3.59.23 \(\int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2))}{-729-4860 x-10800 x^2-8000 x^3+(1215+5400 x+6000 x^2) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx\)

Optimal. Leaf size=25 \[ e^{\left (-2+\frac {2+x}{16+5 \left (-5-4 x+\log ^2(2)\right )}\right )^2} \]

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Rubi [A]  time = 1.61, antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 3, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6688, 12, 6706} \begin {gather*} \exp \left (\frac {\left (41 x+10 \left (2-\log ^2(2)\right )\right )^2}{\left (20 x+9-5 \log ^2(2)\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((400 + 1640*x + 1681*x^2 + (-400 - 820*x)*Log[2]^2 + 100*Log[2]^4)/(81 + 360*x + 400*x^2 + (-90 - 200*
x)*Log[2]^2 + 25*Log[2]^4))*(1240 + 2542*x + (-420 + 410*x)*Log[2]^2 - 100*Log[2]^4))/(-729 - 4860*x - 10800*x
^2 - 8000*x^3 + (1215 + 5400*x + 6000*x^2)*Log[2]^2 + (-675 - 1500*x)*Log[2]^4 + 125*Log[2]^6),x]

[Out]

E^((41*x + 10*(2 - Log[2]^2))^2/(9 + 20*x - 5*Log[2]^2)^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (\frac {\left (20+41 x-10 \log ^2(2)\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}\right ) \left (20+41 x-10 \log ^2(2)\right ) \left (-31-5 \log ^2(2)\right )}{\left (9+20 x-5 \log ^2(2)\right )^3} \, dx\\ &=-\left (\left (2 \left (31+5 \log ^2(2)\right )\right ) \int \frac {\exp \left (\frac {\left (20+41 x-10 \log ^2(2)\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}\right ) \left (20+41 x-10 \log ^2(2)\right )}{\left (9+20 x-5 \log ^2(2)\right )^3} \, dx\right )\\ &=\exp \left (\frac {\left (41 x+10 \left (2-\log ^2(2)\right )\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 49, normalized size = 1.96 \begin {gather*} \frac {2 e^{\frac {\left (20+41 x-10 \log ^2(2)\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}} \left (31+5 \log ^2(2)\right )}{62+10 \log ^2(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((400 + 1640*x + 1681*x^2 + (-400 - 820*x)*Log[2]^2 + 100*Log[2]^4)/(81 + 360*x + 400*x^2 + (-90
- 200*x)*Log[2]^2 + 25*Log[2]^4))*(1240 + 2542*x + (-420 + 410*x)*Log[2]^2 - 100*Log[2]^4))/(-729 - 4860*x - 1
0800*x^2 - 8000*x^3 + (1215 + 5400*x + 6000*x^2)*Log[2]^2 + (-675 - 1500*x)*Log[2]^4 + 125*Log[2]^6),x]

[Out]

(2*E^((20 + 41*x - 10*Log[2]^2)^2/(9 + 20*x - 5*Log[2]^2)^2)*(31 + 5*Log[2]^2))/(62 + 10*Log[2]^2)

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fricas [B]  time = 0.65, size = 58, normalized size = 2.32 \begin {gather*} e^{\left (\frac {100 \, \log \relax (2)^{4} - 20 \, {\left (41 \, x + 20\right )} \log \relax (2)^{2} + 1681 \, x^{2} + 1640 \, x + 400}{25 \, \log \relax (2)^{4} - 10 \, {\left (20 \, x + 9\right )} \log \relax (2)^{2} + 400 \, x^{2} + 360 \, x + 81}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(2)^4+(410*x-420)*log(2)^2+2542*x+1240)*exp((100*log(2)^4+(-820*x-400)*log(2)^2+1681*x^2+16
40*x+400)/(25*log(2)^4+(-200*x-90)*log(2)^2+400*x^2+360*x+81))/(125*log(2)^6+(-1500*x-675)*log(2)^4+(6000*x^2+
5400*x+1215)*log(2)^2-8000*x^3-10800*x^2-4860*x-729),x, algorithm="fricas")

[Out]

e^((100*log(2)^4 - 20*(41*x + 20)*log(2)^2 + 1681*x^2 + 1640*x + 400)/(25*log(2)^4 - 10*(20*x + 9)*log(2)^2 +
400*x^2 + 360*x + 81))

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giac [B]  time = 0.32, size = 217, normalized size = 8.68 \begin {gather*} e^{\left (\frac {100 \, \log \relax (2)^{4}}{25 \, \log \relax (2)^{4} - 200 \, x \log \relax (2)^{2} + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 360 \, x + 81} - \frac {820 \, x \log \relax (2)^{2}}{25 \, \log \relax (2)^{4} - 200 \, x \log \relax (2)^{2} + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 360 \, x + 81} + \frac {1681 \, x^{2}}{25 \, \log \relax (2)^{4} - 200 \, x \log \relax (2)^{2} + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 360 \, x + 81} - \frac {400 \, \log \relax (2)^{2}}{25 \, \log \relax (2)^{4} - 200 \, x \log \relax (2)^{2} + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 360 \, x + 81} + \frac {1640 \, x}{25 \, \log \relax (2)^{4} - 200 \, x \log \relax (2)^{2} + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 360 \, x + 81} + \frac {400}{25 \, \log \relax (2)^{4} - 200 \, x \log \relax (2)^{2} + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 360 \, x + 81}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(2)^4+(410*x-420)*log(2)^2+2542*x+1240)*exp((100*log(2)^4+(-820*x-400)*log(2)^2+1681*x^2+16
40*x+400)/(25*log(2)^4+(-200*x-90)*log(2)^2+400*x^2+360*x+81))/(125*log(2)^6+(-1500*x-675)*log(2)^4+(6000*x^2+
5400*x+1215)*log(2)^2-8000*x^3-10800*x^2-4860*x-729),x, algorithm="giac")

[Out]

e^(100*log(2)^4/(25*log(2)^4 - 200*x*log(2)^2 + 400*x^2 - 90*log(2)^2 + 360*x + 81) - 820*x*log(2)^2/(25*log(2
)^4 - 200*x*log(2)^2 + 400*x^2 - 90*log(2)^2 + 360*x + 81) + 1681*x^2/(25*log(2)^4 - 200*x*log(2)^2 + 400*x^2
- 90*log(2)^2 + 360*x + 81) - 400*log(2)^2/(25*log(2)^4 - 200*x*log(2)^2 + 400*x^2 - 90*log(2)^2 + 360*x + 81)
+ 1640*x/(25*log(2)^4 - 200*x*log(2)^2 + 400*x^2 - 90*log(2)^2 + 360*x + 81) + 400/(25*log(2)^4 - 200*x*log(2
)^2 + 400*x^2 - 90*log(2)^2 + 360*x + 81))

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maple [A]  time = 0.25, size = 29, normalized size = 1.16

method	result	size
risch	\({\mathrm e}^{\frac {\left (10 \ln \relax (2)^{2}-41 x -20\right )^{2}}{\left (5 \ln \relax (2)^{2}-20 x -9\right )^{2}}}\)	\(29\)
gosper	\({\mathrm e}^{\frac {100 \ln \relax (2)^{4}-820 x \ln \relax (2)^{2}-400 \ln \relax (2)^{2}+1681 x^{2}+1640 x +400}{25 \ln \relax (2)^{4}-200 x \ln \relax (2)^{2}-90 \ln \relax (2)^{2}+400 x^{2}+360 x +81}}\)	\(63\)
norman	\(\frac {\left (25 \ln \relax (2)^{4}-90 \ln \relax (2)^{2}+81\right ) {\mathrm e}^{\frac {100 \ln \relax (2)^{4}+\left (-820 x -400\right ) \ln \relax (2)^{2}+1681 x^{2}+1640 x +400}{25 \ln \relax (2)^{4}+\left (-200 x -90\right ) \ln \relax (2)^{2}+400 x^{2}+360 x +81}}+\left (-200 \ln \relax (2)^{2}+360\right ) x \,{\mathrm e}^{\frac {100 \ln \relax (2)^{4}+\left (-820 x -400\right ) \ln \relax (2)^{2}+1681 x^{2}+1640 x +400}{25 \ln \relax (2)^{4}+\left (-200 x -90\right ) \ln \relax (2)^{2}+400 x^{2}+360 x +81}}+400 x^{2} {\mathrm e}^{\frac {100 \ln \relax (2)^{4}+\left (-820 x -400\right ) \ln \relax (2)^{2}+1681 x^{2}+1640 x +400}{25 \ln \relax (2)^{4}+\left (-200 x -90\right ) \ln \relax (2)^{2}+400 x^{2}+360 x +81}}}{\left (5 \ln \relax (2)^{2}-20 x -9\right )^{2}}\)	\(214\)
derivativedivides	\(-\frac {1271 i \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{20 \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {961 \,{\mathrm e}^{\frac {1681}{400}+\frac {\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}}{\left (-5 \ln \relax (2)^{2}+20 x +9\right )^{2}}+\frac {\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}}{-5 \ln \relax (2)^{2}+20 x +9}}}{400 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}+\frac {118203 i \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right ) \ln \relax (2)^{2}}{1600 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {1221431 i \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{8000 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}-\frac {41 i \ln \relax (2)^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{4 \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {31 \ln \relax (2)^{2} {\mathrm e}^{\frac {1681}{400}+\frac {\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}}{\left (-5 \ln \relax (2)^{2}+20 x +9\right )^{2}}+\frac {\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}}{-5 \ln \relax (2)^{2}+20 x +9}}}{40 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}+\frac {3813 i \ln \relax (2)^{4} \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{320 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {\ln \relax (2)^{4} {\mathrm e}^{\frac {1681}{400}+\frac {\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}}{\left (-5 \ln \relax (2)^{2}+20 x +9\right )^{2}}+\frac {\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}}{-5 \ln \relax (2)^{2}+20 x +9}}}{\ln \relax (2)^{4}+\frac {62 \ln \relax (2)^{2}}{5}+\frac {961}{25}}+\frac {41 i \ln \relax (2)^{6} \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{64 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\)	\(987\)
default	\(-\frac {1271 i \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{20 \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {961 \,{\mathrm e}^{\frac {1681}{400}+\frac {\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}}{\left (-5 \ln \relax (2)^{2}+20 x +9\right )^{2}}+\frac {\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}}{-5 \ln \relax (2)^{2}+20 x +9}}}{400 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}+\frac {118203 i \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right ) \ln \relax (2)^{2}}{1600 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {1221431 i \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{8000 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}-\frac {41 i \ln \relax (2)^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{4 \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {31 \ln \relax (2)^{2} {\mathrm e}^{\frac {1681}{400}+\frac {\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}}{\left (-5 \ln \relax (2)^{2}+20 x +9\right )^{2}}+\frac {\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}}{-5 \ln \relax (2)^{2}+20 x +9}}}{40 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}+\frac {3813 i \ln \relax (2)^{4} \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{320 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}+\frac {\ln \relax (2)^{4} {\mathrm e}^{\frac {1681}{400}+\frac {\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}}{\left (-5 \ln \relax (2)^{2}+20 x +9\right )^{2}}+\frac {\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}}{-5 \ln \relax (2)^{2}+20 x +9}}}{\ln \relax (2)^{4}+\frac {62 \ln \relax (2)^{2}}{5}+\frac {961}{25}}+\frac {41 i \ln \relax (2)^{6} \sqrt {\pi }\, {\mathrm e}^{\frac {1681}{400}-\frac {\left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )^{2}}{4 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right )}} \erf \left (\frac {i \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}{-100 \ln \relax (2)^{2}+400 x +180}+\frac {10 i \left (\frac {41 \ln \relax (2)^{2}}{40}+\frac {1271}{200}\right )}{\sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\right )}{64 \left (\frac {\ln \relax (2)^{4}}{16}+\frac {31 \ln \relax (2)^{2}}{40}+\frac {961}{400}\right ) \sqrt {25 \ln \relax (2)^{4}+310 \ln \relax (2)^{2}+961}}\)	\(987\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-100*ln(2)^4+(410*x-420)*ln(2)^2+2542*x+1240)*exp((100*ln(2)^4+(-820*x-400)*ln(2)^2+1681*x^2+1640*x+400)/
(25*ln(2)^4+(-200*x-90)*ln(2)^2+400*x^2+360*x+81))/(125*ln(2)^6+(-1500*x-675)*ln(2)^4+(6000*x^2+5400*x+1215)*l
n(2)^2-8000*x^3-10800*x^2-4860*x-729),x,method=_RETURNVERBOSE)

[Out]

exp((10*ln(2)^2-41*x-20)^2/(5*ln(2)^2-20*x-9)^2)

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maxima [B]  time = 53.07, size = 147, normalized size = 5.88 \begin {gather*} e^{\left (\frac {\log \relax (2)^{4}}{16 \, {\left (25 \, \log \relax (2)^{4} - 40 \, {\left (5 \, \log \relax (2)^{2} - 9\right )} x + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 81\right )}} + \frac {31 \, \log \relax (2)^{2}}{40 \, {\left (25 \, \log \relax (2)^{4} - 40 \, {\left (5 \, \log \relax (2)^{2} - 9\right )} x + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 81\right )}} - \frac {41 \, \log \relax (2)^{2}}{40 \, {\left (5 \, \log \relax (2)^{2} - 20 \, x - 9\right )}} + \frac {961}{400 \, {\left (25 \, \log \relax (2)^{4} - 40 \, {\left (5 \, \log \relax (2)^{2} - 9\right )} x + 400 \, x^{2} - 90 \, \log \relax (2)^{2} + 81\right )}} - \frac {1271}{200 \, {\left (5 \, \log \relax (2)^{2} - 20 \, x - 9\right )}} + \frac {1681}{400}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*log(2)^4+(410*x-420)*log(2)^2+2542*x+1240)*exp((100*log(2)^4+(-820*x-400)*log(2)^2+1681*x^2+16
40*x+400)/(25*log(2)^4+(-200*x-90)*log(2)^2+400*x^2+360*x+81))/(125*log(2)^6+(-1500*x-675)*log(2)^4+(6000*x^2+
5400*x+1215)*log(2)^2-8000*x^3-10800*x^2-4860*x-729),x, algorithm="maxima")

[Out]

e^(1/16*log(2)^4/(25*log(2)^4 - 40*(5*log(2)^2 - 9)*x + 400*x^2 - 90*log(2)^2 + 81) + 31/40*log(2)^2/(25*log(2
)^4 - 40*(5*log(2)^2 - 9)*x + 400*x^2 - 90*log(2)^2 + 81) - 41/40*log(2)^2/(5*log(2)^2 - 20*x - 9) + 961/400/(
25*log(2)^4 - 40*(5*log(2)^2 - 9)*x + 400*x^2 - 90*log(2)^2 + 81) - 1271/200/(5*log(2)^2 - 20*x - 9) + 1681/40
0)

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mupad [B]  time = 5.52, size = 222, normalized size = 8.88 \begin {gather*} {\mathrm {e}}^{\frac {1640\,x}{360\,x-200\,x\,{\ln \relax (2)}^2-90\,{\ln \relax (2)}^2+25\,{\ln \relax (2)}^4+400\,x^2+81}}\,{\mathrm {e}}^{\frac {100\,{\ln \relax (2)}^4}{360\,x-200\,x\,{\ln \relax (2)}^2-90\,{\ln \relax (2)}^2+25\,{\ln \relax (2)}^4+400\,x^2+81}}\,{\mathrm {e}}^{-\frac {400\,{\ln \relax (2)}^2}{360\,x-200\,x\,{\ln \relax (2)}^2-90\,{\ln \relax (2)}^2+25\,{\ln \relax (2)}^4+400\,x^2+81}}\,{\mathrm {e}}^{\frac {1681\,x^2}{360\,x-200\,x\,{\ln \relax (2)}^2-90\,{\ln \relax (2)}^2+25\,{\ln \relax (2)}^4+400\,x^2+81}}\,{\mathrm {e}}^{\frac {400}{360\,x-200\,x\,{\ln \relax (2)}^2-90\,{\ln \relax (2)}^2+25\,{\ln \relax (2)}^4+400\,x^2+81}}\,{\mathrm {e}}^{-\frac {820\,x\,{\ln \relax (2)}^2}{360\,x-200\,x\,{\ln \relax (2)}^2-90\,{\ln \relax (2)}^2+25\,{\ln \relax (2)}^4+400\,x^2+81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((1640*x - log(2)^2*(820*x + 400) + 100*log(2)^4 + 1681*x^2 + 400)/(360*x - log(2)^2*(200*x + 90) + 2
5*log(2)^4 + 400*x^2 + 81))*(2542*x + log(2)^2*(410*x - 420) - 100*log(2)^4 + 1240))/(4860*x + log(2)^4*(1500*
x + 675) - log(2)^2*(5400*x + 6000*x^2 + 1215) - 125*log(2)^6 + 10800*x^2 + 8000*x^3 + 729),x)

[Out]

exp((1640*x)/(360*x - 200*x*log(2)^2 - 90*log(2)^2 + 25*log(2)^4 + 400*x^2 + 81))*exp((100*log(2)^4)/(360*x -
200*x*log(2)^2 - 90*log(2)^2 + 25*log(2)^4 + 400*x^2 + 81))*exp(-(400*log(2)^2)/(360*x - 200*x*log(2)^2 - 90*l
og(2)^2 + 25*log(2)^4 + 400*x^2 + 81))*exp((1681*x^2)/(360*x - 200*x*log(2)^2 - 90*log(2)^2 + 25*log(2)^4 + 40
0*x^2 + 81))*exp(400/(360*x - 200*x*log(2)^2 - 90*log(2)^2 + 25*log(2)^4 + 400*x^2 + 81))*exp(-(820*x*log(2)^2
)/(360*x - 200*x*log(2)^2 - 90*log(2)^2 + 25*log(2)^4 + 400*x^2 + 81))

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sympy [B]  time = 0.77, size = 58, normalized size = 2.32 \begin {gather*} e^{\frac {1681 x^{2} + 1640 x + \left (- 820 x - 400\right ) \log {\relax (2 )}^{2} + 100 \log {\relax (2 )}^{4} + 400}{400 x^{2} + 360 x + \left (- 200 x - 90\right ) \log {\relax (2 )}^{2} + 25 \log {\relax (2 )}^{4} + 81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-100*ln(2)**4+(410*x-420)*ln(2)**2+2542*x+1240)*exp((100*ln(2)**4+(-820*x-400)*ln(2)**2+1681*x**2+1
640*x+400)/(25*ln(2)**4+(-200*x-90)*ln(2)**2+400*x**2+360*x+81))/(125*ln(2)**6+(-1500*x-675)*ln(2)**4+(6000*x*
*2+5400*x+1215)*ln(2)**2-8000*x**3-10800*x**2-4860*x-729),x)

[Out]

exp((1681*x**2 + 1640*x + (-820*x - 400)*log(2)**2 + 100*log(2)**4 + 400)/(400*x**2 + 360*x + (-200*x - 90)*lo
g(2)**2 + 25*log(2)**4 + 81))

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