Integrand size = 14, antiderivative size = 174 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {3 b x}{4 c}+\frac {b \arctan \left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \arctan \left (c x^3\right )\right )-\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}} \]
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Time = 0.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4946, 327, 215, 648, 632, 210, 642, 209} \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{4} x^4 \left (a+b \arctan \left (c x^3\right )\right )+\frac {b \arctan \left (\sqrt [3]{c} x\right )}{4 c^{4/3}}-\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \arctan \left (2 \sqrt [3]{c} x+\sqrt {3}\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \log \left (c^{2/3} x^2-\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (c^{2/3} x^2+\sqrt {3} \sqrt [3]{c} x+1\right )}{16 c^{4/3}}-\frac {3 b x}{4 c} \]
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Rule 209
Rule 210
Rule 215
Rule 327
Rule 632
Rule 642
Rule 648
Rule 4946
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \left (a+b \arctan \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 \arctan \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 \arctan \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 \arctan \left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \arctan \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 \arctan \left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \arctan \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 \text {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 \text {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 \arctan \left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} x^4 \left (a+b \arctan \left (c x^3\right )\right )-\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \arctan \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*}
Time = 0.08 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.03 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=-\frac {3 b x}{4 c}+\frac {a x^4}{4}+\frac {b \arctan \left (\sqrt [3]{c} x\right )}{4 c^{4/3}}+\frac {1}{4} b x^4 \arctan \left (c x^3\right )-\frac {b \arctan \left (\sqrt {3}-2 \sqrt [3]{c} x\right )}{8 c^{4/3}}+\frac {b \arctan \left (\sqrt {3}+2 \sqrt [3]{c} x\right )}{8 c^{4/3}}-\frac {\sqrt {3} b \log \left (1-\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}}+\frac {\sqrt {3} b \log \left (1+\sqrt {3} \sqrt [3]{c} x+c^{2/3} x^2\right )}{16 c^{4/3}} \]
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Time = 0.80 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {a \,x^{4}}{4}+b \left (\frac {x^{4} \arctan \left (c \,x^{3}\right )}{4}-\frac {3 c \left (\frac {x}{c^{2}}-\frac {\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3}}{c^{2}}\right )}{4}\right )\) | \(155\) |
parts | \(\frac {a \,x^{4}}{4}+b \left (\frac {x^{4} \arctan \left (c \,x^{3}\right )}{4}-\frac {3 c \left (\frac {x}{c^{2}}-\frac {\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} x +\left (\frac {1}{c^{2}}\right )^{\frac {1}{3}}\right )}{12}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6}+\frac {\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {1}{c^{2}}\right )^{\frac {1}{6}}}\right )}{3}}{c^{2}}\right )}{4}\right )\) | \(155\) |
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Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (126) = 252\).
Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.55 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {4 \, b c x^{4} \arctan \left (c x^{3}\right ) + 4 \, a c x^{4} + {\left (\sqrt {-3} c + c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b x + \frac {1}{2} \, {\left (\sqrt {-3} c + c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}}\right ) - {\left (\sqrt {-3} c + c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b x - \frac {1}{2} \, {\left (\sqrt {-3} c + c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}}\right ) + {\left (\sqrt {-3} c - c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b x + \frac {1}{2} \, {\left (\sqrt {-3} c - c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}}\right ) - {\left (\sqrt {-3} c - c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b x - \frac {1}{2} \, {\left (\sqrt {-3} c - c\right )} \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}}\right ) + 2 \, c \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b x + c \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}}\right ) - 2 \, c \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}} \log \left (b x - c \left (-\frac {b^{6}}{c^{8}}\right )^{\frac {1}{6}}\right ) - 12 \, b x}{16 \, c} \]
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Time = 24.30 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.47 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\begin {cases} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {atan}{\left (c x^{3} \right )}}{4} - \frac {3 b x}{4 c} - \frac {3 b \sqrt [6]{- \frac {1}{c^{2}}} \log {\left (4 x^{2} - 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{16 c} + \frac {3 b \sqrt [6]{- \frac {1}{c^{2}}} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{16 c} + \frac {\sqrt {3} b \sqrt [6]{- \frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} - \frac {\sqrt {3}}{3} \right )}}{8 c} + \frac {\sqrt {3} b \sqrt [6]{- \frac {1}{c^{2}}} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3 \sqrt [6]{- \frac {1}{c^{2}}}} + \frac {\sqrt {3}}{3} \right )}}{8 c} + \frac {b \operatorname {atan}{\left (c x^{3} \right )}}{4 c^{2} \sqrt [3]{- \frac {1}{c^{2}}}} & \text {for}\: c \neq 0 \\\frac {a x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.85 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{4} \, a x^{4} + \frac {1}{16} \, {\left (4 \, x^{4} \arctan \left (c x^{3}\right ) + c {\left (\frac {\frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} + \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (c^{\frac {2}{3}} x^{2} - \sqrt {3} c^{\frac {1}{3}} x + 1\right )}{c^{\frac {1}{3}}} + \frac {4 \, \arctan \left (c^{\frac {1}{3}} x\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x + \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} + \frac {2 \, \arctan \left (\frac {2 \, c^{\frac {2}{3}} x - \sqrt {3} c^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}}{c^{2}} - \frac {12 \, x}{c^{2}}\right )}\right )} b \]
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Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.96 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{16} \, b c^{7} {\left (\frac {\sqrt {3} \log \left (x^{2} + \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (x^{2} - \frac {\sqrt {3} x}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x + \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} + \frac {2 \, \arctan \left ({\left (2 \, x - \frac {\sqrt {3}}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}} + \frac {4 \, \arctan \left (x {\left | c \right |}^{\frac {1}{3}}\right )}{c^{8} {\left | c \right |}^{\frac {1}{3}}}\right )} + \frac {b c x^{4} \arctan \left (c x^{3}\right ) + a c x^{4} - 3 \, b x}{4 \, c} \]
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Time = 1.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.66 \[ \int x^3 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {a\,x^4}{4}-\frac {b\,\left (\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )-\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\right )}{8\,c^{4/3}}+\frac {b\,x^4\,\mathrm {atan}\left (c\,x^3\right )}{4}-\frac {3\,b\,x}{4\,c}-\frac {\sqrt {3}\,b\,\left (\mathrm {atan}\left (\frac {c^{1/3}\,x\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )+\mathrm {atan}\left ({\left (-1\right )}^{2/3}\,c^{1/3}\,x\right )\right )\,1{}\mathrm {i}}{8\,c^{4/3}} \]
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