3.67.8 \(\int \frac {e^{-6+\frac {-4+4 e^3+e^6 (-1+4 x^2)+(-4 e^3+2 e^6) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} (32-8 x+e^3 (-32+8 x)+e^6 (8-2 x+4 x^2-2 x^3)+(e^3 (32-8 x)+e^6 (-16+4 x)) \log (5)+e^6 (8-2 x) \log ^2(5))}{x^2} \, dx\)

Optimal. Leaf size=27 \[ e^{4-\frac {\left (-1+\frac {2}{e^3}+\log (5)\right )^2}{x^2}} (4-x) x \]

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Rubi [A]  time = 2.57, antiderivative size = 34, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6741, 2288} \begin {gather*} (4-x) x e^{4-\frac {\left (2-e^3 (1-\log (5))\right )^2}{e^6 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-6 + (-4 + 4*E^3 + E^6*(-1 + 4*x^2) + (-4*E^3 + 2*E^6)*Log[5] - E^6*Log[5]^2)/(E^6*x^2))*(32 - 8*x + E
^3*(-32 + 8*x) + E^6*(8 - 2*x + 4*x^2 - 2*x^3) + (E^3*(32 - 8*x) + E^6*(-16 + 4*x))*Log[5] + E^6*(8 - 2*x)*Log
[5]^2))/x^2,x]

[Out]

E^(4 - (2 - E^3*(1 - Log[5]))^2/(E^6*x^2))*(4 - x)*x

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-2+\frac {-4+4 e^3-e^6-4 e^3 \log (5)+2 e^6 \log (5)-e^6 \log ^2(5)}{e^6 x^2}\right ) \left (4 e^6 x^2-2 e^6 x^3+8 \left (2-e^3 (1-\log (5))\right )^2-2 x \left (2-e^3 (1-\log (5))\right )^2\right )}{x^2} \, dx\\ &=e^{4-\frac {\left (2-e^3 (1-\log (5))\right )^2}{e^6 x^2}} (4-x) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 54, normalized size = 2.00 \begin {gather*} -5^{\frac {2 \left (-2+e^3\right )}{e^3 x^2}} e^{\frac {-4+4 e^3+e^6 \left (-1+4 x^2-\log ^2(5)\right )}{e^6 x^2}} (-4+x) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 + (-4 + 4*E^3 + E^6*(-1 + 4*x^2) + (-4*E^3 + 2*E^6)*Log[5] - E^6*Log[5]^2)/(E^6*x^2))*(32 - 8
*x + E^3*(-32 + 8*x) + E^6*(8 - 2*x + 4*x^2 - 2*x^3) + (E^3*(32 - 8*x) + E^6*(-16 + 4*x))*Log[5] + E^6*(8 - 2*
x)*Log[5]^2))/x^2,x]

[Out]

-(5^((2*(-2 + E^3))/(E^3*x^2))*E^((-4 + 4*E^3 + E^6*(-1 + 4*x^2 - Log[5]^2))/(E^6*x^2))*(-4 + x)*x)

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fricas [B]  time = 0.47, size = 53, normalized size = 1.96 \begin {gather*} -{\left (x^{2} - 4 \, x\right )} e^{\left (-\frac {{\left (e^{6} \log \relax (5)^{2} + {\left (2 \, x^{2} + 1\right )} e^{6} - 2 \, {\left (e^{6} - 2 \, e^{3}\right )} \log \relax (5) - 4 \, e^{3} + 4\right )} e^{\left (-6\right )}}{x^{2}} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(3)^2*log(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3))*log(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^
2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3)^2*log(5)^2+(2*exp(3)^2-4*exp(3))*log(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/
x^2/exp(3)^2)/x^2/exp(3)^2,x, algorithm="fricas")

[Out]

-(x^2 - 4*x)*e^(-(e^6*log(5)^2 + (2*x^2 + 1)*e^6 - 2*(e^6 - 2*e^3)*log(5) - 4*e^3 + 4)*e^(-6)/x^2 + 6)

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giac [B]  time = 0.60, size = 99, normalized size = 3.67 \begin {gather*} -x^{2} e^{\left (\frac {{\left (x^{2} e^{6} - e^{6} \log \relax (5)^{2} + 2 \, e^{6} \log \relax (5) - 4 \, e^{3} \log \relax (5) - e^{6} + 4 \, e^{3} - 4\right )} e^{\left (-6\right )}}{x^{2}} + 3\right )} + 4 \, x e^{\left (\frac {{\left (x^{2} e^{6} - e^{6} \log \relax (5)^{2} + 2 \, e^{6} \log \relax (5) - 4 \, e^{3} \log \relax (5) - e^{6} + 4 \, e^{3} - 4\right )} e^{\left (-6\right )}}{x^{2}} + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(3)^2*log(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3))*log(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^
2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3)^2*log(5)^2+(2*exp(3)^2-4*exp(3))*log(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/
x^2/exp(3)^2)/x^2/exp(3)^2,x, algorithm="giac")

[Out]

-x^2*e^((x^2*e^6 - e^6*log(5)^2 + 2*e^6*log(5) - 4*e^3*log(5) - e^6 + 4*e^3 - 4)*e^(-6)/x^2 + 3) + 4*x*e^((x^2
*e^6 - e^6*log(5)^2 + 2*e^6*log(5) - 4*e^3*log(5) - e^6 + 4*e^3 - 4)*e^(-6)/x^2 + 3)

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maple [A]  time = 0.78, size = 54, normalized size = 2.00

method	result	size
risch	\(\left (\frac {1}{625}\right )^{\frac {{\mathrm e}^{-3}}{x^{2}}} 25^{\frac {1}{x^{2}}} \left (-x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}\right ) {\mathrm e}^{\frac {-\ln \relax (5)^{2}-2 x^{2}-4 \,{\mathrm e}^{-6}+4 \,{\mathrm e}^{-3}-1}{x^{2}}}\)	\(54\)
gosper	\(-{\mathrm e}^{-\frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}-4 x^{2} {\mathrm e}^{6}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}} \left (x -4\right ) x\)	\(59\)
norman	\(\frac {\left (4 x^{2} {\mathrm e}^{3} {\mathrm e}^{\frac {\left (-{\mathrm e}^{6} \ln \relax (5)^{2}+\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+\left (4 x^{2}-1\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3}-4\right ) {\mathrm e}^{-6}}{x^{2}}}-x^{3} {\mathrm e}^{3} {\mathrm e}^{\frac {\left (-{\mathrm e}^{6} \ln \relax (5)^{2}+\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+\left (4 x^{2}-1\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3}-4\right ) {\mathrm e}^{-6}}{x^{2}}}\right ) {\mathrm e}^{-3}}{x}\)	\(126\)
default	\({\mathrm e}^{-6} \left (-\frac {4 \,{\mathrm e}^{9} \ln \relax (5)^{2} \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}-\frac {\left (-16 \,{\mathrm e}^{6}+32 \,{\mathrm e}^{3}\right ) \ln \relax (5) \sqrt {\pi }\, {\mathrm e}^{4} {\mathrm e}^{3} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{2 \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}-\frac {4 \,{\mathrm e}^{9} \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}+\frac {16 \,{\mathrm e}^{6} \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}-\frac {16 \sqrt {\pi }\, {\mathrm e}^{4} {\mathrm e}^{3} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}+\frac {\left (-2 \,{\mathrm e}^{6} \ln \relax (5)^{2}+\left (4 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}\right ) \ln \relax (5)-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{3}-8\right ) {\mathrm e}^{4} \expIntegralEi \left (1, \frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}\right )}{2}+4 \,{\mathrm e}^{6} x \,{\mathrm e}^{4-\frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}}+4 \,{\mathrm e}^{3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}\, \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )-{\mathrm e}^{6} x^{2} {\mathrm e}^{4-\frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}}+\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{4} \expIntegralEi \left (1, \frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}\right )\right )\)	\(826\)
derivativedivides	\(-{\mathrm e}^{-6} \left (\frac {4 \,{\mathrm e}^{9} \ln \relax (5)^{2} \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}+\frac {\left (-16 \,{\mathrm e}^{6}+32 \,{\mathrm e}^{3}\right ) \ln \relax (5) \sqrt {\pi }\, {\mathrm e}^{4} {\mathrm e}^{3} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{2 \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}+\frac {4 \,{\mathrm e}^{9} \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}-\frac {16 \,{\mathrm e}^{6} \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}+\frac {16 \sqrt {\pi }\, {\mathrm e}^{4} {\mathrm e}^{3} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}-\frac {\left (-2 \,{\mathrm e}^{6} \ln \relax (5)^{2}+\left (4 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}\right ) \ln \relax (5)-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{3}-8\right ) {\mathrm e}^{4} \expIntegralEi \left (1, \frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}\right )}{2}-4 \,{\mathrm e}^{6} x \,{\mathrm e}^{4-\frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}}-4 \,{\mathrm e}^{3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}\, \sqrt {\pi }\, {\mathrm e}^{4} \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4}}{x}\right )+{\mathrm e}^{6} x^{2} {\mathrm e}^{4-\frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}}-\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{4} \expIntegralEi \left (1, \frac {\left ({\mathrm e}^{6} \ln \relax (5)^{2}+4 \,{\mathrm e}^{3} \ln \relax (5)-2 \ln \relax (5) {\mathrm e}^{6}-4 \,{\mathrm e}^{3}+{\mathrm e}^{6}+4\right ) {\mathrm e}^{-6}}{x^{2}}\right )\right )\)	\(827\)
meijerg	\(-\frac {4 \ln \relax (5)^{2} {\mathrm e}^{7} \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}+\frac {8 \ln \relax (5) {\mathrm e}^{7} \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}-\frac {16 \,{\mathrm e}^{4} \ln \relax (5) \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}-\frac {4 \,{\mathrm e}^{7} \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}+\frac {16 \,{\mathrm e}^{4} \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}-\frac {16 \,{\mathrm e} \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}+{\mathrm e}^{-2} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right ) \left (\frac {x^{2} {\mathrm e}^{6} \left (2-\frac {2 \,{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}\right )}{2 \,{\mathrm e}^{6} \ln \relax (5)^{2}-2 \left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+2 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+8}-\frac {x^{2} {\mathrm e}^{6-\frac {{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}}}{{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}+\ln \left (\frac {{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}\right )+\expIntegralEi \left (1, \frac {{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}\right )+7+2 \ln \relax (x )-\ln \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )-\frac {x^{2} {\mathrm e}^{6}}{{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}\right )-2 \,{\mathrm e} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}\, \left (-\frac {2 x \,{\mathrm e}^{3-\frac {{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}}}{\sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}-2 \sqrt {\pi }\, \erf \left (\frac {{\mathrm e}^{-3} \sqrt {{\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4}}{x}\right )\right )-\frac {{\mathrm e}^{-2} \left (-2 \,{\mathrm e}^{6} \ln \relax (5)^{2}+4 \ln \relax (5) {\mathrm e}^{6}-8 \,{\mathrm e}^{3} \ln \relax (5)-2 \,{\mathrm e}^{6}+8 \,{\mathrm e}^{3}-8\right ) \left (-\ln \left (\frac {{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}\right )-\expIntegralEi \left (1, \frac {{\mathrm e}^{-6} \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )}{x^{2}}\right )-2 \ln \relax (x )-6+\ln \left ({\mathrm e}^{6} \ln \relax (5)^{2}-\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \relax (5)+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right )\right )}{2}\)	\(1069\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+8)*exp(3)^2*ln(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3))*ln(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^2+(8*x-3
2)*exp(3)-8*x+32)*exp((-exp(3)^2*ln(5)^2+(2*exp(3)^2-4*exp(3))*ln(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/x^2/exp(3)
^2)/x^2/exp(3)^2,x,method=_RETURNVERBOSE)

[Out]

(1/625)^(1/x^2*exp(-3))*25^(1/x^2)*(-x^2*exp(6)+4*x*exp(6))*exp((-ln(5)^2-2*x^2-4*exp(-6)+4*exp(-3)-1)/x^2)

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maxima [C]  time = 0.88, size = 450, normalized size = 16.67 \begin {gather*} 2 \, x \sqrt {\frac {1}{x^{2}}} {\left | e^{3} \log \relax (5) - e^{3} + 2 \right |} e \Gamma \left (-\frac {1}{2}, \frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) - {\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-2\right )} \Gamma \left (-1, \frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) + {\rm Ei}\left (-\frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) e^{4} \log \relax (5)^{2} - \frac {4 \, \sqrt {\pi } \operatorname {erf}\left (\frac {{\left ({\left (\log \relax (5) - 1\right )} e^{3} + 2\right )} e^{\left (-3\right )}}{x}\right ) e^{7} \log \relax (5)^{2}}{{\left (\log \relax (5) - 1\right )} e^{3} + 2} - 2 \, {\rm Ei}\left (-\frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) e^{4} \log \relax (5) + 4 \, {\rm Ei}\left (-\frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) e \log \relax (5) + \frac {8 \, \sqrt {\pi } \operatorname {erf}\left (\frac {{\left ({\left (\log \relax (5) - 1\right )} e^{3} + 2\right )} e^{\left (-3\right )}}{x}\right ) e^{7} \log \relax (5)}{{\left (\log \relax (5) - 1\right )} e^{3} + 2} - \frac {16 \, \sqrt {\pi } \operatorname {erf}\left (\frac {{\left ({\left (\log \relax (5) - 1\right )} e^{3} + 2\right )} e^{\left (-3\right )}}{x}\right ) e^{4} \log \relax (5)}{{\left (\log \relax (5) - 1\right )} e^{3} + 2} + {\rm Ei}\left (-\frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) e^{4} - 4 \, {\rm Ei}\left (-\frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) e + 4 \, {\rm Ei}\left (-\frac {{\left (e^{3} \log \relax (5) - e^{3} + 2\right )}^{2} e^{\left (-6\right )}}{x^{2}}\right ) e^{\left (-2\right )} - \frac {4 \, \sqrt {\pi } \operatorname {erf}\left (\frac {{\left ({\left (\log \relax (5) - 1\right )} e^{3} + 2\right )} e^{\left (-3\right )}}{x}\right ) e^{7}}{{\left (\log \relax (5) - 1\right )} e^{3} + 2} + \frac {16 \, \sqrt {\pi } \operatorname {erf}\left (\frac {{\left ({\left (\log \relax (5) - 1\right )} e^{3} + 2\right )} e^{\left (-3\right )}}{x}\right ) e^{4}}{{\left (\log \relax (5) - 1\right )} e^{3} + 2} - \frac {16 \, \sqrt {\pi } \operatorname {erf}\left (\frac {{\left ({\left (\log \relax (5) - 1\right )} e^{3} + 2\right )} e^{\left (-3\right )}}{x}\right ) e}{{\left (\log \relax (5) - 1\right )} e^{3} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(3)^2*log(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3))*log(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^
2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3)^2*log(5)^2+(2*exp(3)^2-4*exp(3))*log(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/
x^2/exp(3)^2)/x^2/exp(3)^2,x, algorithm="maxima")

[Out]

2*x*sqrt(x^(-2))*abs(e^3*log(5) - e^3 + 2)*e*gamma(-1/2, (e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2) - (e^3*log(5) -
e^3 + 2)^2*e^(-2)*gamma(-1, (e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2) + Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^
4*log(5)^2 - 4*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^7*log(5)^2/((log(5) - 1)*e^3 + 2) - 2*Ei(-(e^3*
log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^4*log(5) + 4*Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e*log(5) + 8*sqrt(pi)*
erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^7*log(5)/((log(5) - 1)*e^3 + 2) - 16*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2
)*e^(-3)/x)*e^4*log(5)/((log(5) - 1)*e^3 + 2) + Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^4 - 4*Ei(-(e^3*log(
5) - e^3 + 2)^2*e^(-6)/x^2)*e + 4*Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^(-2) - 4*sqrt(pi)*erf(((log(5) -
1)*e^3 + 2)*e^(-3)/x)*e^7/((log(5) - 1)*e^3 + 2) + 16*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^4/((log(
5) - 1)*e^3 + 2) - 16*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e/((log(5) - 1)*e^3 + 2)

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mupad [B]  time = 4.76, size = 56, normalized size = 2.07 \begin {gather*} -\frac {{25}^{\frac {1}{x^2}}\,x\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-3}}{x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{-6}}{x^2}}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-\frac {{\ln \relax (5)}^2}{x^2}}\,{\mathrm {e}}^{-\frac {1}{x^2}}\,\left (x-4\right )}{{25}^{\frac {2\,{\mathrm {e}}^{-3}}{x^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-6)*exp(-(exp(-6)*(exp(6)*log(5)^2 - exp(6)*(4*x^2 - 1) - 4*exp(3) + log(5)*(4*exp(3) - 2*exp(6)) +
4))/x^2)*(8*x - log(5)*(exp(6)*(4*x - 16) - exp(3)*(8*x - 32)) + exp(6)*(2*x - 4*x^2 + 2*x^3 - 8) - exp(3)*(8*
x - 32) + exp(6)*log(5)^2*(2*x - 8) - 32))/x^2,x)

[Out]

-(25^(1/x^2)*x*exp((4*exp(-3))/x^2)*exp(-(4*exp(-6))/x^2)*exp(4)*exp(-log(5)^2/x^2)*exp(-1/x^2)*(x - 4))/25^((
2*exp(-3))/x^2)

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sympy [B]  time = 0.71, size = 53, normalized size = 1.96 \begin {gather*} \left (- x^{2} + 4 x\right ) e^{\frac {\left (4 x^{2} - 1\right ) e^{6} - e^{6} \log {\relax (5 )}^{2} - 4 + 4 e^{3} + \left (- 4 e^{3} + 2 e^{6}\right ) \log {\relax (5 )}}{x^{2} e^{6}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+8)*exp(3)**2*ln(5)**2+((4*x-16)*exp(3)**2+(-8*x+32)*exp(3))*ln(5)+(-2*x**3+4*x**2-2*x+8)*exp(
3)**2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3)**2*ln(5)**2+(2*exp(3)**2-4*exp(3))*ln(5)+(4*x**2-1)*exp(3)**2+4*exp
(3)-4)/x**2/exp(3)**2)/x**2/exp(3)**2,x)

[Out]

(-x**2 + 4*x)*exp(((4*x**2 - 1)*exp(6) - exp(6)*log(5)**2 - 4 + 4*exp(3) + (-4*exp(3) + 2*exp(6))*log(5))*exp(
-6)/x**2)

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