3.32.16 \(\int \frac {e^x (54+108 x-162 x^2-135 x^3+54 x^4)-27 e^x x \log (x^2)}{16 x^3-8 x^4-15 x^5+4 x^6+4 x^7+(8 x^2-2 x^3-4 x^4) \log (x^2)+x \log ^2(x^2)} \, dx\)

Optimal. Leaf size=25 \[ \frac {27 e^x}{-4 x+x^2+2 x^3-\log \left (x^2\right )} \]

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Rubi [F]  time = 1.71, 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 \left (54+108 x-162 x^2-135 x^3+54 x^4\right )-27 e^x x \log \left (x^2\right )}{16 x^3-8 x^4-15 x^5+4 x^6+4 x^7+\left (8 x^2-2 x^3-4 x^4\right ) \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(54 + 108*x - 162*x^2 - 135*x^3 + 54*x^4) - 27*E^x*x*Log[x^2])/(16*x^3 - 8*x^4 - 15*x^5 + 4*x^6 + 4*x
^7 + (8*x^2 - 2*x^3 - 4*x^4)*Log[x^2] + x*Log[x^2]^2),x]

[Out]

108*Defer[Int][E^x/(-4*x + x^2 + 2*x^3 - Log[x^2])^2, x] + 54*Defer[Int][E^x/(x*(-4*x + x^2 + 2*x^3 - Log[x^2]
)^2), x] - 54*Defer[Int][(E^x*x)/(-4*x + x^2 + 2*x^3 - Log[x^2])^2, x] - 162*Defer[Int][(E^x*x^2)/(-4*x + x^2
+ 2*x^3 - Log[x^2])^2, x] + 27*Defer[Int][E^x/(-4*x + x^2 + 2*x^3 - Log[x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27 e^x \left (2+4 x-6 x^2-5 x^3+2 x^4-x \log \left (x^2\right )\right )}{x \left (x \left (-4+x+2 x^2\right )-\log \left (x^2\right )\right )^2} \, dx\\ &=27 \int \frac {e^x \left (2+4 x-6 x^2-5 x^3+2 x^4-x \log \left (x^2\right )\right )}{x \left (x \left (-4+x+2 x^2\right )-\log \left (x^2\right )\right )^2} \, dx\\ &=27 \int \left (-\frac {2 e^x \left (-1-2 x+x^2+3 x^3\right )}{x \left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2}+\frac {e^x}{-4 x+x^2+2 x^3-\log \left (x^2\right )}\right ) \, dx\\ &=27 \int \frac {e^x}{-4 x+x^2+2 x^3-\log \left (x^2\right )} \, dx-54 \int \frac {e^x \left (-1-2 x+x^2+3 x^3\right )}{x \left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2} \, dx\\ &=27 \int \frac {e^x}{-4 x+x^2+2 x^3-\log \left (x^2\right )} \, dx-54 \int \left (-\frac {2 e^x}{\left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2}-\frac {e^x}{x \left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2}+\frac {e^x x}{\left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2}+\frac {3 e^x x^2}{\left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2}\right ) \, dx\\ &=27 \int \frac {e^x}{-4 x+x^2+2 x^3-\log \left (x^2\right )} \, dx+54 \int \frac {e^x}{x \left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2} \, dx-54 \int \frac {e^x x}{\left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2} \, dx+108 \int \frac {e^x}{\left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2} \, dx-162 \int \frac {e^x x^2}{\left (-4 x+x^2+2 x^3-\log \left (x^2\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.81, size = 25, normalized size = 1.00 \begin {gather*} -\frac {27 e^x}{4 x-x^2-2 x^3+\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(54 + 108*x - 162*x^2 - 135*x^3 + 54*x^4) - 27*E^x*x*Log[x^2])/(16*x^3 - 8*x^4 - 15*x^5 + 4*x^6
+ 4*x^7 + (8*x^2 - 2*x^3 - 4*x^4)*Log[x^2] + x*Log[x^2]^2),x]

[Out]

(-27*E^x)/(4*x - x^2 - 2*x^3 + Log[x^2])

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fricas [A]  time = 0.51, size = 24, normalized size = 0.96 \begin {gather*} \frac {27 \, e^{x}}{2 \, x^{3} + x^{2} - 4 \, x - \log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*x*exp(x)*log(x^2)+(54*x^4-135*x^3-162*x^2+108*x+54)*exp(x))/(x*log(x^2)^2+(-4*x^4-2*x^3+8*x^2)*
log(x^2)+4*x^7+4*x^6-15*x^5-8*x^4+16*x^3),x, algorithm="fricas")

[Out]

27*e^x/(2*x^3 + x^2 - 4*x - log(x^2))

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giac [A]  time = 0.22, size = 24, normalized size = 0.96 \begin {gather*} \frac {27 \, e^{x}}{2 \, x^{3} + x^{2} - 4 \, x - \log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*x*exp(x)*log(x^2)+(54*x^4-135*x^3-162*x^2+108*x+54)*exp(x))/(x*log(x^2)^2+(-4*x^4-2*x^3+8*x^2)*
log(x^2)+4*x^7+4*x^6-15*x^5-8*x^4+16*x^3),x, algorithm="giac")

[Out]

27*e^x/(2*x^3 + x^2 - 4*x - log(x^2))

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maple [C]  time = 0.08, size = 74, normalized size = 2.96

method	result	size
risch	\(-\frac {54 i {\mathrm e}^{x}}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}-4 i x^{3}-2 i x^{2}+8 i x +4 i \ln \relax (x )}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-27*x*exp(x)*ln(x^2)+(54*x^4-135*x^3-162*x^2+108*x+54)*exp(x))/(x*ln(x^2)^2+(-4*x^4-2*x^3+8*x^2)*ln(x^2)+
4*x^7+4*x^6-15*x^5-8*x^4+16*x^3),x,method=_RETURNVERBOSE)

[Out]

-54*I*exp(x)/(Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*x^2)^3-4*I*x^3-2*I*x^2+8*I*x+4
*I*ln(x))

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maxima [A]  time = 0.46, size = 22, normalized size = 0.88 \begin {gather*} \frac {27 \, e^{x}}{2 \, x^{3} + x^{2} - 4 \, x - 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*x*exp(x)*log(x^2)+(54*x^4-135*x^3-162*x^2+108*x+54)*exp(x))/(x*log(x^2)^2+(-4*x^4-2*x^3+8*x^2)*
log(x^2)+4*x^7+4*x^6-15*x^5-8*x^4+16*x^3),x, algorithm="maxima")

[Out]

27*e^x/(2*x^3 + x^2 - 4*x - 2*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (54\,x^4-135\,x^3-162\,x^2+108\,x+54\right )-27\,x\,\ln \left (x^2\right )\,{\mathrm {e}}^x}{x\,{\ln \left (x^2\right )}^2-\ln \left (x^2\right )\,\left (4\,x^4+2\,x^3-8\,x^2\right )+16\,x^3-8\,x^4-15\,x^5+4\,x^6+4\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(108*x - 162*x^2 - 135*x^3 + 54*x^4 + 54) - 27*x*log(x^2)*exp(x))/(x*log(x^2)^2 - log(x^2)*(2*x^3
- 8*x^2 + 4*x^4) + 16*x^3 - 8*x^4 - 15*x^5 + 4*x^6 + 4*x^7),x)

[Out]

int((exp(x)*(108*x - 162*x^2 - 135*x^3 + 54*x^4 + 54) - 27*x*log(x^2)*exp(x))/(x*log(x^2)^2 - log(x^2)*(2*x^3
- 8*x^2 + 4*x^4) + 16*x^3 - 8*x^4 - 15*x^5 + 4*x^6 + 4*x^7), x)

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sympy [A]  time = 0.31, size = 20, normalized size = 0.80 \begin {gather*} \frac {27 e^{x}}{2 x^{3} + x^{2} - 4 x - \log {\left (x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*x*exp(x)*ln(x**2)+(54*x**4-135*x**3-162*x**2+108*x+54)*exp(x))/(x*ln(x**2)**2+(-4*x**4-2*x**3+8
*x**2)*ln(x**2)+4*x**7+4*x**6-15*x**5-8*x**4+16*x**3),x)

[Out]

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

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