Optimal. Leaf size=220 \[ \frac {\sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac {x}{a} \]
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Rubi [A] time = 0.44, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {193, 321, 209, 634, 618, 204, 628, 205} \[ \frac {\sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 193
Rule 204
Rule 205
Rule 209
Rule 321
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{a+\frac {b}{x^6}} \, dx &=\int \frac {x^6}{b+a x^6} \, dx\\ &=\frac {x}{a}-\frac {b \int \frac {1}{b+a x^6} \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt [6]{b} \int \frac {\sqrt [6]{b}-\frac {1}{2} \sqrt {3} \sqrt [6]{a} x}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{3 a}-\frac {\sqrt [6]{b} \int \frac {\sqrt [6]{b}+\frac {1}{2} \sqrt {3} \sqrt [6]{a} x}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}+\sqrt [3]{a} x^2} \, dx}{3 a}\\ &=\frac {x}{a}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{a} x}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{a} x}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{12 a}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{12 a}\\ &=\frac {x}{a}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )}{6 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )}{6 \sqrt {3} a^{7/6}}\\ &=\frac {x}{a}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt {3} a^{7/6}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 182, normalized size = 0.83 \[ \frac {\sqrt {3} \sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )-\sqrt {3} \sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )-4 \sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )+2 \sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )-2 \sqrt [6]{b} \tan ^{-1}\left (\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}+\sqrt {3}\right )+12 \sqrt [6]{a} x}{12 a^{7/6}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.30, size = 314, normalized size = 1.43 \[ -\frac {4 \, \sqrt {3} a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} \sqrt {a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} + a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}} a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - \sqrt {3} b}{3 \, b}\right ) + 4 \, \sqrt {3} a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} \sqrt {a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} - a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}} a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + \sqrt {3} b}{3 \, b}\right ) + a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} + a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) - a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} - a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) + 2 \, a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - 2 \, a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - 12 \, x}{12 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 180, normalized size = 0.82 \[ \frac {x}{a} - \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a^{2}} + \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 164, normalized size = 0.75 \[ \frac {x}{a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 a}-\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.77, size = 194, normalized size = 0.88 \[ -\frac {\frac {\sqrt {3} b^{\frac {1}{6}} \log \left (a^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{a^{\frac {1}{6}}} - \frac {\sqrt {3} b^{\frac {1}{6}} \log \left (a^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{a^{\frac {1}{6}}} + \frac {4 \, b^{\frac {1}{3}} \arctan \left (\frac {a^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, b^{\frac {1}{3}} \arctan \left (\frac {2 \, a^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, b^{\frac {1}{3}} \arctan \left (\frac {2 \, a^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{12 \, a} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 227, normalized size = 1.03 \[ \frac {x}{a}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{1/6}\,x\,1{}\mathrm {i}}{{\left (-b\right )}^{1/6}}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{25/6}\,x\,1{}\mathrm {i}}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}+\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}+\frac {\sqrt {3}\,{\left (-b\right )}^{25/6}\,x}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}+\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{25/6}\,x\,1{}\mathrm {i}}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}-\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}-\frac {\sqrt {3}\,{\left (-b\right )}^{25/6}\,x}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}-\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 22, normalized size = 0.10 \[ \operatorname {RootSum} {\left (46656 t^{6} a^{7} + b, \left (t \mapsto t \log {\left (- 6 t a + x \right )} \right )\right )} + \frac {x}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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