))) + (2*E^(2*a)*x^3)/(1 - E^(2*a)*x^4) + E^(2*a)*( -1/2*ArcTan[E^(a/2)*x]/E^((3*a)/2) + ArcTanh[E^(a/2)*x]/(2*E^((3*a)/2)))
3.2.63.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a *b*e*n*(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*b*n* (p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && N eQ[p, -5/4]) || !RationalQ[m] || (IGtQ[n, 0] && ILtQ[p + 1/2, 0] && LeQ[-1 , m, (-n)*(p + 1)]))
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^2, x_Symbol] :> Simp[c^2*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1)) ), x] - Simp[1/(a*e^n*(m + 1)) Int[(e*x)^(m + n)*(a + b*x^n)^p*Simp[b*c^2 *n*(p + 1) + c*(b*c - 2*a*d)*(m + 1) - a*(m + 1)*d^2*x^n, x], x], x] /; Fre eQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && GtQ[n, 0]
Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*((-1 - E^(2*a*d)*x^(2*b*d))^p/(1 - E^(2*a*d)*x^(2*b*d))^p), x] /; FreeQ[{a, b, d, e, m, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21
method | result | size |
risch | \(\frac {-2 \,{\mathrm e}^{2 a} x^{4}+1}{x \left (-1+{\mathrm e}^{2 a} x^{4}\right )}+\frac {\sqrt {{\mathrm e}^{a}}\, \ln \left (-\left ({\mathrm e}^{a}\right )^{\frac {3}{2}}-{\mathrm e}^{2 a} x \right )}{4}-\frac {\sqrt {{\mathrm e}^{a}}\, \ln \left (\left ({\mathrm e}^{a}\right )^{\frac {3}{2}}-{\mathrm e}^{2 a} x \right )}{4}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{2}+{\mathrm e}^{a}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4}+4 \,{\mathrm e}^{2 a}\right ) x -\textit {\_R}^{3}\right )\right )}{4}\) | \(104\) |
(-2*exp(2*a)*x^4+1)/x/(-1+exp(2*a)*x^4)+1/4*exp(a)^(1/2)*ln(-exp(a)^(3/2)- exp(2*a)*x)-1/4*exp(a)^(1/2)*ln(exp(a)^(3/2)-exp(2*a)*x)+1/4*sum(_R*ln((-5 *_R^4+4*exp(2*a))*x-_R^3),_R=RootOf(_Z^2+exp(a)))
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.13 \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=-\frac {8 \, x^{4} e^{\left (2 \, a\right )} + 2 \, {\left (x^{5} e^{\left (2 \, a\right )} - x\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - {\left (x^{5} e^{\left (2 \, a\right )} - x\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} + 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right ) - 4}{4 \, {\left (x^{5} e^{\left (2 \, a\right )} - x\right )}} \]
-1/4*(8*x^4*e^(2*a) + 2*(x^5*e^(2*a) - x)*arctan(x*e^(1/2*a))*e^(1/2*a) - (x^5*e^(2*a) - x)*e^(1/2*a)*log((x^2*e^a + 2*x*e^(1/2*a) + 1)/(x^2*e^a - 1 )) - 4)/(x^5*e^(2*a) - x)
\[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=\int \frac {\coth ^{2}{\left (a + 2 \log {\left (x \right )} \right )}}{x^{2}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.80 \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=\frac {1}{2} \, \arctan \left (\frac {e^{\left (-\frac {1}{2} \, a\right )}}{x}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {\frac {1}{x} - e^{\left (\frac {1}{2} \, a\right )}}{\frac {1}{x} + e^{\left (\frac {1}{2} \, a\right )}}\right ) - \frac {1}{x} + \frac {e^{\left (2 \, a\right )}}{x {\left (\frac {1}{x^{4}} - e^{\left (2 \, a\right )}\right )}} \]
1/2*arctan(e^(-1/2*a)/x)*e^(1/2*a) - 1/4*e^(1/2*a)*log((1/x - e^(1/2*a))/( 1/x + e^(1/2*a))) - 1/x + e^(2*a)/(x*(1/x^4 - e^(2*a)))
Time = 0.28 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.90 \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=-\frac {1}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) - \frac {2 \, x^{4} e^{\left (2 \, a\right )} - 1}{x^{5} e^{\left (2 \, a\right )} - x} \]
-1/2*arctan(x*e^(1/2*a))*e^(1/2*a) - 1/4*e^(1/2*a)*log(abs(2*x*e^a - 2*e^( 1/2*a))/abs(2*x*e^a + 2*e^(1/2*a))) - (2*x^4*e^(2*a) - 1)/(x^5*e^(2*a) - x )
Time = 1.91 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.70 \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx=\frac {{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2}-\frac {{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2}+\frac {2\,x^4\,{\mathrm {e}}^{2\,a}-1}{x-x^5\,{\mathrm {e}}^{2\,a}} \]