3.94.0 ∫ (125000+30000x+8e4x2x+2400x2 +64x3 +(150000+24000x+960x2) log(2)+(75000+7200x+96x2) log2(2)+ (20000+960x) log3(2)+(3000+48x) log4(2)+240 log5(2)+8 log6(2)+e3x2 (−8−600x−48x2 −240x log(2)−24x log2(2))+ e2x2(600+15048x+2400x2+96x3+(240+12000x+960x2) log(2)+(24+3600x+96x2) log2(2)+480x log3(2)+24x log4(2))+ ex2(−15000−127400x−30096x2 −2400x3 −64x4 +(−12000−150960x−24000x2 −960x3) log(2)+(−3600−75096x− 7200x2 −96x3) log2(2)+(−480−20000x−960x2) log3(2)+(−24−3000x−48x2) log4(2)−240x log5(2)−8x log6(2))) dx [9357]

Integrand size = 288, antiderivative size = 19 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=\left (-e^{x^2}+2 x+(5+\log (2))^2\right )^4 \]

Rubi [C] (veriﬁed)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.25 (sec) , antiderivative size = 370, normalized size of antiderivative = 19.47, number of steps used = 35, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {2240, 6, 2258, 2235, 2243} \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=-24 \sqrt {\pi } \text {erfi}(x)+12 \sqrt {\pi } \left (627+\log ^4(2)+20 \log ^3(2)+150 \log ^2(2)+500 \log (2)\right ) \text {erfi}(x)-12 \sqrt {\pi } (5+\log (2))^4 \text {erfi}(x)+16 x^4+800 x^3+32 x^3 \log ^2(2)+320 x^3 \log (2)+24 e^{2 x^2} x^2+15000 x^2+48 e^{x^2} x-8 e^{3 x^2} x-12 e^{2 x^2}+e^{4 x^2}+3600 x^2 \log ^2(2)-24 e^{x^2} x \left (627+\log ^4(2)+20 \log ^3(2)+150 \log ^2(2)+500 \log (2)\right )-4 e^{x^2} (5+\log (2))^2 \left (637+\log ^4(2)+20 \log ^3(2)+150 \log ^2(2)+500 \log (2)\right )+6 e^{2 x^2} \left (627+\log ^4(2)+20 \log ^3(2)+150 \log ^2(2)+500 \log (2)\right )-48 e^{x^2} x^2 (5+\log (2))^2+12000 x^2 \log (2)+24 e^{2 x^2} x (5+\log (2))^2+48 e^{x^2} (5+\log (2))^2-4 e^{3 x^2} (5+\log (2))^2-32 e^{x^2} x^3+6 (2 x+125)^2 \log ^4(2)+\frac {40}{3} (6 x+125)^2 \log ^3(2)+75000 x \log ^2(2)+8 x \left (15625+\log ^6(2)+30 \log ^5(2)\right )+150000 x \log (2) \]

Rubi steps \begin{align*} \text {integral}& = 15000 x^2+800 x^3+16 x^4+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )+8 \int e^{4 x^2} x \, dx+\log (2) \int \left (150000+24000 x+960 x^2\right ) \, dx+\log ^2(2) \int \left (75000+7200 x+96 x^2\right ) \, dx+\int e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right ) \, dx+\int e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right ) \, dx+\int e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right ) \, dx \\ & = e^{4 x^2}+15000 x^2+800 x^3+16 x^4+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )+\int e^{3 x^2} \left (-8-48 x^2+x (-600-240 \log (2))-24 x \log ^2(2)\right ) \, dx+\int e^{2 x^2} \left (600+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+24 x \log ^4(2)+x \left (15048+480 \log ^3(2)\right )\right ) \, dx+\int e^{x^2} \left (-15000-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-8 x \log ^6(2)+x \left (-127400-240 \log ^5(2)\right )\right ) \, dx \\ & = e^{4 x^2}+15000 x^2+800 x^3+16 x^4+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )+\int e^{3 x^2} \left (-8-48 x^2+x \left (-600-240 \log (2)-24 \log ^2(2)\right )\right ) \, dx+\int e^{2 x^2} \left (600+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+x \left (15048+480 \log ^3(2)+24 \log ^4(2)\right )\right ) \, dx+\int e^{x^2} \left (-15000-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)+x \left (-127400-240 \log ^5(2)-8 \log ^6(2)\right )\right ) \, dx \\ & = e^{4 x^2}+15000 x^2+800 x^3+16 x^4+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )+\int \left (-8 e^{3 x^2}-48 e^{3 x^2} x^2-24 e^{3 x^2} x (5+\log (2))^2\right ) \, dx+\int \left (96 e^{2 x^2} x^3+24 e^{2 x^2} (5+\log (2))^2+96 e^{2 x^2} x^2 (5+\log (2))^2+24 e^{2 x^2} x \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )\right ) \, dx+\int \left (-64 e^{x^2} x^4-96 e^{x^2} x^3 (5+\log (2))^2-24 e^{x^2} (5+\log (2))^4-48 e^{x^2} x^2 \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-8 e^{x^2} x (5+\log (2))^2 \left (637+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )\right ) \, dx \\ & = e^{4 x^2}+15000 x^2+800 x^3+16 x^4+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )-8 \int e^{3 x^2} \, dx-48 \int e^{3 x^2} x^2 \, dx-64 \int e^{x^2} x^4 \, dx+96 \int e^{2 x^2} x^3 \, dx+\left (24 (5+\log (2))^2\right ) \int e^{2 x^2} \, dx-\left (24 (5+\log (2))^2\right ) \int e^{3 x^2} x \, dx+\left (96 (5+\log (2))^2\right ) \int e^{2 x^2} x^2 \, dx-\left (96 (5+\log (2))^2\right ) \int e^{x^2} x^3 \, dx-\left (24 (5+\log (2))^4\right ) \int e^{x^2} \, dx+\left (24 \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )\right ) \int e^{2 x^2} x \, dx-\left (48 \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )\right ) \int e^{x^2} x^2 \, dx-\left (8 (5+\log (2))^2 \left (637+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )\right ) \int e^{x^2} x \, dx \\ & = e^{4 x^2}-8 e^{3 x^2} x+15000 x^2+24 e^{2 x^2} x^2+800 x^3-32 e^{x^2} x^3+16 x^4-4 \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} x\right )+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)-4 e^{3 x^2} (5+\log (2))^2+24 e^{2 x^2} x (5+\log (2))^2-48 e^{x^2} x^2 (5+\log (2))^2+6 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} x\right ) (5+\log (2))^2-12 \sqrt {\pi } \text {erfi}(x) (5+\log (2))^4+6 e^{2 x^2} \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-24 e^{x^2} x \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-4 e^{x^2} (5+\log (2))^2 \left (637+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )+8 \int e^{3 x^2} \, dx-48 \int e^{2 x^2} x \, dx+96 \int e^{x^2} x^2 \, dx-\left (24 (5+\log (2))^2\right ) \int e^{2 x^2} \, dx+\left (96 (5+\log (2))^2\right ) \int e^{x^2} x \, dx+\left (24 \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )\right ) \int e^{x^2} \, dx \\ & = -12 e^{2 x^2}+e^{4 x^2}+48 e^{x^2} x-8 e^{3 x^2} x+15000 x^2+24 e^{2 x^2} x^2+800 x^3-32 e^{x^2} x^3+16 x^4+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+48 e^{x^2} (5+\log (2))^2-4 e^{3 x^2} (5+\log (2))^2+24 e^{2 x^2} x (5+\log (2))^2-48 e^{x^2} x^2 (5+\log (2))^2-12 \sqrt {\pi } \text {erfi}(x) (5+\log (2))^4+6 e^{2 x^2} \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-24 e^{x^2} x \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )+12 \sqrt {\pi } \text {erfi}(x) \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-4 e^{x^2} (5+\log (2))^2 \left (637+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right )-48 \int e^{x^2} \, dx \\ & = -12 e^{2 x^2}+e^{4 x^2}+48 e^{x^2} x-8 e^{3 x^2} x+15000 x^2+24 e^{2 x^2} x^2+800 x^3-32 e^{x^2} x^3+16 x^4-24 \sqrt {\pi } \text {erfi}(x)+150000 x \log (2)+12000 x^2 \log (2)+320 x^3 \log (2)+75000 x \log ^2(2)+3600 x^2 \log ^2(2)+32 x^3 \log ^2(2)+\frac {40}{3} (125+6 x)^2 \log ^3(2)+6 (125+2 x)^2 \log ^4(2)+48 e^{x^2} (5+\log (2))^2-4 e^{3 x^2} (5+\log (2))^2+24 e^{2 x^2} x (5+\log (2))^2-48 e^{x^2} x^2 (5+\log (2))^2-12 \sqrt {\pi } \text {erfi}(x) (5+\log (2))^4+6 e^{2 x^2} \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-24 e^{x^2} x \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )+12 \sqrt {\pi } \text {erfi}(x) \left (627+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )-4 e^{x^2} (5+\log (2))^2 \left (637+500 \log (2)+150 \log ^2(2)+20 \log ^3(2)+\log ^4(2)\right )+8 x \left (15625+30 \log ^5(2)+\log ^6(2)\right ) \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=\left (-e^{x^2}+2 x+(5+\log (2))^2\right )^4 \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(284\) vs. \(2(18)=36\).

Time = 0.24 (sec) , antiderivative size = 285, normalized size of antiderivative = 15.00


method	result	size

norman	\({\mathrm e}^{4 x^{2}}+\left (-4 \ln \left (2\right )^{2}-40 \ln \left (2\right )-100\right ) {\mathrm e}^{3 x^{2}}+\left (32 \ln \left (2\right )^{2}+320 \ln \left (2\right )+800\right ) x^{3}+\left (6 \ln \left (2\right )^{4}+120 \ln \left (2\right )^{3}+900 \ln \left (2\right )^{2}+3000 \ln \left (2\right )+3750\right ) {\mathrm e}^{2 x^{2}}+\left (24 \ln \left (2\right )^{4}+480 \ln \left (2\right )^{3}+3600 \ln \left (2\right )^{2}+12000 \ln \left (2\right )+15000\right ) x^{2}+\left (-62500-4 \ln \left (2\right )^{6}-120 \ln \left (2\right )^{5}-1500 \ln \left (2\right )^{4}-75000 \ln \left (2\right )-37500 \ln \left (2\right )^{2}-10000 \ln \left (2\right )^{3}\right ) {\mathrm e}^{x^{2}}+\left (125000+8 \ln \left (2\right )^{6}+240 \ln \left (2\right )^{5}+3000 \ln \left (2\right )^{4}+150000 \ln \left (2\right )+75000 \ln \left (2\right )^{2}+20000 \ln \left (2\right )^{3}\right ) x +\left (-48 \ln \left (2\right )^{2}-480 \ln \left (2\right )-1200\right ) x^{2} {\mathrm e}^{x^{2}}+\left (24 \ln \left (2\right )^{2}+240 \ln \left (2\right )+600\right ) x \,{\mathrm e}^{2 x^{2}}+\left (-24 \ln \left (2\right )^{4}-480 \ln \left (2\right )^{3}-3600 \ln \left (2\right )^{2}-12000 \ln \left (2\right )-15000\right ) x \,{\mathrm e}^{x^{2}}+16 x^{4}+24 \,{\mathrm e}^{2 x^{2}} x^{2}-32 x^{3} {\mathrm e}^{x^{2}}-8 x \,{\mathrm e}^{3 x^{2}}\)	\(285\)
risch	\({\mathrm e}^{4 x^{2}}+\left (-4 \ln \left (2\right )^{2}-40 \ln \left (2\right )-8 x -100\right ) {\mathrm e}^{3 x^{2}}+\left (6 \ln \left (2\right )^{4}+120 \ln \left (2\right )^{3}+24 x \ln \left (2\right )^{2}+900 \ln \left (2\right )^{2}+240 x \ln \left (2\right )+24 x^{2}+3000 \ln \left (2\right )+600 x +3750\right ) {\mathrm e}^{2 x^{2}}+\left (-4 \ln \left (2\right )^{6}-120 \ln \left (2\right )^{5}-24 x \ln \left (2\right )^{4}-1500 \ln \left (2\right )^{4}-480 x \ln \left (2\right )^{3}-48 x^{2} \ln \left (2\right )^{2}-10000 \ln \left (2\right )^{3}-3600 x \ln \left (2\right )^{2}-480 x^{2} \ln \left (2\right )-32 x^{3}-37500 \ln \left (2\right )^{2}-12000 x \ln \left (2\right )-1200 x^{2}-75000 \ln \left (2\right )-15000 x -62500\right ) {\mathrm e}^{x^{2}}+8 x \ln \left (2\right )^{6}+240 x \ln \left (2\right )^{5}+24 x^{2} \ln \left (2\right )^{4}+3000 x \ln \left (2\right )^{4}+480 x^{2} \ln \left (2\right )^{3}+20000 x \ln \left (2\right )^{3}+32 x^{3} \ln \left (2\right )^{2}+3600 x^{2} \ln \left (2\right )^{2}+75000 x \ln \left (2\right )^{2}+320 x^{3} \ln \left (2\right )+12000 x^{2} \ln \left (2\right )+150000 x \ln \left (2\right )+625000 \ln \left (2\right )+16 x^{4}+800 x^{3}+15000 x^{2}+125000 x\)	\(289\)
default	\(125000 x -12000 x \ln \left (2\right ) {\mathrm e}^{x^{2}}+24 \,{\mathrm e}^{2 x^{2}} x^{2}-75000 \ln \left (2\right ) {\mathrm e}^{x^{2}}-8 x \,{\mathrm e}^{3 x^{2}}+8 x \ln \left (2\right )^{6}-32 x^{3} {\mathrm e}^{x^{2}}+600 x \,{\mathrm e}^{2 x^{2}}-15000 \,{\mathrm e}^{x^{2}} x -100 \,{\mathrm e}^{3 x^{2}}+{\mathrm e}^{4 x^{2}}+3750 \,{\mathrm e}^{2 x^{2}}+240 x \ln \left (2\right )^{5}+24 \ln \left (2\right )^{2} \left (\frac {4}{3} x^{3}+150 x^{2}+3125 x \right )-62500 \,{\mathrm e}^{x^{2}}+16 x^{4}+800 x^{3}+15000 x^{2}-1200 x^{2} {\mathrm e}^{x^{2}}-24 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{4} x -480 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{3} x -3600 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2} x -480 \,{\mathrm e}^{x^{2}} \ln \left (2\right ) x^{2}+24 \ln \left (2\right )^{2} {\mathrm e}^{2 x^{2}} x +240 \ln \left (2\right ) {\mathrm e}^{2 x^{2}} x -48 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2} x^{2}+160 \ln \left (2\right )^{3} \left (3 x^{2}+125 x \right )+24 \ln \left (2\right )^{4} \left (x^{2}+125 x \right )-37500 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2}-1500 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{4}-10000 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{3}-120 \ln \left (2\right )^{5} {\mathrm e}^{x^{2}}-4 \ln \left (2\right )^{6} {\mathrm e}^{x^{2}}+40 \ln \left (2\right ) \left (2 x +25\right )^{3}+900 \ln \left (2\right )^{2} {\mathrm e}^{2 x^{2}}-4 \ln \left (2\right )^{2} {\mathrm e}^{3 x^{2}}+3000 \ln \left (2\right ) {\mathrm e}^{2 x^{2}}-40 \ln \left (2\right ) {\mathrm e}^{3 x^{2}}+6 \ln \left (2\right )^{4} {\mathrm e}^{2 x^{2}}+120 \ln \left (2\right )^{3} {\mathrm e}^{2 x^{2}}\)	\(391\)
parallelrisch	\(-12000 x \ln \left (2\right ) {\mathrm e}^{x^{2}}+24 \,{\mathrm e}^{2 x^{2}} x^{2}-75000 \ln \left (2\right ) {\mathrm e}^{x^{2}}+24 x^{2} \ln \left (2\right )^{4}+480 x^{2} \ln \left (2\right )^{3}-8 x \,{\mathrm e}^{3 x^{2}}+20000 x \ln \left (2\right )^{3}+3000 x \ln \left (2\right )^{4}-32 x^{3} {\mathrm e}^{x^{2}}+32 x^{3} \ln \left (2\right )^{2}+600 x \,{\mathrm e}^{2 x^{2}}+75000 x \ln \left (2\right )^{2}-15000 \,{\mathrm e}^{x^{2}} x +3600 x^{2} \ln \left (2\right )^{2}+150000 x \ln \left (2\right )+12000 x^{2} \ln \left (2\right )+320 x^{3} \ln \left (2\right )-100 \,{\mathrm e}^{3 x^{2}}+{\mathrm e}^{4 x^{2}}+3750 \,{\mathrm e}^{2 x^{2}}-62500 \,{\mathrm e}^{x^{2}}+16 x^{4}+800 x^{3}+15000 x^{2}-1200 x^{2} {\mathrm e}^{x^{2}}-24 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{4} x -480 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{3} x -3600 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2} x -480 \,{\mathrm e}^{x^{2}} \ln \left (2\right ) x^{2}+24 \ln \left (2\right )^{2} {\mathrm e}^{2 x^{2}} x +240 \ln \left (2\right ) {\mathrm e}^{2 x^{2}} x -48 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2} x^{2}-37500 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2}-1500 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{4}-10000 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{3}-120 \ln \left (2\right )^{5} {\mathrm e}^{x^{2}}-4 \ln \left (2\right )^{6} {\mathrm e}^{x^{2}}+\left (125000+8 \ln \left (2\right )^{6}+240 \ln \left (2\right )^{5}\right ) x +900 \ln \left (2\right )^{2} {\mathrm e}^{2 x^{2}}-4 \ln \left (2\right )^{2} {\mathrm e}^{3 x^{2}}+3000 \ln \left (2\right ) {\mathrm e}^{2 x^{2}}-40 \ln \left (2\right ) {\mathrm e}^{3 x^{2}}+6 \ln \left (2\right )^{4} {\mathrm e}^{2 x^{2}}+120 \ln \left (2\right )^{3} {\mathrm e}^{2 x^{2}}\)	\(407\)
parts	\(125000 x -12000 x \ln \left (2\right ) {\mathrm e}^{x^{2}}+24 \,{\mathrm e}^{2 x^{2}} x^{2}-75000 \ln \left (2\right ) {\mathrm e}^{x^{2}}+24 x^{2} \ln \left (2\right )^{4}+480 x^{2} \ln \left (2\right )^{3}-8 x \,{\mathrm e}^{3 x^{2}}+20000 x \ln \left (2\right )^{3}+8 x \ln \left (2\right )^{6}+3000 x \ln \left (2\right )^{4}-32 x^{3} {\mathrm e}^{x^{2}}+32 x^{3} \ln \left (2\right )^{2}+600 x \,{\mathrm e}^{2 x^{2}}+75000 x \ln \left (2\right )^{2}-15000 \,{\mathrm e}^{x^{2}} x +3600 x^{2} \ln \left (2\right )^{2}+150000 x \ln \left (2\right )+12000 x^{2} \ln \left (2\right )+320 x^{3} \ln \left (2\right )-100 \,{\mathrm e}^{3 x^{2}}+{\mathrm e}^{4 x^{2}}+3750 \,{\mathrm e}^{2 x^{2}}+240 x \ln \left (2\right )^{5}-62500 \,{\mathrm e}^{x^{2}}+16 x^{4}+800 x^{3}+15000 x^{2}-1200 x^{2} {\mathrm e}^{x^{2}}-24 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{4} x -480 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{3} x -3600 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2} x -480 \,{\mathrm e}^{x^{2}} \ln \left (2\right ) x^{2}+24 \ln \left (2\right )^{2} {\mathrm e}^{2 x^{2}} x +240 \ln \left (2\right ) {\mathrm e}^{2 x^{2}} x -48 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2} x^{2}-37500 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{2}-1500 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{4}-10000 \,{\mathrm e}^{x^{2}} \ln \left (2\right )^{3}-120 \ln \left (2\right )^{5} {\mathrm e}^{x^{2}}-4 \ln \left (2\right )^{6} {\mathrm e}^{x^{2}}+900 \ln \left (2\right )^{2} {\mathrm e}^{2 x^{2}}-4 \ln \left (2\right )^{2} {\mathrm e}^{3 x^{2}}+3000 \ln \left (2\right ) {\mathrm e}^{2 x^{2}}-40 \ln \left (2\right ) {\mathrm e}^{3 x^{2}}+6 \ln \left (2\right )^{4} {\mathrm e}^{2 x^{2}}+120 \ln \left (2\right )^{3} {\mathrm e}^{2 x^{2}}\)	\(408\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 257 vs. \(2 (18) = 36\).

Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 13.53 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=8 \, x \log \left (2\right )^{6} + 240 \, x \log \left (2\right )^{5} + 24 \, {\left (x^{2} + 125 \, x\right )} \log \left (2\right )^{4} + 16 \, x^{4} + 160 \, {\left (3 \, x^{2} + 125 \, x\right )} \log \left (2\right )^{3} + 800 \, x^{3} + 8 \, {\left (4 \, x^{3} + 450 \, x^{2} + 9375 \, x\right )} \log \left (2\right )^{2} + 15000 \, x^{2} - 4 \, {\left (\log \left (2\right )^{2} + 2 \, x + 10 \, \log \left (2\right ) + 25\right )} e^{\left (3 \, x^{2}\right )} + 6 \, {\left (\log \left (2\right )^{4} + 2 \, {\left (2 \, x + 75\right )} \log \left (2\right )^{2} + 20 \, \log \left (2\right )^{3} + 4 \, x^{2} + 20 \, {\left (2 \, x + 25\right )} \log \left (2\right ) + 100 \, x + 625\right )} e^{\left (2 \, x^{2}\right )} - 4 \, {\left (\log \left (2\right )^{6} + 3 \, {\left (2 \, x + 125\right )} \log \left (2\right )^{4} + 30 \, \log \left (2\right )^{5} + 20 \, {\left (6 \, x + 125\right )} \log \left (2\right )^{3} + 8 \, x^{3} + 3 \, {\left (4 \, x^{2} + 300 \, x + 3125\right )} \log \left (2\right )^{2} + 300 \, x^{2} + 30 \, {\left (4 \, x^{2} + 100 \, x + 625\right )} \log \left (2\right ) + 3750 \, x + 15625\right )} e^{\left (x^{2}\right )} + 80 \, {\left (4 \, x^{3} + 150 \, x^{2} + 1875 \, x\right )} \log \left (2\right ) + 125000 \, x + e^{\left (4 \, x^{2}\right )} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 296, normalized size of antiderivative = 15.58 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=16 x^{4} + x^{3} \cdot \left (32 \log {\left (2 \right )}^{2} + 320 \log {\left (2 \right )} + 800\right ) + x^{2} \cdot \left (24 \log {\left (2 \right )}^{4} + 480 \log {\left (2 \right )}^{3} + 3600 \log {\left (2 \right )}^{2} + 12000 \log {\left (2 \right )} + 15000\right ) + x \left (8 \log {\left (2 \right )}^{6} + 240 \log {\left (2 \right )}^{5} + 3000 \log {\left (2 \right )}^{4} + 20000 \log {\left (2 \right )}^{3} + 75000 \log {\left (2 \right )}^{2} + 150000 \log {\left (2 \right )} + 125000\right ) + \left (- 8 x - 100 - 40 \log {\left (2 \right )} - 4 \log {\left (2 \right )}^{2}\right ) e^{3 x^{2}} + \left (24 x^{2} + 24 x \log {\left (2 \right )}^{2} + 240 x \log {\left (2 \right )} + 600 x + 6 \log {\left (2 \right )}^{4} + 120 \log {\left (2 \right )}^{3} + 900 \log {\left (2 \right )}^{2} + 3000 \log {\left (2 \right )} + 3750\right ) e^{2 x^{2}} + \left (- 32 x^{3} - 1200 x^{2} - 480 x^{2} \log {\left (2 \right )} - 48 x^{2} \log {\left (2 \right )}^{2} - 15000 x - 12000 x \log {\left (2 \right )} - 3600 x \log {\left (2 \right )}^{2} - 480 x \log {\left (2 \right )}^{3} - 24 x \log {\left (2 \right )}^{4} - 62500 - 75000 \log {\left (2 \right )} - 37500 \log {\left (2 \right )}^{2} - 10000 \log {\left (2 \right )}^{3} - 1500 \log {\left (2 \right )}^{4} - 120 \log {\left (2 \right )}^{5} - 4 \log {\left (2 \right )}^{6}\right ) e^{x^{2}} + e^{4 x^{2}} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (18) = 36\).

Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 13.63 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=8 \, x \log \left (2\right )^{6} + 240 \, x \log \left (2\right )^{5} + 24 \, {\left (x^{2} + 125 \, x\right )} \log \left (2\right )^{4} + 16 \, x^{4} + 160 \, {\left (3 \, x^{2} + 125 \, x\right )} \log \left (2\right )^{3} + 800 \, x^{3} + 8 \, {\left (4 \, x^{3} + 450 \, x^{2} + 9375 \, x\right )} \log \left (2\right )^{2} + 15000 \, x^{2} - 4 \, {\left (\log \left (2\right )^{2} + 2 \, x + 10 \, \log \left (2\right ) + 25\right )} e^{\left (3 \, x^{2}\right )} + 6 \, {\left (\log \left (2\right )^{4} + 20 \, \log \left (2\right )^{3} + 4 \, {\left (\log \left (2\right )^{2} + 10 \, \log \left (2\right ) + 25\right )} x + 4 \, x^{2} + 150 \, \log \left (2\right )^{2} + 500 \, \log \left (2\right ) + 625\right )} e^{\left (2 \, x^{2}\right )} - 4 \, {\left (\log \left (2\right )^{6} + 30 \, \log \left (2\right )^{5} + 375 \, \log \left (2\right )^{4} + 12 \, {\left (\log \left (2\right )^{2} + 10 \, \log \left (2\right ) + 25\right )} x^{2} + 8 \, x^{3} + 2500 \, \log \left (2\right )^{3} + 6 \, {\left (\log \left (2\right )^{4} + 20 \, \log \left (2\right )^{3} + 150 \, \log \left (2\right )^{2} + 500 \, \log \left (2\right ) + 625\right )} x + 9375 \, \log \left (2\right )^{2} + 18750 \, \log \left (2\right ) + 15625\right )} e^{\left (x^{2}\right )} + 80 \, {\left (4 \, x^{3} + 150 \, x^{2} + 1875 \, x\right )} \log \left (2\right ) + 125000 \, x + e^{\left (4 \, x^{2}\right )} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 271, normalized size of antiderivative = 14.26 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx=8 \, x \log \left (2\right )^{6} + 240 \, x \log \left (2\right )^{5} + 24 \, {\left (x^{2} + 125 \, x\right )} \log \left (2\right )^{4} + 16 \, x^{4} + 160 \, {\left (3 \, x^{2} + 125 \, x\right )} \log \left (2\right )^{3} + 800 \, x^{3} + 8 \, {\left (4 \, x^{3} + 450 \, x^{2} + 9375 \, x\right )} \log \left (2\right )^{2} + 15000 \, x^{2} - 4 \, {\left (\log \left (2\right )^{2} + 2 \, x + 10 \, \log \left (2\right ) + 25\right )} e^{\left (3 \, x^{2}\right )} + 6 \, {\left (\log \left (2\right )^{4} + 4 \, x \log \left (2\right )^{2} + 20 \, \log \left (2\right )^{3} + 4 \, x^{2} + 40 \, x \log \left (2\right ) + 150 \, \log \left (2\right )^{2} + 100 \, x + 500 \, \log \left (2\right ) + 625\right )} e^{\left (2 \, x^{2}\right )} - 4 \, {\left (\log \left (2\right )^{6} + 6 \, x \log \left (2\right )^{4} + 30 \, \log \left (2\right )^{5} + 12 \, x^{2} \log \left (2\right )^{2} + 120 \, x \log \left (2\right )^{3} + 375 \, \log \left (2\right )^{4} + 8 \, x^{3} + 120 \, x^{2} \log \left (2\right ) + 900 \, x \log \left (2\right )^{2} + 2500 \, \log \left (2\right )^{3} + 300 \, x^{2} + 3000 \, x \log \left (2\right ) + 9375 \, \log \left (2\right )^{2} + 3750 \, x + 18750 \, \log \left (2\right ) + 15625\right )} e^{\left (x^{2}\right )} + 80 \, {\left (4 \, x^{3} + 150 \, x^{2} + 1875 \, x\right )} \log \left (2\right ) + 125000 \, x + e^{\left (4 \, x^{2}\right )} \]

Mupad [B] (veriﬁcation not implemented)

Time = 0.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 12.16 \[ \int \left (125000+30000 x+8 e^{4 x^2} x+2400 x^2+64 x^3+\left (150000+24000 x+960 x^2\right ) \log (2)+\left (75000+7200 x+96 x^2\right ) \log ^2(2)+(20000+960 x) \log ^3(2)+(3000+48 x) \log ^4(2)+240 \log ^5(2)+8 \log ^6(2)+e^{3 x^2} \left (-8-600 x-48 x^2-240 x \log (2)-24 x \log ^2(2)\right )+e^{2 x^2} \left (600+15048 x+2400 x^2+96 x^3+\left (240+12000 x+960 x^2\right ) \log (2)+\left (24+3600 x+96 x^2\right ) \log ^2(2)+480 x \log ^3(2)+24 x \log ^4(2)\right )+e^{x^2} \left (-15000-127400 x-30096 x^2-2400 x^3-64 x^4+\left (-12000-150960 x-24000 x^2-960 x^3\right ) \log (2)+\left (-3600-75096 x-7200 x^2-96 x^3\right ) \log ^2(2)+\left (-480-20000 x-960 x^2\right ) \log ^3(2)+\left (-24-3000 x-48 x^2\right ) \log ^4(2)-240 x \log ^5(2)-8 x \log ^6(2)\right )\right ) \, dx={\mathrm {e}}^{4\,x^2}-4\,{\mathrm {e}}^{3\,x^2}\,{\left (\ln \left (2\right )+5\right )}^2+6\,{\mathrm {e}}^{2\,x^2}\,{\left (\ln \left (2\right )+5\right )}^4+32\,x^3\,{\left (\ln \left (2\right )+5\right )}^2-8\,x\,{\mathrm {e}}^{3\,x^2}-32\,x^3\,{\mathrm {e}}^{x^2}-{\mathrm {e}}^{x^2}\,\left (75000\,\ln \left (2\right )+37500\,{\ln \left (2\right )}^2+10000\,{\ln \left (2\right )}^3+1500\,{\ln \left (2\right )}^4+120\,{\ln \left (2\right )}^5+4\,{\ln \left (2\right )}^6+62500\right )+24\,x^2\,{\mathrm {e}}^{2\,x^2}+16\,x^4+x\,\left (150000\,\ln \left (2\right )+75000\,{\ln \left (2\right )}^2+20000\,{\ln \left (2\right )}^3+3000\,{\ln \left (2\right )}^4+240\,{\ln \left (2\right )}^5+8\,{\ln \left (2\right )}^6+125000\right )+x^2\,\left (12000\,\ln \left (2\right )+3600\,{\ln \left (2\right )}^2+480\,{\ln \left (2\right )}^3+24\,{\ln \left (2\right )}^4+15000\right )-24\,x\,{\mathrm {e}}^{x^2}\,{\left (\ln \left (2\right )+5\right )}^4+24\,x\,{\mathrm {e}}^{2\,x^2}\,{\left (\ln \left (2\right )+5\right )}^2-48\,x^2\,{\mathrm {e}}^{x^2}\,{\left (\ln \left (2\right )+5\right )}^2 \]

Optimal result