Optimal. Leaf size=165 \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5033, 209, 634, 618, 204, 628, 203} \[ -\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 209
Rule 618
Rule 628
Rule 634
Rule 5033
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}\left (c x^3\right )}{x^3} \, dx &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}+\frac {1}{2} (3 b c) \int \frac {1}{1+c^2 x^6} \, dx\\ &=-\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}+\frac {1}{2} (b c) \int \frac {1}{1+c^{2/3} x^2} \, dx+\frac {1}{2} (b c) \int \frac {1-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{2} (b c) \int \frac {1+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=\frac {1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} \left (\sqrt {3} b c^{2/3}\right ) \int \frac {-\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{8} \left (\sqrt {3} b c^{2/3}\right ) \int \frac {\sqrt {3} \sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{8} (b c) \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx+\frac {1}{8} (b c) \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx\\ &=\frac {1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {\left (b c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {\left (b c^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{4 \sqrt {3}}\\ &=\frac {1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {a+b \tan ^{-1}\left (c x^3\right )}{2 x^2}-\frac {1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )-\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )+\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 170, normalized size = 1.03 \[ -\frac {a}{2 x^2}-\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{8} \sqrt {3} b c^{2/3} \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )+\frac {1}{2} b c^{2/3} \tan ^{-1}\left (\sqrt [3]{c} x\right )-\frac {1}{4} b c^{2/3} \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )+\frac {1}{4} b c^{2/3} \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )-\frac {b \tan ^{-1}\left (c x^3\right )}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 505, normalized size = 3.06 \[ \frac {\sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \log \left (4 \, b^{2} c^{2} x^{2} + 4 \, \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} b c x + 4 \, \left (b^{6} c^{4}\right )^{\frac {1}{3}}\right ) - \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \log \left (4 \, b^{2} c^{2} x^{2} - 4 \, \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} b c x + 4 \, \left (b^{6} c^{4}\right )^{\frac {1}{3}}\right ) + \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \log \left (b^{2} c^{2} x^{2} + \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac {1}{3}}\right ) - \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \log \left (b^{2} c^{2} x^{2} - \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac {1}{3}}\right ) - 8 \, \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \arctan \left (-\frac {\sqrt {3} b^{6} c^{4} + 2 \, \left (b^{6} c^{4}\right )^{\frac {5}{6}} b c x - 2 \, \left (b^{6} c^{4}\right )^{\frac {5}{6}} \sqrt {b^{2} c^{2} x^{2} + \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac {1}{3}}}}{b^{6} c^{4}}\right ) - 8 \, \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \arctan \left (\frac {\sqrt {3} b^{6} c^{4} - 2 \, \left (b^{6} c^{4}\right )^{\frac {5}{6}} b c x + 2 \, \left (b^{6} c^{4}\right )^{\frac {5}{6}} \sqrt {b^{2} c^{2} x^{2} - \sqrt {3} \left (b^{6} c^{4}\right )^{\frac {1}{6}} b c x + \left (b^{6} c^{4}\right )^{\frac {1}{3}}}}{b^{6} c^{4}}\right ) - 16 \, \left (b^{6} c^{4}\right )^{\frac {1}{6}} x^{2} \arctan \left (-\frac {\left (b^{6} c^{4}\right )^{\frac {5}{6}} b c x - \left (b^{6} c^{4}\right )^{\frac {5}{6}} \sqrt {b^{2} c^{2} x^{2} + \left (b^{6} c^{4}\right )^{\frac {1}{3}}}}{b^{6} c^{4}}\right ) - 8 \, b \arctan \left (c x^{3}\right ) - 8 \, a}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 137, normalized size = 0.83 \[ \frac {1}{8} \, {\left (\frac {\sqrt {3} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}} + \frac {4 \, \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{{\left | c \right |}^{\frac {1}{3}}}\right )} b c - \frac {b \arctan \left (c x^{3}\right ) + a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 148, normalized size = 0.90 \[ -\frac {a}{2 x^{2}}-\frac {b \arctan \left (c \,x^{3}\right )}{2 x^{2}}+\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2}-\frac {b c \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{8}+\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{4}+\frac {b c \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{8}+\frac {b c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 137, normalized size = 0.83 \[ \frac {1}{8} \, {\left ({\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} + \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}\right )} c - \frac {4 \, \arctan \left (c x^{3}\right )}{x^{2}}\right )} b - \frac {a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 107, normalized size = 0.65 \[ -\frac {a}{2\,x^2}-\frac {b\,c^{2/3}\,\left (\frac {\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )}{2}-\frac {\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )}{2}+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{2}-\frac {b\,\mathrm {atan}\left (c\,x^3\right )}{2\,x^2}-\frac {\sqrt {3}\,b\,c^{2/3}\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )\right )\,1{}\mathrm {i}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 73.75, size = 301, normalized size = 1.82 \[ \begin {cases} - \frac {a}{2 x^{2}} - \frac {\left (-1\right )^{\frac {2}{3}} b \operatorname {atan}{\left (c x^{3} \right )}}{2 \sqrt [3]{\frac {1}{c^{2}}}} - \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{2 x^{2}} - \frac {3 \sqrt [6]{-1} b \log {\left (4 x^{2} - 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{8 c \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}}} + \frac {3 \sqrt [6]{-1} b \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{8 c \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}}} - \frac {\sqrt [6]{-1} \sqrt {3} b \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}}} - \frac {\sqrt [6]{-1} \sqrt {3} b \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}}} & \text {for}\: c \neq 0 \\- \frac {a}{2 x^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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