3.31.92 \(\int \frac {4 e^{15 e x}-x+(4 e^{30 e x}-30 e^{1+15 e x} x) \log (\frac {4}{x^2})-30 e^{1+30 e x} x \log ^2(\frac {4}{x^2})}{4 x+4 x^2+x^3+e^{15 e x} (8 x+4 x^2) \log (\frac {4}{x^2})+e^{30 e x} (8 x+2 x^2) \log ^2(\frac {4}{x^2})+4 e^{45 e x} x \log ^3(\frac {4}{x^2})+e^{60 e x} x \log ^4(\frac {4}{x^2})} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{1+x+\left (1+e^{15 e x} \log \left (\frac {4}{x^2}\right )\right )^2} \]

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Rubi [F]  time = 5.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {4 e^{15 e x}-x+\left (4 e^{30 e x}-30 e^{1+15 e x} x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{4 x+4 x^2+x^3+e^{15 e x} \left (8 x+4 x^2\right ) \log \left (\frac {4}{x^2}\right )+e^{30 e x} \left (8 x+2 x^2\right ) \log ^2\left (\frac {4}{x^2}\right )+4 e^{45 e x} x \log ^3\left (\frac {4}{x^2}\right )+e^{60 e x} x \log ^4\left (\frac {4}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*E^(15*E*x) - x + (4*E^(30*E*x) - 30*E^(1 + 15*E*x)*x)*Log[4/x^2] - 30*E^(1 + 30*E*x)*x*Log[4/x^2]^2)/(4
*x + 4*x^2 + x^3 + E^(15*E*x)*(8*x + 4*x^2)*Log[4/x^2] + E^(30*E*x)*(8*x + 2*x^2)*Log[4/x^2]^2 + 4*E^(45*E*x)*
x*Log[4/x^2]^3 + E^(60*E*x)*x*Log[4/x^2]^4),x]

[Out]

-((1 - 60*E)*Defer[Int][(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^(-2), x]) - 4*Defer[Int][E
^(15*E*x)/(x*(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^2), x] + 30*E*Defer[Int][x/(2 + x + 2
*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^2, x] - 4*Defer[Int][1/(Log[4/x^2]*(2 + x + 2*E^(15*E*x)*Log
[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^2), x] - 8*Defer[Int][1/(x*Log[4/x^2]*(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^
(30*E*x)*Log[4/x^2]^2)^2), x] + 30*Defer[Int][(E^(1 + 15*E*x)*Log[4/x^2])/(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E
^(30*E*x)*Log[4/x^2]^2)^2, x] - 30*E*Defer[Int][(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^(-
1), x] + 4*Defer[Int][1/(x*Log[4/x^2]*(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^{15 e x}-x+2 e^{15 e x} \left (2 e^{15 e x}-15 e x\right ) \log \left (\frac {4}{x^2}\right )-30 e^{1+30 e x} x \log ^2\left (\frac {4}{x^2}\right )}{x \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx\\ &=\int \left (-\frac {2 \left (-2+15 e x \log \left (\frac {4}{x^2}\right )\right )}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )}+\frac {-8-4 x-4 e^{15 e x} \log \left (\frac {4}{x^2}\right )-(1-60 e) x \log \left (\frac {4}{x^2}\right )+30 e x^2 \log \left (\frac {4}{x^2}\right )+30 e^{1+15 e x} x \log ^2\left (\frac {4}{x^2}\right )}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {-2+15 e x \log \left (\frac {4}{x^2}\right )}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )} \, dx\right )+\int \frac {-8-4 x-4 e^{15 e x} \log \left (\frac {4}{x^2}\right )-(1-60 e) x \log \left (\frac {4}{x^2}\right )+30 e x^2 \log \left (\frac {4}{x^2}\right )+30 e^{1+15 e x} x \log ^2\left (\frac {4}{x^2}\right )}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx\\ &=-\left (2 \int \left (\frac {15 e}{2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )}-\frac {2}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )}\right ) \, dx\right )+\int \left (\frac {-1+60 e}{\left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}-\frac {4 e^{15 e x}}{x \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}+\frac {30 e x}{\left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}-\frac {4}{\log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}-\frac {8}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}+\frac {30 e^{1+15 e x} \log \left (\frac {4}{x^2}\right )}{\left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {e^{15 e x}}{x \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx\right )-4 \int \frac {1}{\log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx+4 \int \frac {1}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )} \, dx-8 \int \frac {1}{x \log \left (\frac {4}{x^2}\right ) \left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx+30 \int \frac {e^{1+15 e x} \log \left (\frac {4}{x^2}\right )}{\left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx+(30 e) \int \frac {x}{\left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx-(30 e) \int \frac {1}{2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )} \, dx+(-1+60 e) \int \frac {1}{\left (2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 34, normalized size = 1.55 \begin {gather*} \frac {1}{2+x+2 e^{15 e x} \log \left (\frac {4}{x^2}\right )+e^{30 e x} \log ^2\left (\frac {4}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(15*E*x) - x + (4*E^(30*E*x) - 30*E^(1 + 15*E*x)*x)*Log[4/x^2] - 30*E^(1 + 30*E*x)*x*Log[4/x^2]
^2)/(4*x + 4*x^2 + x^3 + E^(15*E*x)*(8*x + 4*x^2)*Log[4/x^2] + E^(30*E*x)*(8*x + 2*x^2)*Log[4/x^2]^2 + 4*E^(45
*E*x)*x*Log[4/x^2]^3 + E^(60*E*x)*x*Log[4/x^2]^4),x]

[Out]

(2 + x + 2*E^(15*E*x)*Log[4/x^2] + E^(30*E*x)*Log[4/x^2]^2)^(-1)

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fricas [B]  time = 0.94, size = 45, normalized size = 2.05 \begin {gather*} \frac {e^{2}}{e^{\left (30 \, x e + 2\right )} \log \left (\frac {4}{x^{2}}\right )^{2} + {\left (x + 2\right )} e^{2} + 2 \, e^{\left (15 \, x e + 2\right )} \log \left (\frac {4}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*exp(1)*exp(15*x*exp(1))^2*log(4/x^2)^2+(4*exp(15*x*exp(1))^2-30*x*exp(1)*exp(15*x*exp(1)))*lo
g(4/x^2)+4*exp(15*x*exp(1))-x)/(x*exp(15*x*exp(1))^4*log(4/x^2)^4+4*x*exp(15*x*exp(1))^3*log(4/x^2)^3+(2*x^2+8
*x)*exp(15*x*exp(1))^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*x*exp(1))*log(4/x^2)+x^3+4*x^2+4*x),x, algorithm="frica
s")

[Out]

e^2/(e^(30*x*e + 2)*log(4/x^2)^2 + (x + 2)*e^2 + 2*e^(15*x*e + 2)*log(4/x^2))

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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.

[In]

integrate((-30*x*exp(1)*exp(15*x*exp(1))^2*log(4/x^2)^2+(4*exp(15*x*exp(1))^2-30*x*exp(1)*exp(15*x*exp(1)))*lo
g(4/x^2)+4*exp(15*x*exp(1))-x)/(x*exp(15*x*exp(1))^4*log(4/x^2)^4+4*x*exp(15*x*exp(1))^3*log(4/x^2)^3+(2*x^2+8
*x)*exp(15*x*exp(1))^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*x*exp(1))*log(4/x^2)+x^3+4*x^2+4*x),x, algorithm="giac"
)

[Out]

Timed out

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maple [C]  time = 2.91, size = 403, normalized size = 18.32

method	result	size
risch	\(-\frac {4}{-8-4 x +{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \,{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}+6 \,{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}-4 \,{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}-16 \ln \relax (2) {\mathrm e}^{15 x \,{\mathrm e}}+16 \,{\mathrm e}^{15 x \,{\mathrm e}} \ln \relax (x )-16 \ln \relax (2)^{2} {\mathrm e}^{30 x \,{\mathrm e}}-16 \,{\mathrm e}^{30 x \,{\mathrm e}} \ln \relax (x )^{2}-4 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{15 x \,{\mathrm e}}+32 \,{\mathrm e}^{30 x \,{\mathrm e}} \ln \relax (2) \ln \relax (x )-16 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{30 x \,{\mathrm e}} \ln \relax (x )-8 i \pi \ln \relax (2) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{30 x \,{\mathrm e}}-8 i \pi \ln \relax (2) \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{30 x \,{\mathrm e}}+8 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{30 x \,{\mathrm e}} \ln \relax (x )+{\mathrm e}^{30 x \,{\mathrm e}} \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}+8 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{30 x \,{\mathrm e}} \ln \relax (x )+8 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{15 x \,{\mathrm e}}-4 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{15 x \,{\mathrm e}}+16 i \pi \ln \relax (2) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{30 x \,{\mathrm e}}}\)	\(403\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-30*x*exp(1)*exp(15*x*exp(1))^2*ln(4/x^2)^2+(4*exp(15*x*exp(1))^2-30*x*exp(1)*exp(15*x*exp(1)))*ln(4/x^2)
+4*exp(15*x*exp(1))-x)/(x*exp(15*x*exp(1))^4*ln(4/x^2)^4+4*x*exp(15*x*exp(1))^3*ln(4/x^2)^3+(2*x^2+8*x)*exp(15
*x*exp(1))^2*ln(4/x^2)^2+(4*x^2+8*x)*exp(15*x*exp(1))*ln(4/x^2)+x^3+4*x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

-4/(-8-4*x+exp(30*x*exp(1))*Pi^2*csgn(I*x)^4*csgn(I*x^2)^2-4*exp(30*x*exp(1))*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3+6
*exp(30*x*exp(1))*Pi^2*csgn(I*x)^2*csgn(I*x^2)^4-4*exp(30*x*exp(1))*Pi^2*csgn(I*x)*csgn(I*x^2)^5-16*ln(2)*exp(
15*x*exp(1))+16*exp(15*x*exp(1))*ln(x)-16*ln(2)^2*exp(30*x*exp(1))-16*exp(30*x*exp(1))*ln(x)^2-4*I*Pi*csgn(I*x
)^2*csgn(I*x^2)*exp(15*x*exp(1))+32*exp(30*x*exp(1))*ln(2)*ln(x)-16*I*Pi*csgn(I*x)*csgn(I*x^2)^2*exp(30*x*exp(
1))*ln(x)-8*I*Pi*ln(2)*csgn(I*x)^2*csgn(I*x^2)*exp(30*x*exp(1))-8*I*Pi*ln(2)*csgn(I*x^2)^3*exp(30*x*exp(1))+8*
I*Pi*csgn(I*x)^2*csgn(I*x^2)*exp(30*x*exp(1))*ln(x)+exp(30*x*exp(1))*Pi^2*csgn(I*x^2)^6+8*I*Pi*csgn(I*x^2)^3*e
xp(30*x*exp(1))*ln(x)+8*I*Pi*csgn(I*x)*csgn(I*x^2)^2*exp(15*x*exp(1))-4*I*Pi*csgn(I*x^2)^3*exp(15*x*exp(1))+16
*I*Pi*ln(2)*csgn(I*x)*csgn(I*x^2)^2*exp(30*x*exp(1)))

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maxima [A]  time = 0.99, size = 43, normalized size = 1.95 \begin {gather*} \frac {1}{4 \, {\left (\log \relax (2)^{2} - 2 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (30 \, x e\right )} + 4 \, {\left (\log \relax (2) - \log \relax (x)\right )} e^{\left (15 \, x e\right )} + x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*exp(1)*exp(15*x*exp(1))^2*log(4/x^2)^2+(4*exp(15*x*exp(1))^2-30*x*exp(1)*exp(15*x*exp(1)))*lo
g(4/x^2)+4*exp(15*x*exp(1))-x)/(x*exp(15*x*exp(1))^4*log(4/x^2)^4+4*x*exp(15*x*exp(1))^3*log(4/x^2)^3+(2*x^2+8
*x)*exp(15*x*exp(1))^2*log(4/x^2)^2+(4*x^2+8*x)*exp(15*x*exp(1))*log(4/x^2)+x^3+4*x^2+4*x),x, algorithm="maxim
a")

[Out]

1/(4*(log(2)^2 - 2*log(2)*log(x) + log(x)^2)*e^(30*x*e) + 4*(log(2) - log(x))*e^(15*x*e) + x + 2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {30\,x\,\mathrm {e}\,{\mathrm {e}}^{30\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^2+\left (30\,x\,\mathrm {e}\,{\mathrm {e}}^{15\,x\,\mathrm {e}}-4\,{\mathrm {e}}^{30\,x\,\mathrm {e}}\right )\,\ln \left (\frac {4}{x^2}\right )+x-4\,{\mathrm {e}}^{15\,x\,\mathrm {e}}}{4\,x+4\,x^2+x^3+{\mathrm {e}}^{15\,x\,\mathrm {e}}\,\ln \left (\frac {4}{x^2}\right )\,\left (4\,x^2+8\,x\right )+{\mathrm {e}}^{30\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^2\,\left (2\,x^2+8\,x\right )+4\,x\,{\mathrm {e}}^{45\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^3+x\,{\mathrm {e}}^{60\,x\,\mathrm {e}}\,{\ln \left (\frac {4}{x^2}\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - 4*exp(15*x*exp(1)) - log(4/x^2)*(4*exp(30*x*exp(1)) - 30*x*exp(1)*exp(15*x*exp(1))) + 30*x*exp(1)*ex
p(30*x*exp(1))*log(4/x^2)^2)/(4*x + 4*x^2 + x^3 + exp(15*x*exp(1))*log(4/x^2)*(8*x + 4*x^2) + exp(30*x*exp(1))
*log(4/x^2)^2*(8*x + 2*x^2) + 4*x*exp(45*x*exp(1))*log(4/x^2)^3 + x*exp(60*x*exp(1))*log(4/x^2)^4),x)

[Out]

int(-(x - 4*exp(15*x*exp(1)) - log(4/x^2)*(4*exp(30*x*exp(1)) - 30*x*exp(1)*exp(15*x*exp(1))) + 30*x*exp(1)*ex
p(30*x*exp(1))*log(4/x^2)^2)/(4*x + 4*x^2 + x^3 + exp(15*x*exp(1))*log(4/x^2)*(8*x + 4*x^2) + exp(30*x*exp(1))
*log(4/x^2)^2*(8*x + 2*x^2) + 4*x*exp(45*x*exp(1))*log(4/x^2)^3 + x*exp(60*x*exp(1))*log(4/x^2)^4), x)

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sympy [A]  time = 0.47, size = 37, normalized size = 1.68 \begin {gather*} \frac {1}{x + e^{30 e x} \log {\left (\frac {4}{x^{2}} \right )}^{2} + 2 e^{15 e x} \log {\left (\frac {4}{x^{2}} \right )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-30*x*exp(1)*exp(15*x*exp(1))**2*ln(4/x**2)**2+(4*exp(15*x*exp(1))**2-30*x*exp(1)*exp(15*x*exp(1)))
*ln(4/x**2)+4*exp(15*x*exp(1))-x)/(x*exp(15*x*exp(1))**4*ln(4/x**2)**4+4*x*exp(15*x*exp(1))**3*ln(4/x**2)**3+(
2*x**2+8*x)*exp(15*x*exp(1))**2*ln(4/x**2)**2+(4*x**2+8*x)*exp(15*x*exp(1))*ln(4/x**2)+x**3+4*x**2+4*x),x)

[Out]

1/(x + exp(30*E*x)*log(4/x**2)**2 + 2*exp(15*E*x)*log(4/x**2) + 2)

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