Optimal. Leaf size=34 \[ -\frac {(1-x) \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n}{1+n} \]
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Rubi [F]
time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n \, dx &=\int \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n \, dx\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 31, normalized size = 0.91 \begin {gather*} \frac {(-1+x) \left (1+x+x^2+x^3\right )^{-n} \left (1-x^4\right )^n}{1+n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 32, normalized size = 0.94
method | result | size |
gosper | \(\frac {\left (-1+x \right ) \left (-x^{4}+1\right )^{n} \left (x^{3}+x^{2}+x +1\right )^{-n}}{1+n}\) | \(32\) |
norman | \(\left (\frac {x \,{\mathrm e}^{n \ln \left (-x^{4}+1\right )}}{1+n}-\frac {{\mathrm e}^{n \ln \left (-x^{4}+1\right )}}{1+n}\right ) {\mathrm e}^{-n \ln \left (x^{3}+x^{2}+x +1\right )}\) | \(54\) |
risch | \(\frac {\left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )^{-n} {\mathrm e}^{\frac {n \left (i \pi \mathrm {csgn}\left (i \left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )\right )^{3}+i \mathrm {csgn}\left (i \left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (-1+x \right )\right ) \pi +i \mathrm {csgn}\left (i \left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{3}+x^{2}+x +1\right )\right ) \pi -i \mathrm {csgn}\left (i \left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )\right ) \mathrm {csgn}\left (i \left (-1+x \right )\right ) \mathrm {csgn}\left (i \left (x^{3}+x^{2}+x +1\right )\right ) \pi -2 i \pi \mathrm {csgn}\left (i \left (-1+x \right ) \left (x^{3}+x^{2}+x +1\right )\right )^{2}+2 i \pi +2 \ln \left (x^{3}+x^{2}+x +1\right )+2 \ln \left (-1+x \right )\right )}{2}}}{1+n}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 16, normalized size = 0.47 \begin {gather*} \frac {{\left (x - 1\right )} {\left (-x + 1\right )}^{n}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 31, normalized size = 0.91 \begin {gather*} \frac {{\left (-x^{4} + 1\right )}^{n} {\left (x - 1\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n} {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (24) = 48\).
time = 20.91, size = 73, normalized size = 2.15 \begin {gather*} \begin {cases} \frac {x \left (1 - x^{4}\right )^{n}}{n \left (x^{3} + x^{2} + x + 1\right )^{n} + \left (x^{3} + x^{2} + x + 1\right )^{n}} - \frac {\left (1 - x^{4}\right )^{n}}{n \left (x^{3} + x^{2} + x + 1\right )^{n} + \left (x^{3} + x^{2} + x + 1\right )^{n}} & \text {for}\: n \neq -1 \\- \log {\left (x - 1 \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 81 vs.
\(2 (31) = 62\).
time = 2.58, size = 81, normalized size = 2.38 \begin {gather*} \frac {\frac {x e^{\left (n \log \left (x^{3} + x^{2} + x + 1\right ) + n \log \left (-x + 1\right )\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n}} - \frac {e^{\left (n \log \left (x^{3} + x^{2} + x + 1\right ) + n \log \left (-x + 1\right )\right )}}{{\left (x^{3} + x^{2} + x + 1\right )}^{n}}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.45, size = 31, normalized size = 0.91 \begin {gather*} \frac {{\left (1-x^4\right )}^n\,\left (x-1\right )}{\left (n+1\right )\,{\left (x^3+x^2+x+1\right )}^n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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