Integrand size = 13, antiderivative size = 133 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=-\frac {1}{2 a x^2}+\frac {\sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{4/3}}+\frac {\sqrt [3]{b} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{6 a^{4/3}}-\frac {\sqrt [3]{b} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{12 a^{4/3}} \]
-1/2/a/x^2+1/6*b^(1/3)*ln(a^(1/3)+b^(1/3)*x^2)/a^(4/3)-1/12*b^(1/3)*ln(a^( 2/3)-a^(1/3)*b^(1/3)*x^2+b^(2/3)*x^4)/a^(4/3)+1/6*b^(1/3)*arctan(1/3*(a^(1 /3)-2*b^(1/3)*x^2)/a^(1/3)*3^(1/2))/a^(4/3)*3^(1/2)
Time = 0.04 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.53 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=\frac {-6 \sqrt [3]{a}+2 \sqrt {3} \sqrt [3]{b} x^2 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt {3} \sqrt [3]{b} x^2 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt [3]{b} x^2 \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 a^{4/3} x^2} \]
(-6*a^(1/3) + 2*Sqrt[3]*b^(1/3)*x^2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6) ] + 2*Sqrt[3]*b^(1/3)*x^2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + 2*b^(1 /3)*x^2*Log[a^(1/3) + b^(1/3)*x^2] - b^(1/3)*x^2*Log[a^(1/3) - Sqrt[3]*a^( 1/6)*b^(1/6)*x + b^(1/3)*x^2] - b^(1/3)*x^2*Log[a^(1/3) + Sqrt[3]*a^(1/6)* b^(1/6)*x + b^(1/3)*x^2])/(12*a^(4/3)*x^2)
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {807, 847, 821, 16, 1142, 25, 27, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^6+a\right )}dx^2\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \int \frac {x^2}{b x^6+a}dx^2}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x^2+\sqrt [3]{a}}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\int \frac {\sqrt [3]{b} x^2+\sqrt [3]{a}}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x^2\right )}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {3 \int \frac {1}{-x^4-3}d\left (1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x^2}{b^{2/3} x^4-\sqrt [3]{a} \sqrt [3]{b} x^2+a^{2/3}}dx^2-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (-\frac {b \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x^2+b^{2/3} x^4\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x^2}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{a}-\frac {1}{a x^2}\right )\) |
(-(1/(a*x^2)) - (b*(-1/3*Log[a^(1/3) + b^(1/3)*x^2]/(a^(1/3)*b^(2/3)) + (- ((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x^2)/a^(1/3))/Sqrt[3]])/b^(1/3)) + Log[a^ (2/3) - a^(1/3)*b^(1/3)*x^2 + b^(2/3)*x^4]/(2*b^(1/3)))/(3*a^(1/3)*b^(1/3) )))/a)/2
3.14.29.3.1 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_)*(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_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.54 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(-\frac {1}{2 a \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{4} \textit {\_Z}^{3}-b \right )}{\sum }\textit {\_R} \ln \left (\left (-7 a^{4} \textit {\_R}^{3}+6 b \right ) x^{2}-a^{3} \textit {\_R}^{2}\right )\right )}{6}\) | \(55\) |
| default | \(-\frac {b \left (-\frac {\ln \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{4}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x^{2}+\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x^{2}}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{2 a}-\frac {1}{2 a \,x^{2}}\) | \(112\) |
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{4} - a x^{2} \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, x^{2} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6}{12 \, a x^{2}} \]
-1/12*(2*sqrt(3)*x^2*(b/a)^(1/3)*arctan(2/3*sqrt(3)*x^2*(b/a)^(1/3) - 1/3* sqrt(3)) + x^2*(b/a)^(1/3)*log(b*x^4 - a*x^2*(b/a)^(2/3) + a*(b/a)^(1/3)) - 2*x^2*(b/a)^(1/3)*log(b*x^2 + a*(b/a)^(2/3)) + 6)/(a*x^2)
Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=\operatorname {RootSum} {\left (216 t^{3} a^{4} - b, \left ( t \mapsto t \log {\left (\frac {36 t^{2} a^{3}}{b} + x^{2} \right )} \right )\right )} - \frac {1}{2 a x^{2}} \]
Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\log \left (x^{4} - x^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{6 \, a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {1}{2 \, a x^{2}} \]
-1/6*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - (a/b)^(1/3))/(a/b)^(1/3))/(a*(a/b )^(1/3)) - 1/12*log(x^4 - x^2*(a/b)^(1/3) + (a/b)^(2/3))/(a*(a/b)^(1/3)) + 1/6*log(x^2 + (a/b)^(1/3))/(a*(a/b)^(1/3)) - 1/2/(a*x^2)
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=\frac {b \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x^{2} - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{6 \, a^{2}} + \frac {\sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{6 \, a^{2} b} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{4} + x^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{12 \, a^{2} b} - \frac {1}{2 \, a x^{2}} \]
1/6*b*(-a/b)^(2/3)*log(abs(x^2 - (-a/b)^(1/3)))/a^2 + 1/6*sqrt(3)*(-a*b^2) ^(2/3)*arctan(1/3*sqrt(3)*(2*x^2 + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b) - 1 /12*(-a*b^2)^(2/3)*log(x^4 + x^2*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b) - 1/ 2/(a*x^2)
Time = 5.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x^3 \left (a+b x^6\right )} \, dx=\frac {b^{1/3}\,\ln \left (a^{1/3}+b^{1/3}\,x^2\right )}{6\,a^{4/3}}-\frac {1}{2\,a\,x^2}+\frac {b^{1/3}\,\ln \left (a^4\,b^6+a^{11/3}\,b^{19/3}\,x^2\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,a^{4/3}}-\frac {b^{1/3}\,\ln \left (a^4\,b^6-a^{11/3}\,b^{19/3}\,x^2\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{6\,a^{4/3}} \]