Integrand size = 18, antiderivative size = 73 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=-\frac {f \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{2 a \sqrt {4 a c-f^2}}+\frac {\log (x)}{a}-\frac {\log \left (a+f x^2+c x^4\right )}{4 a} \] Output:
-1/2*f*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1/2))/a/(4*a*c-f^2)^(1/2)+ln(x)/a-1 /4*ln(c*x^4+f*x^2+a)/a
Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.55 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=\frac {4 \sqrt {-4 a c+f^2} \log (x)-\left (f+\sqrt {-4 a c+f^2}\right ) \log \left (f-\sqrt {-4 a c+f^2}+2 c x^2\right )+\left (f-\sqrt {-4 a c+f^2}\right ) \log \left (f+\sqrt {-4 a c+f^2}+2 c x^2\right )}{4 a \sqrt {-4 a c+f^2}} \] Input:
Integrate[1/(x*(a + f*x^2 + c*x^4)),x]
Output:
(4*Sqrt[-4*a*c + f^2]*Log[x] - (f + Sqrt[-4*a*c + f^2])*Log[f - Sqrt[-4*a* c + f^2] + 2*c*x^2] + (f - Sqrt[-4*a*c + f^2])*Log[f + Sqrt[-4*a*c + f^2] + 2*c*x^2])/(4*a*Sqrt[-4*a*c + f^2])
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {1434, 1144, 25, 1142, 1083, 217, 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.
\(\displaystyle \int \frac {1}{x \left (a+c x^4+f x^2\right )} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (c x^4+f x^2+a\right )}dx^2\) |
\(\Big \downarrow \) 1144 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {c x^2+f}{c x^4+f x^2+a}dx^2}{a}+\frac {\log \left (x^2\right )}{a}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a}-\frac {\int \frac {c x^2+f}{c x^4+f x^2+a}dx^2}{a}\right )\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} f \int \frac {1}{c x^4+f x^2+a}dx^2+\frac {1}{2} \int \frac {2 c x^2+f}{c x^4+f x^2+a}dx^2}{a}\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \int \frac {2 c x^2+f}{c x^4+f x^2+a}dx^2-f \int \frac {1}{-x^4+f^2-4 a c}d\left (2 c x^2+f\right )}{a}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {1}{2} \int \frac {2 c x^2+f}{c x^4+f x^2+a}dx^2+\frac {f \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{\sqrt {4 a c-f^2}}}{a}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\log \left (x^2\right )}{a}-\frac {\frac {f \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{\sqrt {4 a c-f^2}}+\frac {1}{2} \log \left (a+c x^4+f x^2\right )}{a}\right )\) |
Input:
Int[1/(x*(a + f*x^2 + c*x^4)),x]
Output:
(Log[x^2]/a - ((f*ArcTan[(f + 2*c*x^2)/Sqrt[4*a*c - f^2]])/Sqrt[4*a*c - f^ 2] + Log[a + f*x^2 + c*x^4]/2)/a)/2
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[((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]
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {\frac {\ln \left (c \,x^{4}+f \,x^{2}+a \right )}{2}+\frac {f \arctan \left (\frac {2 c \,x^{2}+f}{\sqrt {4 a c -f^{2}}}\right )}{\sqrt {4 a c -f^{2}}}}{2 a}+\frac {\ln \left (x \right )}{a}\) | \(65\) |
risch | \(\frac {\ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 c \,a^{2}-a \,f^{2}\right ) \textit {\_Z}^{2}+\left (4 a c -f^{2}\right ) \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a c -3 f^{2}\right ) \textit {\_R} +5 c \right ) x^{2}-a f \textit {\_R} +2 f \right )\right )}{2}\) | \(77\) |
Input:
int(1/x/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2/a*(1/2*ln(c*x^4+f*x^2+a)+f/(4*a*c-f^2)^(1/2)*arctan((2*c*x^2+f)/(4*a* c-f^2)^(1/2)))+ln(x)/a
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.04 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=\left [-\frac {\sqrt {-4 \, a c + f^{2}} f \log \left (\frac {2 \, c^{2} x^{4} + 2 \, c f x^{2} - 2 \, a c + f^{2} + {\left (2 \, c x^{2} + f\right )} \sqrt {-4 \, a c + f^{2}}}{c x^{4} + f x^{2} + a}\right ) + {\left (4 \, a c - f^{2}\right )} \log \left (c x^{4} + f x^{2} + a\right ) - 4 \, {\left (4 \, a c - f^{2}\right )} \log \left (x\right )}{4 \, {\left (4 \, a^{2} c - a f^{2}\right )}}, \frac {2 \, \sqrt {4 \, a c - f^{2}} f \arctan \left (-\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right ) - {\left (4 \, a c - f^{2}\right )} \log \left (c x^{4} + f x^{2} + a\right ) + 4 \, {\left (4 \, a c - f^{2}\right )} \log \left (x\right )}{4 \, {\left (4 \, a^{2} c - a f^{2}\right )}}\right ] \] Input:
integrate(1/x/(c*x^4+f*x^2+a),x, algorithm="fricas")
Output:
[-1/4*(sqrt(-4*a*c + f^2)*f*log((2*c^2*x^4 + 2*c*f*x^2 - 2*a*c + f^2 + (2* c*x^2 + f)*sqrt(-4*a*c + f^2))/(c*x^4 + f*x^2 + a)) + (4*a*c - f^2)*log(c* x^4 + f*x^2 + a) - 4*(4*a*c - f^2)*log(x))/(4*a^2*c - a*f^2), 1/4*(2*sqrt( 4*a*c - f^2)*f*arctan(-(2*c*x^2 + f)/sqrt(4*a*c - f^2)) - (4*a*c - f^2)*lo g(c*x^4 + f*x^2 + a) + 4*(4*a*c - f^2)*log(x))/(4*a^2*c - a*f^2)]
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (60) = 120\).
Time = 2.78 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.47 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=\left (- \frac {f \sqrt {- 4 a c + f^{2}}}{4 a \left (4 a c - f^{2}\right )} - \frac {1}{4 a}\right ) \log {\left (x^{2} + \frac {- 8 a^{2} c \left (- \frac {f \sqrt {- 4 a c + f^{2}}}{4 a \left (4 a c - f^{2}\right )} - \frac {1}{4 a}\right ) - 2 a c + 2 a f^{2} \left (- \frac {f \sqrt {- 4 a c + f^{2}}}{4 a \left (4 a c - f^{2}\right )} - \frac {1}{4 a}\right ) + f^{2}}{c f} \right )} + \left (\frac {f \sqrt {- 4 a c + f^{2}}}{4 a \left (4 a c - f^{2}\right )} - \frac {1}{4 a}\right ) \log {\left (x^{2} + \frac {- 8 a^{2} c \left (\frac {f \sqrt {- 4 a c + f^{2}}}{4 a \left (4 a c - f^{2}\right )} - \frac {1}{4 a}\right ) - 2 a c + 2 a f^{2} \left (\frac {f \sqrt {- 4 a c + f^{2}}}{4 a \left (4 a c - f^{2}\right )} - \frac {1}{4 a}\right ) + f^{2}}{c f} \right )} + \frac {\log {\left (x \right )}}{a} \] Input:
integrate(1/x/(c*x**4+f*x**2+a),x)
Output:
(-f*sqrt(-4*a*c + f**2)/(4*a*(4*a*c - f**2)) - 1/(4*a))*log(x**2 + (-8*a** 2*c*(-f*sqrt(-4*a*c + f**2)/(4*a*(4*a*c - f**2)) - 1/(4*a)) - 2*a*c + 2*a* f**2*(-f*sqrt(-4*a*c + f**2)/(4*a*(4*a*c - f**2)) - 1/(4*a)) + f**2)/(c*f) ) + (f*sqrt(-4*a*c + f**2)/(4*a*(4*a*c - f**2)) - 1/(4*a))*log(x**2 + (-8* a**2*c*(f*sqrt(-4*a*c + f**2)/(4*a*(4*a*c - f**2)) - 1/(4*a)) - 2*a*c + 2* a*f**2*(f*sqrt(-4*a*c + f**2)/(4*a*(4*a*c - f**2)) - 1/(4*a)) + f**2)/(c*f )) + log(x)/a
Exception generated. \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x/(c*x^4+f*x^2+a),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(f^2-4*a*c>0)', see `assume?` for more deta
Time = 0.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=-\frac {f \arctan \left (\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right )}{2 \, \sqrt {4 \, a c - f^{2}} a} - \frac {\log \left (c x^{4} + f x^{2} + a\right )}{4 \, a} + \frac {\log \left (x^{2}\right )}{2 \, a} \] Input:
integrate(1/x/(c*x^4+f*x^2+a),x, algorithm="giac")
Output:
-1/2*f*arctan((2*c*x^2 + f)/sqrt(4*a*c - f^2))/(sqrt(4*a*c - f^2)*a) - 1/4 *log(c*x^4 + f*x^2 + a)/a + 1/2*log(x^2)/a
Time = 18.00 (sec) , antiderivative size = 1012, normalized size of antiderivative = 13.86 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx =\text {Too large to display} \] Input:
int(1/(x*(a + c*x^4 + f*x^2)),x)
Output:
log(x)/a - (log(a + c*x^4 + f*x^2)*(8*a*c - 2*f^2))/(2*(16*a^2*c - 4*a*f^2 )) - (f*atan((2*(3*f^3 - 8*a*c*f)*(4*a*c - f^2)^(3/2)*((f^2*(4*c^2*f^2 + ( 2*a*c^2*f^2*(8*a*c - 2*f^2))/(16*a^2*c - 4*a*f^2)))/(16*a^2*(4*a*c - f^2)) - ((8*a*c - 2*f^2)^2*(4*c^2*f^2 + (2*a*c^2*f^2*(8*a*c - 2*f^2))/(16*a^2*c - 4*a*f^2)))/(4*(16*a^2*c - 4*a*f^2)^2) + (c^2*f^4*(8*a*c - 2*f^2))/(4*a* (16*a^2*c - 4*a*f^2)*(4*a*c - f^2))))/(c^4*f^2*(25*a*c - 6*f^2)) - (2*(4*a *c - f^2)*(3*f^4 + 10*a^2*c^2 - 14*a*c*f^2)*((c^2*f^3*(8*a*c - 2*f^2)^2)/( 4*(16*a^2*c - 4*a*f^2)^2*(4*a*c - f^2)^(1/2)) - (c^2*f^5)/(16*a^2*(4*a*c - f^2)^(3/2)) + (f*(8*a*c - 2*f^2)*(4*c^2*f^2 + (2*a*c^2*f^2*(8*a*c - 2*f^2 ))/(16*a^2*c - 4*a*f^2)))/(4*a*(16*a^2*c - 4*a*f^2)*(4*a*c - f^2)^(1/2)))) /(c^4*f^2*(25*a*c - 6*f^2)) + (16*a^3*x^2*(((3*f^3 - 8*a*c*f)*((f^2*(10*c^ 3*f + ((12*c^2*f^3 - 40*a*c^3*f)*(8*a*c - 2*f^2))/(2*(16*a^2*c - 4*a*f^2)) ))/(16*a^2*(4*a*c - f^2)) - ((8*a*c - 2*f^2)^2*(10*c^3*f + ((12*c^2*f^3 - 40*a*c^3*f)*(8*a*c - 2*f^2))/(2*(16*a^2*c - 4*a*f^2))))/(4*(16*a^2*c - 4*a *f^2)^2) + (f^2*(12*c^2*f^3 - 40*a*c^3*f)*(8*a*c - 2*f^2))/(16*a^2*(16*a^2 *c - 4*a*f^2)*(4*a*c - f^2))))/(8*a^3*c^2*(25*a*c - 6*f^2)) - ((3*f^4 + 10 *a^2*c^2 - 14*a*c*f^2)*((f*(12*c^2*f^3 - 40*a*c^3*f)*(8*a*c - 2*f^2)^2)/(1 6*a*(16*a^2*c - 4*a*f^2)^2*(4*a*c - f^2)^(1/2)) - (f^3*(12*c^2*f^3 - 40*a* c^3*f))/(64*a^3*(4*a*c - f^2)^(3/2)) + (f*(8*a*c - 2*f^2)*(10*c^3*f + ((12 *c^2*f^3 - 40*a*c^3*f)*(8*a*c - 2*f^2))/(2*(16*a^2*c - 4*a*f^2))))/(4*a...
Time = 0.18 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.38 \[ \int \frac {1}{x \left (a+f x^2+c x^4\right )} \, dx=\frac {2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f +2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f -4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c +\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{2}-4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c +\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{2}+16 \,\mathrm {log}\left (x \right ) a c -4 \,\mathrm {log}\left (x \right ) f^{2}}{4 a \left (4 a c -f^{2}\right )} \] Input:
int(1/x/(c*x^4+f*x^2+a),x)
Output:
(2*sqrt(2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sq rt(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f + 2*sqrt( 2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sqrt(c)*sq rt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f - 4*log( - sqrt(2 *sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c + log( - sqrt(2*sqrt (c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**2 - 4*log(sqrt(2*sqrt(c)*s qrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c + log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**2 + 16*log(x)*a*c - 4*log(x)*f**2)/(4*a *(4*a*c - f**2))