Optimal. Leaf size=215 \[ -\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.40, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {209, 634, 618, 204, 628, 205} \[ -\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 205
Rule 209
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{a+b x^6} \, dx &=\frac {\int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{3 a^{5/6}}+\frac {\int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{3 a^{5/6}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{3 a^{2/3}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a^{2/3}}-\frac {\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} a^{5/6} \sqrt [6]{b}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} a^{5/6} \sqrt [6]{b}}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 154, normalized size = 0.72 \[ \frac {-\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )}{12 a^{5/6} \sqrt [6]{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.84, size = 334, normalized size = 1.55 \[ \frac {1}{3} \, \sqrt {3} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a^{4} b x \left (-\frac {1}{a^{5} b}\right )^{\frac {5}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{2} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{3}} + a x \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x^{2}} a^{4} b \left (-\frac {1}{a^{5} b}\right )^{\frac {5}{6}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{3} \, \sqrt {3} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a^{4} b x \left (-\frac {1}{a^{5} b}\right )^{\frac {5}{6}} + \frac {2}{3} \, \sqrt {3} \sqrt {a^{2} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{3}} - a x \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x^{2}} a^{4} b \left (-\frac {1}{a^{5} b}\right )^{\frac {5}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{12} \, \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{3}} + a x \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x^{2}\right ) - \frac {1}{12} \, \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{3}} - a x \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x^{2}\right ) + \frac {1}{6} \, \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (a \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{6} \, \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (-a \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 190, normalized size = 0.88 \[ \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a b} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a b} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 159, normalized size = 0.74 \[ \frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.46, size = 184, normalized size = 0.86 \[ \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {\arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{3 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.12, size = 219, normalized size = 1.02 \[ -\frac {\mathrm {atanh}\left (\frac {b^{1/6}\,x}{{\left (-a\right )}^{1/6}}\right )}{3\,{\left (-a\right )}^{5/6}\,b^{1/6}}+\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}+\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}+\frac {\sqrt {3}\,b^{29/6}\,x}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}+\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,{\left (-a\right )}^{5/6}\,b^{1/6}}-\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}-\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}-\frac {\sqrt {3}\,b^{29/6}\,x}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}-\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,{\left (-a\right )}^{5/6}\,b^{1/6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.42, size = 20, normalized size = 0.09 \[ \operatorname {RootSum} {\left (46656 t^{6} a^{5} b + 1, \left (t \mapsto t \log {\left (6 t a + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________