Integrand size = 18, antiderivative size = 95 \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=-\frac {1}{2 a x^2}-\frac {\left (2 a c-f^2\right ) \arctan \left (\frac {f+2 c x^2}{\sqrt {4 a c-f^2}}\right )}{2 a^2 \sqrt {4 a c-f^2}}-\frac {f \log (x)}{a^2}+\frac {f \log \left (a+f x^2+c x^4\right )}{4 a^2} \] Output:
-1/2/a/x^2-1/2*(2*a*c-f^2)*arctan((2*c*x^2+f)/(4*a*c-f^2)^(1/2))/a^2/(4*a* c-f^2)^(1/2)-f*ln(x)/a^2+1/4*f*ln(c*x^4+f*x^2+a)/a^2
Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\frac {-\frac {2 a}{x^2}-4 f \log (x)+\frac {\left (-2 a c+f \left (f+\sqrt {-4 a c+f^2}\right )\right ) \log \left (f-\sqrt {-4 a c+f^2}+2 c x^2\right )}{\sqrt {-4 a c+f^2}}+\frac {\left (2 a c+f \left (-f+\sqrt {-4 a c+f^2}\right )\right ) \log \left (f+\sqrt {-4 a c+f^2}+2 c x^2\right )}{\sqrt {-4 a c+f^2}}}{4 a^2} \] Input:
Integrate[1/(x^3*(a + f*x^2 + c*x^4)),x]
Output:
((-2*a)/x^2 - 4*f*Log[x] + ((-2*a*c + f*(f + Sqrt[-4*a*c + f^2]))*Log[f - Sqrt[-4*a*c + f^2] + 2*c*x^2])/Sqrt[-4*a*c + f^2] + ((2*a*c + f*(-f + Sqrt [-4*a*c + f^2]))*Log[f + Sqrt[-4*a*c + f^2] + 2*c*x^2])/Sqrt[-4*a*c + f^2] )/(4*a^2)
Time = 0.52 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1434, 1145, 25, 1200, 2009}
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.
\(\displaystyle \int \frac {1}{x^3 \left (a+c x^4+f x^2\right )} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (c x^4+f x^2+a\right )}dx^2\) |
\(\Big \downarrow \) 1145 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {c x^2+f}{x^2 \left (c x^4+f x^2+a\right )}dx^2}{a}-\frac {1}{a x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {c x^2+f}{x^2 \left (c x^4+f x^2+a\right )}dx^2}{a}-\frac {1}{a x^2}\right )\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \left (\frac {f}{a x^2}+\frac {-f^2-c x^2 f+a c}{a \left (c x^4+f x^2+a\right )}\right )dx^2}{a}-\frac {1}{a x^2}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\frac {\frac {\left (2 a c-f^2\right ) \arctan \left (\frac {2 c x^2+f}{\sqrt {4 a c-f^2}}\right )}{a \sqrt {4 a c-f^2}}-\frac {f \log \left (a+c x^4+f x^2\right )}{2 a}+\frac {f \log \left (x^2\right )}{a}}{a}-\frac {1}{a x^2}\right )\) |
Input:
Int[1/(x^3*(a + f*x^2 + c*x^4)),x]
Output:
(-(1/(a*x^2)) - (((2*a*c - f^2)*ArcTan[(f + 2*c*x^2)/Sqrt[4*a*c - f^2]])/( a*Sqrt[4*a*c - f^2]) + (f*Log[x^2])/a - (f*Log[a + f*x^2 + c*x^4])/(2*a))/ a)/2
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp [1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[m, -1]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
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 = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {-\frac {f \ln \left (c \,x^{4}+f \,x^{2}+a \right )}{2}+\frac {2 \left (a c -\frac {f^{2}}{2}\right ) \arctan \left (\frac {2 c \,x^{2}+f}{\sqrt {4 a c -f^{2}}}\right )}{\sqrt {4 a c -f^{2}}}}{2 a^{2}}-\frac {1}{2 a \,x^{2}}-\frac {f \ln \left (x \right )}{a^{2}}\) | \(85\) |
risch | \(-\frac {1}{2 a \,x^{2}}-\frac {f \ln \left (x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4 a^{3} c -f^{2} a^{2}\right ) \textit {\_Z}^{2}+\left (-4 a c f +f^{3}\right ) \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (10 a^{3} c -3 f^{2} a^{2}\right ) \textit {\_R}^{2}-4 \textit {\_R} a c f +2 c^{2}\right ) x^{2}-a^{3} f \,\textit {\_R}^{2}+\left (c \,a^{2}-2 a \,f^{2}\right ) \textit {\_R} +2 c f \right )\right )}{2}\) | \(124\) |
Input:
int(1/x^3/(c*x^4+f*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/2/a^2*(-1/2*f*ln(c*x^4+f*x^2+a)+2*(a*c-1/2*f^2)/(4*a*c-f^2)^(1/2)*arcta n((2*c*x^2+f)/(4*a*c-f^2)^(1/2)))-1/2/a/x^2-f*ln(x)/a^2
Time = 0.10 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.12 \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\left [\frac {{\left (2 \, a c - f^{2}\right )} \sqrt {-4 \, a c + f^{2}} x^{2} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, c f x^{2} - 2 \, a c + f^{2} - {\left (2 \, c x^{2} + f\right )} \sqrt {-4 \, a c + f^{2}}}{c x^{4} + f x^{2} + a}\right ) + {\left (4 \, a c f - f^{3}\right )} x^{2} \log \left (c x^{4} + f x^{2} + a\right ) - 4 \, {\left (4 \, a c f - f^{3}\right )} x^{2} \log \left (x\right ) - 8 \, a^{2} c + 2 \, a f^{2}}{4 \, {\left (4 \, a^{3} c - a^{2} f^{2}\right )} x^{2}}, \frac {2 \, \sqrt {4 \, a c - f^{2}} {\left (2 \, a c - f^{2}\right )} x^{2} \arctan \left (-\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right ) + {\left (4 \, a c f - f^{3}\right )} x^{2} \log \left (c x^{4} + f x^{2} + a\right ) - 4 \, {\left (4 \, a c f - f^{3}\right )} x^{2} \log \left (x\right ) - 8 \, a^{2} c + 2 \, a f^{2}}{4 \, {\left (4 \, a^{3} c - a^{2} f^{2}\right )} x^{2}}\right ] \] Input:
integrate(1/x^3/(c*x^4+f*x^2+a),x, algorithm="fricas")
Output:
[1/4*((2*a*c - f^2)*sqrt(-4*a*c + f^2)*x^2*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 - f^3)*x^2*log(c*x^4 + f*x^2 + a) - 4*(4*a*c*f - f^3)*x^2*log(x) - 8*a ^2*c + 2*a*f^2)/((4*a^3*c - a^2*f^2)*x^2), 1/4*(2*sqrt(4*a*c - f^2)*(2*a*c - f^2)*x^2*arctan(-(2*c*x^2 + f)/sqrt(4*a*c - f^2)) + (4*a*c*f - f^3)*x^2 *log(c*x^4 + f*x^2 + a) - 4*(4*a*c*f - f^3)*x^2*log(x) - 8*a^2*c + 2*a*f^2 )/((4*a^3*c - a^2*f^2)*x^2)]
Timed out. \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(c*x**4+f*x**2+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^3/(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 = 0.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\frac {f \log \left (c x^{4} + f x^{2} + a\right )}{4 \, a^{2}} - \frac {f \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {{\left (2 \, a c - f^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + f}{\sqrt {4 \, a c - f^{2}}}\right )}{2 \, \sqrt {4 \, a c - f^{2}} a^{2}} + \frac {f x^{2} - a}{2 \, a^{2} x^{2}} \] Input:
integrate(1/x^3/(c*x^4+f*x^2+a),x, algorithm="giac")
Output:
1/4*f*log(c*x^4 + f*x^2 + a)/a^2 - 1/2*f*log(x^2)/a^2 - 1/2*(2*a*c - f^2)* arctan((2*c*x^2 + f)/sqrt(4*a*c - f^2))/(sqrt(4*a*c - f^2)*a^2) + 1/2*(f*x ^2 - a)/(a^2*x^2)
Time = 19.25 (sec) , antiderivative size = 2033, normalized size of antiderivative = 21.40 \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^3*(a + c*x^4 + f*x^2)),x)
Output:
(atan((16*a^6*x^2*(4*a*c - f^2)^(3/2)*(((3*f^4 + a^2*c^2 - 9*a*c*f^2)*(c^5 /a^3 + ((2*f^3 - 8*a*c*f)*((6*c^4*f)/a^2 + (((20*a^3*c^4 + 2*a^2*c^3*f^2)/ a^3 + ((2*f^3 - 8*a*c*f)*(40*a^4*c^3*f - 12*a^3*c^2*f^3))/(2*a^3*(16*a^3*c - 4*a^2*f^2)))*(2*f^3 - 8*a*c*f))/(2*(16*a^3*c - 4*a^2*f^2))))/(2*(16*a^3 *c - 4*a^2*f^2)) - (((((20*a^3*c^4 + 2*a^2*c^3*f^2)/a^3 + ((2*f^3 - 8*a*c* f)*(40*a^4*c^3*f - 12*a^3*c^2*f^3))/(2*a^3*(16*a^3*c - 4*a^2*f^2)))*(2*a*c - f^2))/(4*a^2*(4*a*c - f^2)^(1/2)) + ((2*f^3 - 8*a*c*f)*(40*a^4*c^3*f - 12*a^3*c^2*f^3)*(2*a*c - f^2))/(8*a^5*(4*a*c - f^2)^(1/2)*(16*a^3*c - 4*a^ 2*f^2)))*(2*a*c - f^2))/(4*a^2*(4*a*c - f^2)^(1/2)) - ((2*f^3 - 8*a*c*f)*( 40*a^4*c^3*f - 12*a^3*c^2*f^3)*(2*a*c - f^2)^2)/(32*a^7*(4*a*c - f^2)*(16* a^3*c - 4*a^2*f^2))))/(8*a^3*c^2*(a^2*c^2 - 6*f^4 + 24*a*c*f^2)) + ((((2*f ^3 - 8*a*c*f)*((((20*a^3*c^4 + 2*a^2*c^3*f^2)/a^3 + ((2*f^3 - 8*a*c*f)*(40 *a^4*c^3*f - 12*a^3*c^2*f^3))/(2*a^3*(16*a^3*c - 4*a^2*f^2)))*(2*a*c - f^2 ))/(4*a^2*(4*a*c - f^2)^(1/2)) + ((2*f^3 - 8*a*c*f)*(40*a^4*c^3*f - 12*a^3 *c^2*f^3)*(2*a*c - f^2))/(8*a^5*(4*a*c - f^2)^(1/2)*(16*a^3*c - 4*a^2*f^2) )))/(2*(16*a^3*c - 4*a^2*f^2)) + (((6*c^4*f)/a^2 + (((20*a^3*c^4 + 2*a^2*c ^3*f^2)/a^3 + ((2*f^3 - 8*a*c*f)*(40*a^4*c^3*f - 12*a^3*c^2*f^3))/(2*a^3*( 16*a^3*c - 4*a^2*f^2)))*(2*f^3 - 8*a*c*f))/(2*(16*a^3*c - 4*a^2*f^2)))*(2* a*c - f^2))/(4*a^2*(4*a*c - f^2)^(1/2)) - ((40*a^4*c^3*f - 12*a^3*c^2*f^3) *(2*a*c - f^2)^3)/(64*a^9*(4*a*c - f^2)^(3/2)))*(3*f^5 + 13*a^2*c^2*f -...
Time = 0.17 (sec) , antiderivative size = 409, normalized size of antiderivative = 4.31 \[ \int \frac {1}{x^3 \left (a+f x^2+c x^4\right )} \, dx=\frac {4 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c \,x^{2}-2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}-2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{2} x^{2}+4 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) a c \,x^{2}-2 \sqrt {2 \sqrt {c}\, \sqrt {a}+f}\, \sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, \mathit {atan} \left (\frac {\sqrt {2 \sqrt {c}\, \sqrt {a}-f}+2 \sqrt {c}\, x}{\sqrt {2 \sqrt {c}\, \sqrt {a}+f}}\right ) f^{2} x^{2}+4 \,\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c f \,x^{2}-\mathrm {log}\left (-\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{3} x^{2}+4 \,\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a c f \,x^{2}-\mathrm {log}\left (\sqrt {2 \sqrt {c}\, \sqrt {a}-f}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) f^{3} x^{2}-16 \,\mathrm {log}\left (x \right ) a c f \,x^{2}+4 \,\mathrm {log}\left (x \right ) f^{3} x^{2}-8 a^{2} c +2 a \,f^{2}}{4 a^{2} x^{2} \left (4 a c -f^{2}\right )} \] Input:
int(1/x^3/(c*x^4+f*x^2+a),x)
Output:
(4*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))*a*c*x**2 - 2*sqrt(2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sqr t(c)*sqrt(a) - f) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f**2*x**2 + 4*sqrt(2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sqr t(c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*a*c*x**2 - 2 *sqrt(2*sqrt(c)*sqrt(a) + f)*sqrt(2*sqrt(c)*sqrt(a) - f)*atan((sqrt(2*sqrt (c)*sqrt(a) - f) + 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + f))*f**2*x**2 + 4 *log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f*x**2 - log( - sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**3*x** 2 + 4*log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*a*c*f*x* *2 - log(sqrt(2*sqrt(c)*sqrt(a) - f)*x + sqrt(a) + sqrt(c)*x**2)*f**3*x**2 - 16*log(x)*a*c*f*x**2 + 4*log(x)*f**3*x**2 - 8*a**2*c + 2*a*f**2)/(4*a** 2*x**2*(4*a*c - f**2))