Optimal. Leaf size=174 \[ \frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{8 c^{4/3}}-\frac {3 b x}{4 c} \]
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Rubi [A] time = 0.32, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5033, 321, 209, 634, 618, 204, 628, 203} \[ \frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{8 c^{4/3}}-\frac {3 b x}{4 c} \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 209
Rule 321
Rule 618
Rule 628
Rule 634
Rule 5033
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}\left (c x^3\right )\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {1}{4} (3 b c) \int \frac {x^6}{1+c^2 x^6} \, dx\\ &=-\frac {3 b x}{4 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {(3 b) \int \frac {1}{1+c^2 x^6} \, dx}{4 c}\\ &=-\frac {3 b x}{4 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right )+\frac {b \int \frac {1}{1+c^{2/3} x^2} \, dx}{4 c}+\frac {b \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}{4 c}+\frac {b \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}{4 c}\\ &=-\frac {3 b x}{4 c}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \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}{16 c^{4/3}}+\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}{16 c^{4/3}}+\frac {b \int \frac {1}{1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{16 c}+\frac {b \int \frac {1}{1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2} \, dx}{16 c}\\ &=-\frac {3 b x}{4 c}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \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 )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 \sqrt {3} c^{4/3}}-\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{8 \sqrt {3} c^{4/3}}\\ &=-\frac {3 b x}{4 c}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^3\right )\right )-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \tan ^{-1}\left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 179, normalized size = 1.03 \[ \frac {a x^4}{4}-\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {b \tan ^{-1}\left (\sqrt [3]{c} x\right )}{4 c^{4/3}}-\frac {b \tan ^{-1}\left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \tan ^{-1}\left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{8 c^{4/3}}+\frac {1}{4} b x^4 \tan ^{-1}\left (c x^3\right )-\frac {3 b x}{4 c} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 399, normalized size = 2.29 \[ \frac {4 \, b c x^{4} \arctan \left (c x^{3}\right ) + 4 \, a c x^{4} + \sqrt {3} c \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b^{2} x^{2} + \sqrt {3} b c x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} + c^{2} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{3}}\right ) - \sqrt {3} c \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b^{2} x^{2} - \sqrt {3} b c x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} + c^{2} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{3}}\right ) - 4 \, c \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, b c^{7} x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {5}{6}} - 2 \, \sqrt {b^{2} x^{2} + \sqrt {3} b c x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} + c^{2} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{3}}} c^{7} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {5}{6}} + \sqrt {3} b^{6}}{b^{6}}\right ) - 4 \, c \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, b c^{7} x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {5}{6}} - 2 \, \sqrt {b^{2} x^{2} - \sqrt {3} b c x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} + c^{2} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{3}}} c^{7} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {5}{6}} - \sqrt {3} b^{6}}{b^{6}}\right ) - 8 \, c \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \arctan \left (-\frac {b c^{7} x \left (\frac {b^{6}}{c^{8}}\right )^{\frac {5}{6}} - \sqrt {b^{2} x^{2} + c^{2} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {1}{3}}} c^{7} \left (\frac {b^{6}}{c^{8}}\right )^{\frac {5}{6}}}{b^{6}}\right ) - 12 \, b x}{16 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.78, size = 167, normalized size = 0.96 \[ \frac {1}{16} \, b c^{7} {\left (\frac {\sqrt {3} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} + \frac {4 \, \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}}\right )} + \frac {b c x^{4} \arctan \left (c x^{3}\right ) + a c x^{4} - 3 \, b x}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 165, normalized size = 0.95 \[ \frac {x^{4} a}{4}+\frac {b \,x^{4} \arctan \left (c \,x^{3}\right )}{4}-\frac {3 b x}{4 c}+\frac {b \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{4 c}-\frac {b \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{16 c}+\frac {b \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{8 c}+\frac {b \sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{16 c}+\frac {b \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{8 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 148, normalized size = 0.85 \[ \frac {1}{4} \, a x^{4} + \frac {1}{16} \, {\left (4 \, x^{4} \arctan \left (c x^{3}\right ) + c {\left (\frac {\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} + \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}}{c^{2}} - \frac {12 \, x}{c^{2}}\right )}\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 114, normalized size = 0.66 \[ \frac {a\,x^4}{4}-\frac {b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{8\,c^{4/3}}+\frac {b\,x^4\,\mathrm {atan}\left (c\,x^3\right )}{4}-\frac {3\,b\,x}{4\,c}-\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )\right )\,1{}\mathrm {i}}{8\,c^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 58.10, size = 311, normalized size = 1.79 \[ \begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {atan}{\left (c x^{3} \right )}}{4} - \frac {3 b x}{4 c} - \frac {3 \sqrt [6]{-1} b \sqrt [6]{\frac {1}{c^{2}}} \log {\left (4 x^{2} - 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{16 c} + \frac {3 \sqrt [6]{-1} b \sqrt [6]{\frac {1}{c^{2}}} \log {\left (4 x^{2} + 4 \sqrt [6]{-1} x \sqrt [6]{\frac {1}{c^{2}}} + 4 \sqrt [3]{-1} \sqrt [3]{\frac {1}{c^{2}}} \right )}}{16 c} - \frac {\sqrt [6]{-1} \sqrt {3} b \sqrt [6]{\frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{8 c} - \frac {\sqrt [6]{-1} \sqrt {3} b \sqrt [6]{\frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {2 \left (-1\right )^{\frac {5}{6}} \sqrt {3} x}{3 \sqrt [6]{\frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{8 c} - \frac {\left (-1\right )^{\frac {2}{3}} b \operatorname {atan}{\left (c x^{3} \right )}}{4 c^{2} \sqrt [3]{\frac {1}{c^{2}}}} & \text {for}\: c \neq 0 \\\frac {a x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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