3.102.0 ∫ 12x−24x2+8x3+3x4+e6(7−12x+3x2+2x3)+(26x−22x2−6x3+e6(14−10x−4x2)) log(3)+(14x+3x2+e6(7+2x)) log2(3) _________________ −2+4x+4x2−12x3+5x4+x5+e6(7x−13x2+5x3+x4)+(−4+4x+13x2−12x3−2x4+e6(14x−12x2−2x3)) log(3)+(−2+7x2+x3+e6(7x+x2)) log2(3) dx [10117]

Integrand size = 206, antiderivative size = 26 \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=\log \left (-2+x \left ((7+x) \left (e^6+x\right )+\frac {x}{-1+x-\log (3)}\right )\right ) \]

Rubi [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. $68$ vs. $2(26)=52$.

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62, number of steps used = 3, number of rules used = 2, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.010, Rules used = {2099, 1601} \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=\log \left (x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+2+\log (9)\right )-\log (-x+1+\log (3)) \]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{1-x+\log (3)}+\frac {-2+4 x^3+3 x^2 \left (6+e^6-\log (3)\right )-7 e^6 (1+\log (3))-2 x \left (6-e^6 (6-\log (3))+7 \log (3)\right )}{2+x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+\log (9)}\right ) \, dx \\ & = -\log (1-x+\log (3))+\int \frac {-2+4 x^3+3 x^2 \left (6+e^6-\log (3)\right )-7 e^6 (1+\log (3))-2 x \left (6-e^6 (6-\log (3))+7 \log (3)\right )}{2+x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+\log (9)} \, dx \\ & = -\log (1-x+\log (3))+\log \left (2+x^4+x^3 \left (6+e^6-\log (3)\right )-x^2 \left (6-e^6 (6-\log (3))+7 \log (3)\right )-x \left (2+7 e^6 (1+\log (3))\right )+\log (9)\right ) \\ \end{align*}

Mathematica [C] (veriﬁed)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.89 (sec) , antiderivative size = 1010, normalized size of antiderivative = 38.85 \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=-\log (-1+x-\log (3))+\frac {\text {RootSum}\left [1+\log ^2(3)+\log (9)+8 \text {$\#$1}+8 e^6 \text {$\#$1}+19 \log (3) \text {$\#$1}+9 e^6 \log (3) \text {$\#$1}+10 \log ^2(3) \text {$\#$1}+e^6 \log ^2(3) \text {$\#$1}+\log ^3(3) \text {$\#$1}+18 \text {$\#$1}^2+9 e^6 \text {$\#$1}^2+20 \log (3) \text {$\#$1}^2+3 \log ^2(3) \text {$\#$1}^2+e^6 \log (9) \text {$\#$1}^2+10 \text {$\#$1}^3+e^6 \text {$\#$1}^3+\log (27) \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {8 \log (-1+x-\log (3)-\text {$\#$1})+8 e^6 \log (-1+x-\log (3)-\text {$\#$1})+35 \log (3) \log (-1+x-\log (3)-\text {$\#$1})+25 e^6 \log (3) \log (-1+x-\log (3)-\text {$\#$1})+56 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1})+27 e^6 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1})+40 \log ^3(3) \log (-1+x-\log (3)-\text {$\#$1})+11 e^6 \log ^3(3) \log (-1+x-\log (3)-\text {$\#$1})+12 \log ^4(3) \log (-1+x-\log (3)-\text {$\#$1})+e^6 \log ^4(3) \log (-1+x-\log (3)-\text {$\#$1})+\log ^5(3) \log (-1+x-\log (3)-\text {$\#$1})+36 \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+18 e^6 \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+76 \log (3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+22 e^6 \log (3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+82 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+22 e^6 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+46 \log ^3(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+4 e^6 \log ^3(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+6 \log ^4(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+18 \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+9 e^6 \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+20 \log (3) \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+3 \log ^2(3) \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+e^6 \log ^2(9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}+30 \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+3 e^6 \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+29 \log (3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+36 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+3 e^6 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+9 \log ^3(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+20 \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+e^6 \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+6 \log (3) \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+e^6 \log (81) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^2+4 \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^3+4 \log ^2(3) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^3+4 \log (9) \log (-1+x-\log (3)-\text {$\#$1}) \text {$\#$1}^3}{8+8 e^6+19 \log (3)+9 e^6 \log (3)+10 \log ^2(3)+e^6 \log ^2(3)+\log ^3(3)+36 \text {$\#$1}+18 e^6 \text {$\#$1}+40 \log (3) \text {$\#$1}+6 \log ^2(3) \text {$\#$1}+e^6 \log (81) \text {$\#$1}+30 \text {$\#$1}^2+3 e^6 \text {$\#$1}^2+9 \log (3) \text {$\#$1}^2+4 \text {$\#$1}^3}\&\right ]}{1+\log ^2(3)+\log (9)} \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. $67$ vs. $2(27)=54$.

Time = 0.88 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62


method	result	size

risch	$-\ln \left (1+\ln \left (3\right )-x \right )+\ln \left (x^{4}+\left (-\ln \left (3\right )+{\mathrm e}^{6}+6\right ) x^{3}+\left (-{\mathrm e}^{6} \ln \left (3\right )-7 \ln \left (3\right )+6 \,{\mathrm e}^{6}-6\right ) x^{2}+\left (-7 \,{\mathrm e}^{6} \ln \left (3\right )-7 \,{\mathrm e}^{6}-2\right ) x +2 \ln \left (3\right )+2\right )$	$68$
default	$-\ln \left (x -\ln \left (3\right )-1\right )+\ln \left (-x^{2} {\mathrm e}^{6} \ln \left (3\right )+x^{3} {\mathrm e}^{6}-x^{3} \ln \left (3\right )+x^{4}-7 x \,{\mathrm e}^{6} \ln \left (3\right )+6 x^{2} {\mathrm e}^{6}-7 x^{2} \ln \left (3\right )+6 x^{3}-7 x \,{\mathrm e}^{6}-6 x^{2}+2 \ln \left (3\right )-2 x +2\right )$	$83$
parallelrisch	$-\ln \left (x -\ln \left (3\right )-1\right )+\ln \left (-x^{2} {\mathrm e}^{6} \ln \left (3\right )+x^{3} {\mathrm e}^{6}-x^{3} \ln \left (3\right )+x^{4}-7 x \,{\mathrm e}^{6} \ln \left (3\right )+6 x^{2} {\mathrm e}^{6}-7 x^{2} \ln \left (3\right )+6 x^{3}-7 x \,{\mathrm e}^{6}-6 x^{2}+2 \ln \left (3\right )-2 x +2\right )$	$93$
norman	$-\ln \left (1+\ln \left (3\right )-x \right )+\ln \left (x^{2} {\mathrm e}^{6} \ln \left (3\right )-x^{3} {\mathrm e}^{6}+x^{3} \ln \left (3\right )-x^{4}+7 x \,{\mathrm e}^{6} \ln \left (3\right )-6 x^{2} {\mathrm e}^{6}+7 x^{2} \ln \left (3\right )-6 x^{3}+7 x \,{\mathrm e}^{6}+6 x^{2}-2 \ln \left (3\right )+2 x -2\right )$	$94$

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. $2 (25) = 50$.

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=\log \left (x^{4} + 6 \, x^{3} - 6 \, x^{2} + {\left (x^{3} + 6 \, x^{2} - 7 \, x\right )} e^{6} - {\left (x^{3} + 7 \, x^{2} + {\left (x^{2} + 7 \, x\right )} e^{6} - 2\right )} \log \left (3\right ) - 2 \, x + 2\right ) - \log \left (x - \log \left (3\right ) - 1\right ) \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. $2 (20) = 40$.

Time = 128.69 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.73 \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=- \log {\left (x - \log {\left (3 \right )} - 1 \right )} + \log {\left (x^{4} + x^{3} \left (- \log {\left (3 \right )} + 6 + e^{6}\right ) + x^{2} \left (- e^{6} \log {\left (3 \right )} - 7 \log {\left (3 \right )} - 6 + 6 e^{6}\right ) + x \left (- 7 e^{6} \log {\left (3 \right )} - 7 e^{6} - 2\right ) + 2 + 2 \log {\left (3 \right )} \right )} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. $2 (25) = 50$.

Time = 0.22 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=\log \left (x^{4} + x^{3} {\left (e^{6} - \log \left (3\right ) + 6\right )} - {\left ({\left (e^{6} + 7\right )} \log \left (3\right ) - 6 \, e^{6} + 6\right )} x^{2} - {\left (7 \, e^{6} \log \left (3\right ) + 7 \, e^{6} + 2\right )} x + 2 \, \log \left (3\right ) + 2\right ) - \log \left (x - \log \left (3\right ) - 1\right ) \]

Giac [F(-1)]

Timed out. \[ \int \frac {12 x-24 x^2+8 x^3+3 x^4+e^6 \left (7-12 x+3 x^2+2 x^3\right )+\left (26 x-22 x^2-6 x^3+e^6 \left (14-10 x-4 x^2\right )\right ) \log (3)+\left (14 x+3 x^2+e^6 (7+2 x)\right ) \log ^2(3)}{-2+4 x+4 x^2-12 x^3+5 x^4+x^5+e^6 \left (7 x-13 x^2+5 x^3+x^4\right )+\left (-4+4 x+13 x^2-12 x^3-2 x^4+e^6 \left (14 x-12 x^2-2 x^3\right )\right ) \log (3)+\left (-2+7 x^2+x^3+e^6 \left (7 x+x^2\right )\right ) \log ^2(3)} \, dx=\text {Timed out} \]

Mupad [F(-1)]


method	result	size

risch	\(-\ln \left (1+\ln \left (3\right )-x \right )+\ln \left (x^{4}+\left (-\ln \left (3\right )+{\mathrm e}^{6}+6\right ) x^{3}+\left (-{\mathrm e}^{6} \ln \left (3\right )-7 \ln \left (3\right )+6 \,{\mathrm e}^{6}-6\right ) x^{2}+\left (-7 \,{\mathrm e}^{6} \ln \left (3\right )-7 \,{\mathrm e}^{6}-2\right ) x +2 \ln \left (3\right )+2\right )\)	\(68\)
default	\(-\ln \left (x -\ln \left (3\right )-1\right )+\ln \left (-x^{2} {\mathrm e}^{6} \ln \left (3\right )+x^{3} {\mathrm e}^{6}-x^{3} \ln \left (3\right )+x^{4}-7 x \,{\mathrm e}^{6} \ln \left (3\right )+6 x^{2} {\mathrm e}^{6}-7 x^{2} \ln \left (3\right )+6 x^{3}-7 x \,{\mathrm e}^{6}-6 x^{2}+2 \ln \left (3\right )-2 x +2\right )\)	\(83\)
parallelrisch	\(-\ln \left (x -\ln \left (3\right )-1\right )+\ln \left (-x^{2} {\mathrm e}^{6} \ln \left (3\right )+x^{3} {\mathrm e}^{6}-x^{3} \ln \left (3\right )+x^{4}-7 x \,{\mathrm e}^{6} \ln \left (3\right )+6 x^{2} {\mathrm e}^{6}-7 x^{2} \ln \left (3\right )+6 x^{3}-7 x \,{\mathrm e}^{6}-6 x^{2}+2 \ln \left (3\right )-2 x +2\right )\)	\(93\)
norman	\(-\ln \left (1+\ln \left (3\right )-x \right )+\ln \left (x^{2} {\mathrm e}^{6} \ln \left (3\right )-x^{3} {\mathrm e}^{6}+x^{3} \ln \left (3\right )-x^{4}+7 x \,{\mathrm e}^{6} \ln \left (3\right )-6 x^{2} {\mathrm e}^{6}+7 x^{2} \ln \left (3\right )-6 x^{3}+7 x \,{\mathrm e}^{6}+6 x^{2}-2 \ln \left (3\right )+2 x -2\right )\)	\(94\)

Optimal result