Integrand size = 13, antiderivative size = 73 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}-\frac {\arctan \left (\frac {a-2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^7}-\frac {\log (a+x)}{3 a^7}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7} \]
-1/4/a^3/x^4+1/a^6/x-1/3*ln(a+x)/a^7+1/6*ln(a^2-a*x+x^2)/a^7-1/3*arctan(1/ 3*(a-2*x)/a*3^(1/2))/a^7*3^(1/2)
Time = 0.02 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {1}{4 a^3 x^4}+\frac {1}{a^6 x}+\frac {\arctan \left (\frac {-a+2 x}{\sqrt {3} a}\right )}{\sqrt {3} a^7}-\frac {\log (a+x)}{3 a^7}+\frac {\log \left (a^2-a x+x^2\right )}{6 a^7} \]
-1/4*1/(a^3*x^4) + 1/(a^6*x) + ArcTan[(-a + 2*x)/(Sqrt[3]*a)]/(Sqrt[3]*a^7 ) - Log[a + x]/(3*a^7) + Log[a^2 - a*x + x^2]/(6*a^7)
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {847, 847, 821, 16, 1142, 25, 1082, 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.
\(\displaystyle \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {\int \frac {1}{x^2 \left (a^3+x^3\right )}dx}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle -\frac {-\frac {\int \frac {x}{a^3+x^3}dx}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 821 |
\(\displaystyle -\frac {-\frac {\frac {\int \frac {a+x}{a^2-x a+x^2}dx}{3 a}-\frac {\int \frac {1}{a+x}dx}{3 a}}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -\frac {-\frac {\frac {\int \frac {a+x}{a^2-x a+x^2}dx}{3 a}-\frac {\log (a+x)}{3 a}}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle -\frac {-\frac {\frac {\frac {3}{2} a \int \frac {1}{a^2-x a+x^2}dx+\frac {1}{2} \int -\frac {a-2 x}{a^2-x a+x^2}dx}{3 a}-\frac {\log (a+x)}{3 a}}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {-\frac {\frac {\frac {3}{2} a \int \frac {1}{a^2-x a+x^2}dx-\frac {1}{2} \int \frac {a-2 x}{a^2-x a+x^2}dx}{3 a}-\frac {\log (a+x)}{3 a}}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {-\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 x}{a}\right )^2-3}d\left (1-\frac {2 x}{a}\right )-\frac {1}{2} \int \frac {a-2 x}{a^2-x a+x^2}dx}{3 a}-\frac {\log (a+x)}{3 a}}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {-\frac {\frac {-\frac {1}{2} \int \frac {a-2 x}{a^2-x a+x^2}dx-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{a}}{\sqrt {3}}\right )}{3 a}-\frac {\log (a+x)}{3 a}}{a^3}-\frac {1}{a^3 x}}{a^3}-\frac {1}{4 a^3 x^4}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {1}{4 a^3 x^4}-\frac {-\frac {1}{a^3 x}-\frac {\frac {\frac {1}{2} \log \left (a^2-a x+x^2\right )-\sqrt {3} \arctan \left (\frac {1-\frac {2 x}{a}}{\sqrt {3}}\right )}{3 a}-\frac {\log (a+x)}{3 a}}{a^3}}{a^3}\) |
-1/4*1/(a^3*x^4) - (-(1/(a^3*x)) - (-1/3*Log[a + x]/a + (-(Sqrt[3]*ArcTan[ (1 - (2*x)/a)/Sqrt[3]]) + Log[a^2 - a*x + x^2]/2)/(3*a))/a^3)/a^3
3.2.25.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2 *x^2), x], x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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_))/((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]
Time = 0.23 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {1}{4 a^{3} x^{4}}+\frac {1}{a^{6} x}+\frac {\frac {\ln \left (a^{2}-a x +x^{2}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (-a +2 x \right ) \sqrt {3}}{3 a}\right )}{3 a^{7}}-\frac {\ln \left (a +x \right )}{3 a^{7}}\) | \(66\) |
risch | \(\frac {\frac {x^{3}}{a^{6}}-\frac {1}{4 a^{3}}}{x^{4}}-\frac {\ln \left (a +x \right )}{3 a^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{14} \textit {\_Z}^{2}-a^{7} \textit {\_Z} +1\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{21}-3\right ) x +a^{15} \textit {\_R}^{2}\right )\right )}{3}\) | \(72\) |
-1/4/a^3/x^4+1/a^6/x+1/3/a^7*(1/2*ln(a^2-a*x+x^2)+3^(1/2)*arctan(1/3*(-a+2 *x)*3^(1/2)/a))-1/3*ln(a+x)/a^7
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {4 \, \sqrt {3} x^{4} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right ) + 2 \, x^{4} \log \left (a^{2} - a x + x^{2}\right ) - 4 \, x^{4} \log \left (a + x\right ) - 3 \, a^{4} + 12 \, a x^{3}}{12 \, a^{7} x^{4}} \]
1/12*(4*sqrt(3)*x^4*arctan(-1/3*sqrt(3)*(a - 2*x)/a) + 2*x^4*log(a^2 - a*x + x^2) - 4*x^4*log(a + x) - 3*a^4 + 12*a*x^3)/(a^7*x^4)
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {- a^{3} + 4 x^{3}}{4 a^{6} x^{4}} + \frac {- \frac {\log {\left (a + x \right )}}{3} + \left (\frac {1}{6} - \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (\frac {1}{6} - \frac {\sqrt {3} i}{6}\right )^{2} + x \right )} + \left (\frac {1}{6} + \frac {\sqrt {3} i}{6}\right ) \log {\left (9 a \left (\frac {1}{6} + \frac {\sqrt {3} i}{6}\right )^{2} + x \right )}}{a^{7}} \]
(-a**3 + 4*x**3)/(4*a**6*x**4) + (-log(a + x)/3 + (1/6 - sqrt(3)*I/6)*log( 9*a*(1/6 - sqrt(3)*I/6)**2 + x) + (1/6 + sqrt(3)*I/6)*log(9*a*(1/6 + sqrt( 3)*I/6)**2 + x))/a**7
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.90 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac {\log \left (a + x\right )}{3 \, a^{7}} - \frac {a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \]
1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a^7 + 1/6*log(a^2 - a*x + x^2 )/a^7 - 1/3*log(a + x)/a^7 - 1/4*(a^3 - 4*x^3)/(a^6*x^4)
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=\frac {\sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (a - 2 \, x\right )}}{3 \, a}\right )}{3 \, a^{7}} + \frac {\log \left (a^{2} - a x + x^{2}\right )}{6 \, a^{7}} - \frac {\log \left ({\left | a + x \right |}\right )}{3 \, a^{7}} - \frac {a^{3} - 4 \, x^{3}}{4 \, a^{6} x^{4}} \]
1/3*sqrt(3)*arctan(-1/3*sqrt(3)*(a - 2*x)/a)/a^7 + 1/6*log(a^2 - a*x + x^2 )/a^7 - 1/3*log(abs(a + x))/a^7 - 1/4*(a^3 - 4*x^3)/(a^6*x^4)
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^5 \left (a^3+x^3\right )} \, dx=-\frac {\frac {1}{4\,a^3}-\frac {x^3}{a^6}}{x^4}-\frac {\ln \left (a+x\right )}{3\,a^7}-\frac {\ln \left (\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^7}{4}+x\,a^6\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^7}+\frac {\ln \left (\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,a^7}{4}+x\,a^6\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^7} \]