Integrand size = 7, antiderivative size = 16 \[ \int \left (d x^3\right )^n \, dx=\frac {x \left (d x^3\right )^n}{1+3 n} \]
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Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \[ \int \left (d x^3\right )^n \, dx=\frac {x \left (d x^3\right )^n}{3 n+1} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \left (x^{-3 n} \left (d x^3\right )^n\right ) \int x^{3 n} \, dx \\ & = \frac {x \left (d x^3\right )^n}{1+3 n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {x \left (d x^3\right )^n}{1+3 n} \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
gosper | \(\frac {x \left (x^{3} d \right )^{n}}{1+3 n}\) | \(17\) |
risch | \(\frac {x \left (x^{3} d \right )^{n}}{1+3 n}\) | \(17\) |
parallelrisch | \(\frac {x \left (x^{3} d \right )^{n}}{1+3 n}\) | \(17\) |
norman | \(\frac {x \,{\mathrm e}^{n \ln \left (x^{3} d \right )}}{1+3 n}\) | \(19\) |
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Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {\left (d x^{3}\right )^{n} x}{3 \, n + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \left (d x^3\right )^n \, dx=\begin {cases} \frac {x \left (d x^{3}\right )^{n}}{3 n + 1} & \text {for}\: n \neq - \frac {1}{3} \\\frac {x \log {\left (x \right )}}{\sqrt [3]{d x^{3}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06 \[ \int \left (d x^3\right )^n \, dx=\frac {d^{n} x x^{3 \, n}}{3 \, n + 1} \]
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Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {\left (d x^{3}\right )^{n} x}{3 \, n + 1} \]
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Time = 9.84 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (d x^3\right )^n \, dx=\frac {x\,{\left (d\,x^3\right )}^n}{3\,n+1} \]
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