Optimal. Leaf size=52 \[ 3^n (-1+x) \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},-\frac {2 (-1+x)^2}{1+\sqrt {13}},\frac {2 (-1+x)^2}{-1+\sqrt {13}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(52)=104\).
time = 0.08, antiderivative size = 115, normalized size of antiderivative = 2.21, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1120, 1119,
440} \begin {gather*} (x-1) \left (\frac {2 (x-1)^2}{1-\sqrt {13}}+1\right )^{-n} \left (\frac {2 (x-1)^2}{1+\sqrt {13}}+1\right )^{-n} \left (-(x-1)^4-(x-1)^2+3\right )^n \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},-\frac {2 (x-1)^2}{1+\sqrt {13}},-\frac {2 (x-1)^2}{1-\sqrt {13}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 1119
Rule 1120
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \left (3-x^2-x^4\right )^n \, dx,x,-1+x\right )\\ &=\left (\left (1-\frac {2 (-1+x)^2}{-1-\sqrt {13}}\right )^{-n} \left (1-\frac {2 (-1+x)^2}{-1+\sqrt {13}}\right )^{-n} \left (3-(-1+x)^2-(-1+x)^4\right )^n\right ) \text {Subst}\left (\int \left (1-\frac {2 x^2}{-1-\sqrt {13}}\right )^n \left (1-\frac {2 x^2}{-1+\sqrt {13}}\right )^n \, dx,x,-1+x\right )\\ &=-\left (1+\frac {2 (1-x)^2}{1-\sqrt {13}}\right )^{-n} \left (1+\frac {2 (1-x)^2}{1+\sqrt {13}}\right )^{-n} \left (3-(1-x)^2-(1-x)^4\right )^n (1-x) \operatorname {AppellF1}\left (\frac {1}{2},-n,-n,\frac {3}{2},-\frac {2 (1-x)^2}{1+\sqrt {13}},-\frac {2 (1-x)^2}{1-\sqrt {13}}\right )\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 0.02, size = 0, normalized size = 0.00 \begin {gather*} \int \left (1+6 x-7 x^2+4 x^3-x^4\right )^n \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (-x^{4}+4 x^{3}-7 x^{2}+6 x +1\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.59, size = 24, normalized size = 0.46 \begin {gather*} {\rm integral}\left ({\left (-x^{4} + 4 \, x^{3} - 7 \, x^{2} + 6 \, x + 1\right )}^{n}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- x^{4} + 4 x^{3} - 7 x^{2} + 6 x + 1\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (-x^4+4\,x^3-7\,x^2+6\,x+1\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}
Warning: Unable to verify antiderivative.
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