Integrand size = 22, antiderivative size = 77 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 (a-b) \sqrt {b}}+\frac {\log (x)}{a-b}-\frac {\log \left (a-b+2 a x^2+a x^4\right )}{4 (a-b)} \] Output:
1/2*a^(1/2)*arctanh(a^(1/2)*(x^2+1)/b^(1/2))/(a-b)/b^(1/2)+ln(x)/(a-b)-ln( a*x^4+2*a*x^2+a-b)/(4*a-4*b)
Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.17 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {-4 \sqrt {b} \log (x)+\left (\sqrt {a}+\sqrt {b}\right ) \log \left (-\sqrt {b}+\sqrt {a} \left (1+x^2\right )\right )+\left (-\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt {b}+\sqrt {a} \left (1+x^2\right )\right )}{4 \sqrt {b} (-a+b)} \] Input:
Integrate[1/(x*(a - b + 2*a*x^2 + a*x^4)),x]
Output:
(-4*Sqrt[b]*Log[x] + (Sqrt[a] + Sqrt[b])*Log[-Sqrt[b] + Sqrt[a]*(1 + x^2)] + (-Sqrt[a] + Sqrt[b])*Log[Sqrt[b] + Sqrt[a]*(1 + x^2)])/(4*Sqrt[b]*(-a + b))
Time = 0.47 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1434, 1144, 25, 27, 1142, 27, 1083, 219, 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.
\(\displaystyle \int \frac {1}{x \left (a x^4+2 a x^2+a-b\right )} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (a x^4+2 a x^2+a-b\right )}dx^2\) |
\(\Big \downarrow \) 1144 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {a \left (x^2+2\right )}{a x^4+2 a x^2+a-b}dx^2}{a-b}+\frac {\log \left (x^2\right )}{a-b}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {\int \frac {a \left (x^2+2\right )}{a x^4+2 a x^2+a-b}dx^2}{a-b}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {a \int \frac {x^2+2}{a x^4+2 a x^2+a-b}dx^2}{a-b}\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {a \left (\int \frac {1}{a x^4+2 a x^2+a-b}dx^2+\frac {\int \frac {2 a \left (x^2+1\right )}{a x^4+2 a x^2+a-b}dx^2}{2 a}\right )}{a-b}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {a \left (\int \frac {1}{a x^4+2 a x^2+a-b}dx^2+\int \frac {x^2+1}{a x^4+2 a x^2+a-b}dx^2\right )}{a-b}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {a \left (\int \frac {x^2+1}{a x^4+2 a x^2+a-b}dx^2-2 \int \frac {1}{4 a b-x^4}d\left (2 a x^2+2 a\right )\right )}{a-b}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {a \left (\int \frac {x^2+1}{a x^4+2 a x^2+a-b}dx^2-\frac {\text {arctanh}\left (\frac {2 a x^2+2 a}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right )}{a-b}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a-b}-\frac {a \left (\frac {\log \left (a x^4+2 a x^2+a-b\right )}{2 a}-\frac {\text {arctanh}\left (\frac {2 a x^2+2 a}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right )}{a-b}\right )\) |
Input:
Int[1/(x*(a - b + 2*a*x^2 + a*x^4)),x]
Output:
(Log[x^2]/(a - b) - (a*(-(ArcTanh[(2*a + 2*a*x^2)/(2*Sqrt[a]*Sqrt[b])]/(Sq rt[a]*Sqrt[b])) + Log[a - b + 2*a*x^2 + a*x^4]/(2*a)))/(a - b))/2
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/(((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[e*(Log[RemoveContent[d + e*x, x]]/(c*d^2 - b*d*e + a*e^2)), x] + S imp[1/(c*d^2 - b*d*e + a*e^2) Int[(c*d - b*e - c*e*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.83
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a -b}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (-1+\left (a b -b^{2}\right ) \textit {\_Z}^{2}+2 b \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a -5 b \right ) \textit {\_R} +5\right ) x^{2}+\left (-a +b \right ) \textit {\_R} +4\right )\right )}{4}\) | \(64\) |
default | \(\frac {\ln \left (x \right )}{a -b}-\frac {a \left (\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a -b \right )}{2 a}-\frac {\operatorname {arctanh}\left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{\sqrt {a b}}\right )}{2 \left (a -b \right )}\) | \(70\) |
Input:
int(1/x/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)
Output:
ln(x)/(a-b)+1/4*sum(_R*ln(((-a-5*b)*_R+5)*x^2+(-a+b)*_R+4),_R=RootOf(-1+(a *b-b^2)*_Z^2+2*b*_Z))
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=\left [-\frac {\sqrt {\frac {a}{b}} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, {\left (b x^{2} + b\right )} \sqrt {\frac {a}{b}} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \, {\left (a - b\right )}}, -\frac {2 \, \sqrt {-\frac {a}{b}} \arctan \left ({\left (x^{2} + 1\right )} \sqrt {-\frac {a}{b}}\right ) + \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 4 \, \log \left (x\right )}{4 \, {\left (a - b\right )}}\right ] \] Input:
integrate(1/x/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
Output:
[-1/4*(sqrt(a/b)*log((a*x^4 + 2*a*x^2 - 2*(b*x^2 + b)*sqrt(a/b) + a + b)/( a*x^4 + 2*a*x^2 + a - b)) + log(a*x^4 + 2*a*x^2 + a - b) - 4*log(x))/(a - b), -1/4*(2*sqrt(-a/b)*arctan((x^2 + 1)*sqrt(-a/b)) + log(a*x^4 + 2*a*x^2 + a - b) - 4*log(x))/(a - b)]
Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (61) = 122\).
Time = 2.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.39 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=\left (- \frac {1}{4 \left (a - b\right )} - \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{4 \left (a - b\right )} - \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac {1}{4 \left (a - b\right )} - \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \left (- \frac {1}{4 \left (a - b\right )} + \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) \log {\left (x^{2} + \frac {4 a b \left (- \frac {1}{4 \left (a - b\right )} + \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + a - 4 b^{2} \left (- \frac {1}{4 \left (a - b\right )} + \frac {\sqrt {a b}}{4 b \left (a - b\right )}\right ) + b}{a} \right )} + \frac {\log {\left (x \right )}}{a - b} \] Input:
integrate(1/x/(a*x**4+2*a*x**2+a-b),x)
Output:
(-1/(4*(a - b)) - sqrt(a*b)/(4*b*(a - b)))*log(x**2 + (4*a*b*(-1/(4*(a - b )) - sqrt(a*b)/(4*b*(a - b))) + a - 4*b**2*(-1/(4*(a - b)) - sqrt(a*b)/(4* b*(a - b))) + b)/a) + (-1/(4*(a - b)) + sqrt(a*b)/(4*b*(a - b)))*log(x**2 + (4*a*b*(-1/(4*(a - b)) + sqrt(a*b)/(4*b*(a - b))) + a - 4*b**2*(-1/(4*(a - b)) + sqrt(a*b)/(4*b*(a - b))) + b)/a) + log(x)/(a - b)
Time = 0.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=-\frac {a \log \left (\frac {a x^{2} + a - \sqrt {a b}}{a x^{2} + a + \sqrt {a b}}\right )}{4 \, \sqrt {a b} {\left (a - b\right )}} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, {\left (a - b\right )}} + \frac {\log \left (x^{2}\right )}{2 \, {\left (a - b\right )}} \] Input:
integrate(1/x/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")
Output:
-1/4*a*log((a*x^2 + a - sqrt(a*b))/(a*x^2 + a + sqrt(a*b)))/(sqrt(a*b)*(a - b)) - 1/4*log(a*x^4 + 2*a*x^2 + a - b)/(a - b) + 1/2*log(x^2)/(a - b)
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=-\frac {a \arctan \left (\frac {a x^{2} + a}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b} {\left (a - b\right )}} - \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, {\left (a - b\right )}} + \frac {\log \left (x^{2}\right )}{2 \, {\left (a - b\right )}} \] Input:
integrate(1/x/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")
Output:
-1/2*a*arctan((a*x^2 + a)/sqrt(-a*b))/(sqrt(-a*b)*(a - b)) - 1/4*log(a*x^4 + 2*a*x^2 + a - b)/(a - b) + 1/2*log(x^2)/(a - b)
Time = 19.70 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.38 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {\ln \left (x\right )}{a-b}-\frac {\ln \left (16\,a^4+20\,a^4\,x^2+\frac {\left (b-\sqrt {a\,b}\right )\,\left (x^2\,\left (16\,a^5+80\,b\,a^4\right )-16\,a^4\,b+16\,a^5\right )}{4\,\left (a\,b-b^2\right )}\right )\,\left (b-\sqrt {a\,b}\right )}{4\,\left (a\,b-b^2\right )}-\frac {\ln \left (16\,a^4+20\,a^4\,x^2+\frac {\left (b+\sqrt {a\,b}\right )\,\left (x^2\,\left (16\,a^5+80\,b\,a^4\right )-16\,a^4\,b+16\,a^5\right )}{4\,\left (a\,b-b^2\right )}\right )\,\left (b+\sqrt {a\,b}\right )}{4\,\left (a\,b-b^2\right )} \] Input:
int(1/(x*(a - b + 2*a*x^2 + a*x^4)),x)
Output:
log(x)/(a - b) - (log(16*a^4 + 20*a^4*x^2 + ((b - (a*b)^(1/2))*(x^2*(80*a^ 4*b + 16*a^5) - 16*a^4*b + 16*a^5))/(4*(a*b - b^2)))*(b - (a*b)^(1/2)))/(4 *(a*b - b^2)) - (log(16*a^4 + 20*a^4*x^2 + ((b + (a*b)^(1/2))*(x^2*(80*a^4 *b + 16*a^5) - 16*a^4*b + 16*a^5))/(4*(a*b - b^2)))*(b + (a*b)^(1/2)))/(4* (a*b - b^2))
Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.79 \[ \int \frac {1}{x \left (a-b+2 a x^2+a x^4\right )} \, dx=\frac {-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right )-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right )+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right )-\mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b -\mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b -\mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) b +4 \,\mathrm {log}\left (x \right ) b}{4 b \left (a -b \right )} \] Input:
int(1/x/(a*x^4+2*a*x^2+a-b),x)
Output:
( - sqrt(b)*sqrt(a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x) - sqrt(b )*sqrt(a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x) + sqrt(b)*sqrt(a)*log (sqrt(b)*sqrt(a) + a*x**2 + a) - log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a )*x)*b - log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b - log(sqrt(b)*sqrt(a ) + a*x**2 + a)*b + 4*log(x)*b)/(4*b*(a - b))