Optimal. Leaf size=24 \[ \frac {x^4}{-3+\left (4+\frac {4}{\left (1+e^3\right ) x}\right )^2} \]
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Rubi [A] time = 0.28, antiderivative size = 35, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 4, integrand size = 168, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 1680, 12, 1590} \begin {gather*} \frac {\left (1+e^3\right )^2 x^6}{13 \left (1+e^3\right )^2 x^2+32 \left (1+e^3\right ) x+16} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 1590
Rule 1680
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+\left (169+169 e^{12}\right ) x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx\\ &=\int \frac {96 x^5+160 x^6+\left (52+52 e^{12}\right ) x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+\left (169+169 e^{12}\right ) x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {4 \left (16-13 \left (1+e^3\right ) x\right )^5 \left (72-104 \left (1+e^3\right ) x-169 \left (1+e^3\right )^2 x^2\right )}{28561 \left (1+e^3\right )^3 \left (48-169 \left (1+e^3\right )^2 x^2\right )^2} \, dx,x,\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}+x\right )\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {\left (16-13 \left (1+e^3\right ) x\right )^5 \left (72-104 \left (1+e^3\right ) x-169 \left (1+e^3\right )^2 x^2\right )}{\left (48-169 \left (1+e^3\right )^2 x^2\right )^2} \, dx,x,\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}+x\right )}{28561 \left (1+e^3\right )^3}\\ &=\frac {\left (1+e^3\right )^2 x^6}{16+32 \left (1+e^3\right ) x+13 \left (1+e^3\right )^2 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.05, size = 69, normalized size = 2.88 \begin {gather*} -\frac {7245824+14491648 \left (1+e^3\right ) x+5887232 \left (1+e^3\right )^2 x^2-371293 \left (1+e^3\right )^6 x^6}{371293 \left (1+e^3\right )^4 \left (16+32 \left (1+e^3\right ) x+13 \left (1+e^3\right )^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 164, normalized size = 6.83 \begin {gather*} \frac {371293 \, x^{6} e^{18} + 2227758 \, x^{6} e^{15} + 5569395 \, x^{6} e^{12} + 7425860 \, x^{6} e^{9} + 371293 \, x^{6} - 5887232 \, x^{2} + 13 \, {\left (428415 \, x^{6} - 452864 \, x^{2}\right )} e^{6} + 2 \, {\left (1113879 \, x^{6} - 5887232 \, x^{2} - 7245824 \, x\right )} e^{3} - 14491648 \, x - 7245824}{371293 \, {\left (13 \, x^{2} e^{18} + 13 \, x^{2} + 2 \, {\left (39 \, x^{2} + 16 \, x\right )} e^{15} + {\left (195 \, x^{2} + 160 \, x + 16\right )} e^{12} + 4 \, {\left (65 \, x^{2} + 80 \, x + 16\right )} e^{9} + {\left (195 \, x^{2} + 320 \, x + 96\right )} e^{6} + 2 \, {\left (39 \, x^{2} + 80 \, x + 32\right )} e^{3} + 32 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 48, normalized size = 2.00
method | result | size |
gosper | \(\frac {x^{6} \left ({\mathrm e}^{6}+2 \,{\mathrm e}^{3}+1\right )}{13 x^{2} {\mathrm e}^{6}+26 x^{2} {\mathrm e}^{3}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16}\) | \(48\) |
norman | \(\frac {x^{6} \left ({\mathrm e}^{6}+2 \,{\mathrm e}^{3}+1\right )}{13 x^{2} {\mathrm e}^{6}+26 x^{2} {\mathrm e}^{3}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16}\) | \(48\) |
risch | \(\frac {6591 \,{\mathrm e}^{6} x^{4}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {2197 x^{4} {\mathrm e}^{9}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {6591 \,{\mathrm e}^{3} x^{4}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {5408 x^{3} {\mathrm e}^{6}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {10816 \,{\mathrm e}^{3} x^{3}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {2197 x^{4}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {10608 x^{2} {\mathrm e}^{3}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {5408 x^{3}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {10608 x^{2}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {19456 x}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {-\frac {10444800 x}{169}-\frac {7245824}{169 \left ({\mathrm e}^{3}+1\right )}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2} \left (x^{2} {\mathrm e}^{6}+2 x^{2} {\mathrm e}^{3}+\frac {32 x \,{\mathrm e}^{3}}{13}+x^{2}+\frac {32 x}{13}+\frac {16}{13}\right )}\) | \(279\) |
default | \(\text {Expression too large to display}\) | \(2908\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 143, normalized size = 5.96 \begin {gather*} -\frac {4096 \, {\left (2550 \, x {\left (e^{3} + 1\right )} + 1769\right )}}{371293 \, {\left (13 \, x^{2} {\left (e^{18} + 6 \, e^{15} + 15 \, e^{12} + 20 \, e^{9} + 15 \, e^{6} + 6 \, e^{3} + 1\right )} + 32 \, x {\left (e^{15} + 5 \, e^{12} + 10 \, e^{9} + 10 \, e^{6} + 5 \, e^{3} + 1\right )} + 16 \, e^{12} + 64 \, e^{9} + 96 \, e^{6} + 64 \, e^{3} + 16\right )}} + \frac {2197 \, x^{4} {\left (e^{9} + 3 \, e^{6} + 3 \, e^{3} + 1\right )} - 5408 \, x^{3} {\left (e^{6} + 2 \, e^{3} + 1\right )} + 10608 \, x^{2} {\left (e^{3} + 1\right )} - 19456 \, x}{28561 \, {\left (e^{9} + 3 \, e^{6} + 3 \, e^{3} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.48, size = 133, normalized size = 5.54 \begin {gather*} \frac {816\,x^2}{2197\,{\left ({\mathrm {e}}^3+1\right )}^2}-\frac {32\,x^3}{169\,\left ({\mathrm {e}}^3+1\right )}+x\,\left (\frac {33792}{28561\,{\left ({\mathrm {e}}^3+1\right )}^3}-\frac {4\,\left (1024\,{\mathrm {e}}^3+1024\right )}{2197\,{\left ({\mathrm {e}}^3+1\right )}^4}\right )+\frac {x^4}{13}-\frac {\frac {10444800\,x}{13}+\frac {7245824}{13\,\left ({\mathrm {e}}^3+1\right )}}{\left (1856465\,{\mathrm {e}}^3+3712930\,{\mathrm {e}}^6+3712930\,{\mathrm {e}}^9+1856465\,{\mathrm {e}}^{12}+371293\,{\mathrm {e}}^{15}+371293\right )\,x^2+\left (3655808\,{\mathrm {e}}^3+5483712\,{\mathrm {e}}^6+3655808\,{\mathrm {e}}^9+913952\,{\mathrm {e}}^{12}+913952\right )\,x+1370928\,{\mathrm {e}}^3+1370928\,{\mathrm {e}}^6+456976\,{\mathrm {e}}^9+456976} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.28, size = 151, normalized size = 6.29 \begin {gather*} \frac {x^{4}}{13} - \frac {32 x^{3}}{169 + 169 e^{3}} + \frac {816 x^{2}}{2197 + 4394 e^{3} + 2197 e^{6}} - \frac {19456 x}{28561 + 85683 e^{3} + 85683 e^{6} + 28561 e^{9}} + \frac {x \left (- 10444800 e^{3} - 10444800\right ) - 7245824}{x^{2} \left (4826809 + 28960854 e^{3} + 72402135 e^{6} + 96536180 e^{9} + 72402135 e^{12} + 28960854 e^{15} + 4826809 e^{18}\right ) + x \left (11881376 + 59406880 e^{3} + 118813760 e^{6} + 118813760 e^{9} + 59406880 e^{12} + 11881376 e^{15}\right ) + 5940688 + 23762752 e^{3} + 35644128 e^{6} + 23762752 e^{9} + 5940688 e^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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