Optimal. Leaf size=165 \[ -\frac {b \text {ArcTan}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} x^2 \left (a+b \text {ArcTan}\left (c x^3\right )\right )+\frac {b \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \text {ArcTan}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}} \]
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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 = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4946, 301,
648, 632, 210, 642, 209} \begin {gather*} \frac {1}{2} x^2 \left (a+b \text {ArcTan}\left (c x^3\right )\right )-\frac {b \text {ArcTan}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {b \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \text {ArcTan}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{4 c^{2/3}}-\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}}+\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{8 c^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 210
Rule 301
Rule 632
Rule 642
Rule 648
Rule 4946
Rubi steps
\begin {align*} \int x \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{2} (3 b c) \int \frac {x^4}{1+c^2 x^6} \, dx\\ &=\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {b \int \frac {1}{1+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac {b \int \frac {-\frac {1}{2}+\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}-\frac {b \int \frac {-\frac {1}{2}-\frac {1}{2} \sqrt {3} \sqrt [3]{c} x}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 \sqrt [3]{c}}\\ &=-\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {\left (\sqrt {3} b\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}{8 c^{2/3}}+\frac {\left (\sqrt {3} b\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}{8 c^{2/3}}-\frac {b \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}-\frac {b \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{8 \sqrt [3]{c}}\\ &=-\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}-\frac {b \text {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} c^{2/3}}+\frac {b \text {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} c^{2/3}}\\ &=-\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} x^2 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 170, normalized size = 1.03 \begin {gather*} \frac {a x^2}{2}-\frac {b \text {ArcTan}\left (\sqrt [3]{c} x\right )}{2 c^{2/3}}+\frac {1}{2} b x^2 \text {ArcTan}\left (c x^3\right )+\frac {b \text {ArcTan}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {b \text {ArcTan}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{4 c^{2/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{8 c^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 154, normalized size = 0.93
method | result | size |
default | \(\frac {a \,x^{2}}{2}+\frac {b \,x^{2} \arctan \left (c \,x^{3}\right )}{2}-\frac {b c \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{8}-\frac {b \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\frac {b c \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{8}-\frac {b \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{4 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\frac {b \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{2 c \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 137, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, a x^{2} + \frac {1}{8} \, {\left (4 \, x^{2} \arctan \left (c x^{3}\right ) + c {\left (\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {5}{3}}} - \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {5}{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 {5}{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 {5}{3}}}\right )}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 408 vs.
\(2 (119) = 238\).
time = 1.68, size = 408, normalized size = 2.47 \begin {gather*} \frac {1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) + \frac {1}{2} \, a x^{2} + \frac {1}{8} \, \sqrt {3} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \log \left (b^{10} x^{2} + \sqrt {3} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}} b^{5} c^{3} x + \left (\frac {b^{6}}{c^{4}}\right )^{\frac {2}{3}} b^{6} c^{2}\right ) - \frac {1}{8} \, \sqrt {3} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \log \left (b^{10} x^{2} - \sqrt {3} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}} b^{5} c^{3} x + \left (\frac {b^{6}}{c^{4}}\right )^{\frac {2}{3}} b^{6} c^{2}\right ) + \frac {1}{2} \, \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} b^{5} c x + \sqrt {3} b^{6} - 2 \, \sqrt {b^{10} x^{2} + \sqrt {3} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}} b^{5} c^{3} x + \left (\frac {b^{6}}{c^{4}}\right )^{\frac {2}{3}} b^{6} c^{2}} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} c}{b^{6}}\right ) + \frac {1}{2} \, \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} b^{5} c x - \sqrt {3} b^{6} - 2 \, \sqrt {b^{10} x^{2} - \sqrt {3} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {5}{6}} b^{5} c^{3} x + \left (\frac {b^{6}}{c^{4}}\right )^{\frac {2}{3}} b^{6} c^{2}} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} c}{b^{6}}\right ) + \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} \arctan \left (-\frac {\left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} b^{5} c x - \sqrt {b^{10} x^{2} + \left (\frac {b^{6}}{c^{4}}\right )^{\frac {2}{3}} b^{6} c^{2}} \left (\frac {b^{6}}{c^{4}}\right )^{\frac {1}{6}} c}{b^{6}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 17.09, size = 246, normalized size = 1.49 \begin {gather*} \begin {cases} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {atan}{\left (c x^{3} \right )}}{2} - \frac {3 b \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{8 c \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {3 b \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{8 c \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{4 c \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3} b \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{4 c \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{2 c^{2} \left (- \frac {1}{c^{2}}\right )^{\frac {2}{3}}} & \text {for}\: c \neq 0 \\\frac {a x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 157, normalized size = 0.95 \begin {gather*} \frac {1}{8} \, b c^{5} {\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 )}{c^{4} {\left | c \right |}^{\frac {5}{3}}} - \frac {\sqrt {3} {\left | c \right |}^{\frac {1}{3}} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{6}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{6}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{6}} - \frac {4 \, {\left | c \right |}^{\frac {1}{3}} \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{6}}\right )} + \frac {1}{2} \, b x^{2} \arctan \left (c x^{3}\right ) + \frac {1}{2} \, a x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.69, size = 113, normalized size = 0.68 \begin {gather*} \frac {a\,x^2}{2}+\frac {b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )+\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,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 )}{4\,c^{2/3}}+\frac {b\,x^2\,\mathrm {atan}\left (c\,x^3\right )}{2}-\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )\,1{}\mathrm {i}}{4\,c^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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