3.44.35 \(\int \frac {-e^x+3 e^{5 x}+e^{5 x} (1+5 x) \log (x^3)}{-2-e^x+e^{5 x} x \log (x^3)} \, dx\)

Optimal. Leaf size=19 \[ \log \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right ) \]

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Rubi [F]  time = 1.90, 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 {-e^x+3 e^{5 x}+e^{5 x} (1+5 x) \log \left (x^3\right )}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

5*x + Log[x] + Log[Log[x^3]] + 10*Defer[Int][(-2 - E^x + E^(5*x)*x*Log[x^3])^(-1), x] + 4*Defer[Int][E^x/(-2 -
E^x + E^(5*x)*x*Log[x^3]), x] + 2*Defer[Int][1/(x*(-2 - E^x + E^(5*x)*x*Log[x^3])), x] + Defer[Int][E^x/(x*(-
2 - E^x + E^(5*x)*x*Log[x^3])), x] + 6*Defer[Int][1/(x*Log[x^3]*(-2 - E^x + E^(5*x)*x*Log[x^3])), x] + 3*Defer
[Int][E^x/(x*Log[x^3]*(-2 - E^x + E^(5*x)*x*Log[x^3])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3+\log \left (x^3\right )+5 x \log \left (x^3\right )}{x \log \left (x^3\right )}+\frac {6+3 e^x+2 \log \left (x^3\right )+e^x \log \left (x^3\right )+10 x \log \left (x^3\right )+4 e^x x \log \left (x^3\right )}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )}\right ) \, dx\\ &=\int \frac {3+\log \left (x^3\right )+5 x \log \left (x^3\right )}{x \log \left (x^3\right )} \, dx+\int \frac {6+3 e^x+2 \log \left (x^3\right )+e^x \log \left (x^3\right )+10 x \log \left (x^3\right )+4 e^x x \log \left (x^3\right )}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx\\ &=\int \left (5+\frac {1}{x}+\frac {3}{x \log \left (x^3\right )}\right ) \, dx+\int \left (\frac {10}{-2-e^x+e^{5 x} x \log \left (x^3\right )}+\frac {4 e^x}{-2-e^x+e^{5 x} x \log \left (x^3\right )}+\frac {2}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )}+\frac {e^x}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )}+\frac {6}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )}+\frac {3 e^x}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )}\right ) \, dx\\ &=5 x+\log (x)+2 \int \frac {1}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+3 \int \frac {1}{x \log \left (x^3\right )} \, dx+3 \int \frac {e^x}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+4 \int \frac {e^x}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx+6 \int \frac {1}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+10 \int \frac {1}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx+\int \frac {e^x}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx\\ &=5 x+\log (x)+2 \int \frac {1}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+3 \int \frac {e^x}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+4 \int \frac {e^x}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx+6 \int \frac {1}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+10 \int \frac {1}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx+\int \frac {e^x}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (x^3\right )\right )\\ &=5 x+\log (x)+\log \left (\log \left (x^3\right )\right )+2 \int \frac {1}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+3 \int \frac {e^x}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+4 \int \frac {e^x}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx+6 \int \frac {1}{x \log \left (x^3\right ) \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx+10 \int \frac {1}{-2-e^x+e^{5 x} x \log \left (x^3\right )} \, dx+\int \frac {e^x}{x \left (-2-e^x+e^{5 x} x \log \left (x^3\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.35, size = 18, normalized size = 0.95 \begin {gather*} \log \left (2+e^x-e^{5 x} x \log \left (x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Log[2 + E^x - E^(5*x)*x*Log[x^3]]

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fricas [B]  time = 0.59, size = 35, normalized size = 1.84 \begin {gather*} 5 \, x + \frac {1}{3} \, \log \left (x^{3}\right ) + \log \left (\frac {{\left (x e^{\left (5 \, x\right )} \log \left (x^{3}\right ) - e^{x} - 2\right )} e^{\left (-5 \, x\right )}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 16, normalized size = 0.84 \begin {gather*} \log \left (3 \, x e^{\left (5 \, x\right )} \log \relax (x) - e^{x} - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(3*x*e^(5*x)*log(x) - e^x - 2)

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maple [C]  time = 0.08, size = 176, normalized size = 9.26

method	result	size
risch	\(5 x +\ln \relax (x )+\ln \left (\ln \relax (x )-\frac {i \left ({\mathrm e}^{5 x} x \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \,{\mathrm e}^{5 x} x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+{\mathrm e}^{5 x} x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )-{\mathrm e}^{5 x} x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+{\mathrm e}^{5 x} x \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-{\mathrm e}^{5 x} x \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+{\mathrm e}^{5 x} x \pi \mathrm {csgn}\left (i x^{3}\right )^{3}-2 i {\mathrm e}^{x}-4 i\right ) {\mathrm e}^{-5 x}}{6 x}\right )\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*x+ln(x)+ln(ln(x)-1/6*I*(exp(5*x)*x*Pi*csgn(I*x)^2*csgn(I*x^2)-2*exp(5*x)*x*Pi*csgn(I*x)*csgn(I*x^2)^2+exp(5*
x)*x*Pi*csgn(I*x)*csgn(I*x^2)*csgn(I*x^3)-exp(5*x)*x*Pi*csgn(I*x)*csgn(I*x^3)^2+exp(5*x)*x*Pi*csgn(I*x^2)^3-ex
p(5*x)*x*Pi*csgn(I*x^2)*csgn(I*x^3)^2+exp(5*x)*x*Pi*csgn(I*x^3)^3-2*I*exp(x)-4*I)*exp(-5*x)/x)

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maxima [A]  time = 0.38, size = 31, normalized size = 1.63 \begin {gather*} \log \relax (x) + \log \left (\frac {3 \, x e^{\left (5 \, x\right )} \log \relax (x) - e^{x} - 2}{3 \, x \log \relax (x)}\right ) + \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.46, size = 29, normalized size = 1.53 \begin {gather*} 5\,x+\ln \relax (x)+\ln \left (\frac {{\mathrm {e}}^{-4\,x}+2\,{\mathrm {e}}^{-5\,x}-x\,\ln \left (x^3\right )}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*exp(5*x) - exp(x) + log(x^3)*exp(5*x)*(5*x + 1))/(exp(x) - x*log(x^3)*exp(5*x) + 2),x)

[Out]

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

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sympy [A]  time = 0.85, size = 34, normalized size = 1.79 \begin {gather*} \log {\relax (x )} + \log {\left (e^{5 x} - \frac {e^{x}}{x \log {\left (x^{3} \right )}} - \frac {2}{x \log {\left (x^{3} \right )}} \right )} + \log {\left (\log {\left (x^{3} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+5*x)*exp(5*x)*ln(x**3)+3*exp(5*x)-exp(x))/(x*exp(5*x)*ln(x**3)-2-exp(x)),x)

[Out]

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

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