Integrand size = 11, antiderivative size = 147 \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=\frac {1}{x}-\frac {e^{a/2} \arctan \left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}+\frac {e^{a/2} \arctan \left (1+\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}+\frac {e^{a/2} \log \left (1-\sqrt {2} e^{a/2} x+e^a x^2\right )}{2 \sqrt {2}}-\frac {e^{a/2} \log \left (1+\sqrt {2} e^{a/2} x+e^a x^2\right )}{2 \sqrt {2}} \]
1/x+1/2*exp(1/2*a)*arctan(-1+exp(1/2*a)*x*2^(1/2))*2^(1/2)+1/2*exp(1/2*a)* arctan(1+exp(1/2*a)*x*2^(1/2))*2^(1/2)+1/4*exp(1/2*a)*ln(1+exp(a)*x^2-exp( 1/2*a)*x*2^(1/2))*2^(1/2)-1/4*exp(1/2*a)*ln(1+exp(a)*x^2+exp(1/2*a)*x*2^(1 /2))*2^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.40 \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=\frac {2-x \text {RootSum}\left [\cosh (a)+\sinh (a)+\cosh (a) \text {$\#$1}^4-\sinh (a) \text {$\#$1}^4\&,\frac {\log (x)+\log \left (\frac {1}{x}-\text {$\#$1}\right )}{\text {$\#$1}^3}\&\right ] (\cosh (a)+\sinh (a))^2}{2 x} \]
(2 - x*RootSum[Cosh[a] + Sinh[a] + Cosh[a]*#1^4 - Sinh[a]*#1^4 & , (Log[x] + Log[x^(-1) - #1])/#1^3 & ]*(Cosh[a] + Sinh[a])^2)/(2*x)
Time = 0.42 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {6071, 955, 826, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
\(\displaystyle \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx\) |
\(\Big \downarrow \) 6071 |
\(\displaystyle \int \frac {e^{2 a} x^4-1}{x^2 \left (e^{2 a} x^4+1\right )}dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle 2 e^{2 a} \int \frac {x^2}{e^{2 a} x^4+1}dx+\frac {1}{x}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \int \frac {e^a x^2+1}{e^{2 a} x^4+1}dx-\frac {1}{2} e^{-a} \int \frac {1-e^a x^2}{e^{2 a} x^4+1}dx\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {1}{2} e^{-a} \int \frac {1}{x^2-\sqrt {2} e^{-a/2} x+e^{-a}}dx+\frac {1}{2} e^{-a} \int \frac {1}{x^2+\sqrt {2} e^{-a/2} x+e^{-a}}dx\right )-\frac {1}{2} e^{-a} \int \frac {1-e^a x^2}{e^{2 a} x^4+1}dx\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \int \frac {1}{-\left (1-\sqrt {2} e^{a/2} x\right )^2-1}d\left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}-\frac {e^{-a/2} \int \frac {1}{-\left (\sqrt {2} e^{a/2} x+1\right )^2-1}d\left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}\right )-\frac {1}{2} e^{-a} \int \frac {1-e^a x^2}{e^{2 a} x^4+1}dx\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \arctan \left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}-\frac {e^{-a/2} \arctan \left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}\right )-\frac {1}{2} e^{-a} \int \frac {1-e^a x^2}{e^{2 a} x^4+1}dx\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \arctan \left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}-\frac {e^{-a/2} \arctan \left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}\right )-\frac {1}{2} e^{-a} \left (-\frac {e^{-a/2} \int -\frac {\sqrt {2} e^{-a/2}-2 x}{x^2-\sqrt {2} e^{-a/2} x+e^{-a}}dx}{2 \sqrt {2}}-\frac {e^{-a/2} \int -\frac {\sqrt {2} \left (\sqrt {2} x+e^{-a/2}\right )}{x^2+\sqrt {2} e^{-a/2} x+e^{-a}}dx}{2 \sqrt {2}}\right )\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \arctan \left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}-\frac {e^{-a/2} \arctan \left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}\right )-\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \int \frac {\sqrt {2} e^{-a/2}-2 x}{x^2-\sqrt {2} e^{-a/2} x+e^{-a}}dx}{2 \sqrt {2}}+\frac {e^{-a/2} \int \frac {\sqrt {2} \left (\sqrt {2} x+e^{-a/2}\right )}{x^2+\sqrt {2} e^{-a/2} x+e^{-a}}dx}{2 \sqrt {2}}\right )\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \arctan \left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}-\frac {e^{-a/2} \arctan \left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}\right )-\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \int \frac {\sqrt {2} e^{-a/2}-2 x}{x^2-\sqrt {2} e^{-a/2} x+e^{-a}}dx}{2 \sqrt {2}}+\frac {1}{2} e^{-a/2} \int \frac {\sqrt {2} x+e^{-a/2}}{x^2+\sqrt {2} e^{-a/2} x+e^{-a}}dx\right )\right )+\frac {1}{x}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle 2 e^{2 a} \left (\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \arctan \left (\sqrt {2} e^{a/2} x+1\right )}{\sqrt {2}}-\frac {e^{-a/2} \arctan \left (1-\sqrt {2} e^{a/2} x\right )}{\sqrt {2}}\right )-\frac {1}{2} e^{-a} \left (\frac {e^{-a/2} \log \left (e^a x^2+\sqrt {2} e^{a/2} x+1\right )}{2 \sqrt {2}}-\frac {e^{-a/2} \log \left (e^a x^2-\sqrt {2} e^{a/2} x+1\right )}{2 \sqrt {2}}\right )\right )+\frac {1}{x}\) |
x^(-1) + 2*E^(2*a)*((-(ArcTan[1 - Sqrt[2]*E^(a/2)*x]/(Sqrt[2]*E^(a/2))) + ArcTan[1 + Sqrt[2]*E^(a/2)*x]/(Sqrt[2]*E^(a/2)))/(2*E^a) - (-1/2*Log[1 - S qrt[2]*E^(a/2)*x + E^a*x^2]/(Sqrt[2]*E^(a/2)) + Log[1 + Sqrt[2]*E^(a/2)*x + E^a*x^2]/(2*Sqrt[2]*E^(a/2)))/(2*E^a))
3.2.51.3.1 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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[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[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((e_.)*(x_))^(m_.)*Tanh[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), 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 = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.29
method | result | size |
risch | \(\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+{\mathrm e}^{2 a}\right )}{\sum }\textit {\_R} \ln \left (\left (5 \textit {\_R}^{4}+4 \,{\mathrm e}^{2 a}\right ) x -\textit {\_R}^{3}\right )\right )}{2}\) | \(42\) |
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.82 \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=\frac {x \left (-e^{\left (2 \, a\right )}\right )^{\frac {1}{4}} \log \left (x e^{\left (2 \, a\right )} + \left (-e^{\left (2 \, a\right )}\right )^{\frac {3}{4}}\right ) - i \, x \left (-e^{\left (2 \, a\right )}\right )^{\frac {1}{4}} \log \left (x e^{\left (2 \, a\right )} + i \, \left (-e^{\left (2 \, a\right )}\right )^{\frac {3}{4}}\right ) + i \, x \left (-e^{\left (2 \, a\right )}\right )^{\frac {1}{4}} \log \left (x e^{\left (2 \, a\right )} - i \, \left (-e^{\left (2 \, a\right )}\right )^{\frac {3}{4}}\right ) - x \left (-e^{\left (2 \, a\right )}\right )^{\frac {1}{4}} \log \left (x e^{\left (2 \, a\right )} - \left (-e^{\left (2 \, a\right )}\right )^{\frac {3}{4}}\right ) + 2}{2 \, x} \]
1/2*(x*(-e^(2*a))^(1/4)*log(x*e^(2*a) + (-e^(2*a))^(3/4)) - I*x*(-e^(2*a)) ^(1/4)*log(x*e^(2*a) + I*(-e^(2*a))^(3/4)) + I*x*(-e^(2*a))^(1/4)*log(x*e^ (2*a) - I*(-e^(2*a))^(3/4)) - x*(-e^(2*a))^(1/4)*log(x*e^(2*a) - (-e^(2*a) )^(3/4)) + 2)/x
\[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=\int \frac {\tanh {\left (a + 2 \log {\left (x \right )} \right )}}{x^{2}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.85 \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (\frac {1}{2} \, a\right )} + \frac {2}{x}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (\frac {1}{2} \, a\right )} - \frac {2}{x}\right )} e^{\left (-\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {\sqrt {2} e^{\left (\frac {1}{2} \, a\right )}}{x} + \frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (-\frac {\sqrt {2} e^{\left (\frac {1}{2} \, a\right )}}{x} + \frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{x} \]
-1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*e^(1/2*a) + 2/x)*e^(-1/2*a))*e^(1 /2*a) - 1/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*e^(1/2*a) - 2/x)*e^(-1/2* a))*e^(1/2*a) - 1/4*sqrt(2)*e^(1/2*a)*log(sqrt(2)*e^(1/2*a)/x + 1/x^2 + e^ a) + 1/4*sqrt(2)*e^(1/2*a)*log(-sqrt(2)*e^(1/2*a)/x + 1/x^2 + e^a) + 1/x
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.82 \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} + 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + \frac {1}{2} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} e^{\left (-\frac {1}{2} \, a\right )} - 2 \, x\right )} e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {1}{4} \, \sqrt {2} e^{\left (\frac {1}{2} \, a\right )} \log \left (-\sqrt {2} x e^{\left (-\frac {1}{2} \, a\right )} + x^{2} + e^{\left (-a\right )}\right ) + \frac {1}{x} \]
1/2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*e^(-1/2*a) + 2*x)*e^(1/2*a))*e^(1/ 2*a) + 1/2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*e^(-1/2*a) - 2*x)*e^(1/2*a ))*e^(1/2*a) - 1/4*sqrt(2)*e^(1/2*a)*log(sqrt(2)*x*e^(-1/2*a) + x^2 + e^(- a)) + 1/4*sqrt(2)*e^(1/2*a)*log(-sqrt(2)*x*e^(-1/2*a) + x^2 + e^(-a)) + 1/ x
Time = 1.73 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.31 \[ \int \frac {\tanh (a+2 \log (x))}{x^2} \, dx=\mathrm {atan}\left (x\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}\right )\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}-\mathrm {atanh}\left (x\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}\right )\,{\left (-{\mathrm {e}}^{2\,a}\right )}^{1/4}+\frac {1}{x} \]