Integrand size = 14, antiderivative size = 36 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} x^3 \left (a+b \arctan \left (c x^3\right )\right )-\frac {b \log \left (1+c^2 x^6\right )}{6 c} \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4946, 266} \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} x^3 \left (a+b \arctan \left (c x^3\right )\right )-\frac {b \log \left (c^2 x^6+1\right )}{6 c} \]
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Rule 266
Rule 4946
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \left (a+b \arctan \left (c x^3\right )\right )-(b c) \int \frac {x^5}{1+c^2 x^6} \, dx \\ & = \frac {1}{3} x^3 \left (a+b \arctan \left (c x^3\right )\right )-\frac {b \log \left (1+c^2 x^6\right )}{6 c} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.14 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {a x^3}{3}+\frac {1}{3} b x^3 \arctan \left (c x^3\right )-\frac {b \log \left (1+c^2 x^6\right )}{6 c} \]
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Time = 0.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {x^{3} a}{3}+\frac {b \arctan \left (c \,x^{3}\right ) x^{3}}{3}-\frac {b \ln \left (c^{2} x^{6}+1\right )}{6 c}\) | \(36\) |
derivativedivides | \(\frac {a c \,x^{3}+b \left (c \,x^{3} \arctan \left (c \,x^{3}\right )-\frac {\ln \left (c^{2} x^{6}+1\right )}{2}\right )}{3 c}\) | \(39\) |
default | \(\frac {a c \,x^{3}+b \left (c \,x^{3} \arctan \left (c \,x^{3}\right )-\frac {\ln \left (c^{2} x^{6}+1\right )}{2}\right )}{3 c}\) | \(39\) |
parallelrisch | \(-\frac {-2 x^{3} \arctan \left (c \,x^{3}\right ) b c -2 a c \,x^{3}+b \ln \left (c^{2} x^{6}+1\right )}{6 c}\) | \(39\) |
risch | \(-\frac {i x^{3} b \ln \left (i c \,x^{3}+1\right )}{6}+\frac {i b \,x^{3} \ln \left (-i c \,x^{3}+1\right )}{6}+\frac {x^{3} a}{3}-\frac {b \ln \left (-c^{2} x^{6}-1\right )}{6 c}\) | \(59\) |
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.08 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {2 \, b c x^{3} \arctan \left (c x^{3}\right ) + 2 \, a c x^{3} - b \log \left (c^{2} x^{6} + 1\right )}{6 \, c} \]
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Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (29) = 58\).
Time = 20.58 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.83 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\begin {cases} \frac {a x^{3}}{3} + \frac {b x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{3} + \frac {b \sqrt {- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {b \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{3 c} - \frac {b \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{3 c} & \text {for}\: c \neq 0 \\\frac {a x^{3}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {1}{3} \, a x^{3} + \frac {{\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \]
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Time = 0.27 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.11 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {2 \, a c x^{3} + {\left (2 \, c x^{3} \arctan \left (c x^{3}\right ) - \log \left (c^{2} x^{6} + 1\right )\right )} b}{6 \, c} \]
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Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int x^2 \left (a+b \arctan \left (c x^3\right )\right ) \, dx=\frac {a\,x^3}{3}-\frac {b\,\ln \left (c^2\,x^6+1\right )}{6\,c}+\frac {b\,x^3\,\mathrm {atan}\left (c\,x^3\right )}{3} \]
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