3.1.58 \(\int \frac {-270 x^2-54 e^x x^2+(405 x^2+e^x (81 x^2-27 x^3)) \log (x)+(25+10 e^x+e^{2 x}) \log ^3(x)}{(25+10 e^x+e^{2 x}) \log ^3(x)} \, dx\)

Optimal. Leaf size=19 \[ -1+x+\frac {27 x^3}{\left (5+e^x\right ) \log ^2(x)} \]

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Rubi [F]  time = 1.70, 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 {-270 x^2-54 e^x x^2+\left (405 x^2+e^x \left (81 x^2-27 x^3\right )\right ) \log (x)+\left (25+10 e^x+e^{2 x}\right ) \log ^3(x)}{\left (25+10 e^x+e^{2 x}\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-270*x^2 - 54*E^x*x^2 + (405*x^2 + E^x*(81*x^2 - 27*x^3))*Log[x] + (25 + 10*E^x + E^(2*x))*Log[x]^3)/((25
+ 10*E^x + E^(2*x))*Log[x]^3),x]

[Out]

x - 54*Defer[Int][x^2/((5 + E^x)*Log[x]^3), x] + 81*Defer[Int][x^2/((5 + E^x)*Log[x]^2), x] + 135*Defer[Int][x
^3/((5 + E^x)^2*Log[x]^2), x] - 27*Defer[Int][x^3/((5 + E^x)*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {54 x^2}{\left (5+e^x\right ) \log ^3(x)}-\frac {27 \left (-15+e^x (-3+x)\right ) x^2}{\left (5+e^x\right )^2 \log ^2(x)}\right ) \, dx\\ &=x-27 \int \frac {\left (-15+e^x (-3+x)\right ) x^2}{\left (5+e^x\right )^2 \log ^2(x)} \, dx-54 \int \frac {x^2}{\left (5+e^x\right ) \log ^3(x)} \, dx\\ &=x-27 \int \left (\frac {(-3+x) x^2}{\left (5+e^x\right ) \log ^2(x)}-\frac {5 x^3}{\left (5+e^x\right )^2 \log ^2(x)}\right ) \, dx-54 \int \frac {x^2}{\left (5+e^x\right ) \log ^3(x)} \, dx\\ &=x-27 \int \frac {(-3+x) x^2}{\left (5+e^x\right ) \log ^2(x)} \, dx-54 \int \frac {x^2}{\left (5+e^x\right ) \log ^3(x)} \, dx+135 \int \frac {x^3}{\left (5+e^x\right )^2 \log ^2(x)} \, dx\\ &=x-27 \int \left (-\frac {3 x^2}{\left (5+e^x\right ) \log ^2(x)}+\frac {x^3}{\left (5+e^x\right ) \log ^2(x)}\right ) \, dx-54 \int \frac {x^2}{\left (5+e^x\right ) \log ^3(x)} \, dx+135 \int \frac {x^3}{\left (5+e^x\right )^2 \log ^2(x)} \, dx\\ &=x-27 \int \frac {x^3}{\left (5+e^x\right ) \log ^2(x)} \, dx-54 \int \frac {x^2}{\left (5+e^x\right ) \log ^3(x)} \, dx+81 \int \frac {x^2}{\left (5+e^x\right ) \log ^2(x)} \, dx+135 \int \frac {x^3}{\left (5+e^x\right )^2 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.37, size = 18, normalized size = 0.95 \begin {gather*} x+\frac {27 x^3}{\left (5+e^x\right ) \log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-270*x^2 - 54*E^x*x^2 + (405*x^2 + E^x*(81*x^2 - 27*x^3))*Log[x] + (25 + 10*E^x + E^(2*x))*Log[x]^3
)/((25 + 10*E^x + E^(2*x))*Log[x]^3),x]

[Out]

x + (27*x^3)/((5 + E^x)*Log[x]^2)

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fricas [A]  time = 0.64, size = 30, normalized size = 1.58 \begin {gather*} \frac {27 \, x^{3} + {\left (x e^{x} + 5 \, x\right )} \log \relax (x)^{2}}{{\left (e^{x} + 5\right )} \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)^2+10*exp(x)+25)*log(x)^3+((-27*x^3+81*x^2)*exp(x)+405*x^2)*log(x)-54*exp(x)*x^2-270*x^2)/(e
xp(x)^2+10*exp(x)+25)/log(x)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.42, size = 38, normalized size = 2.00 \begin {gather*} \frac {x e^{x} \log \relax (x)^{2} + 27 \, x^{3} + 5 \, x \log \relax (x)^{2}}{e^{x} \log \relax (x)^{2} + 5 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)^2+10*exp(x)+25)*log(x)^3+((-27*x^3+81*x^2)*exp(x)+405*x^2)*log(x)-54*exp(x)*x^2-270*x^2)/(e
xp(x)^2+10*exp(x)+25)/log(x)^3,x, algorithm="giac")

[Out]

(x*e^x*log(x)^2 + 27*x^3 + 5*x*log(x)^2)/(e^x*log(x)^2 + 5*log(x)^2)

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maple [A]  time = 0.04, size = 18, normalized size = 0.95

method	result	size
risch	\(x +\frac {27 x^{3}}{\ln \relax (x )^{2} \left ({\mathrm e}^{x}+5\right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)^2+10*exp(x)+25)*ln(x)^3+((-27*x^3+81*x^2)*exp(x)+405*x^2)*ln(x)-54*exp(x)*x^2-270*x^2)/(exp(x)^2+
10*exp(x)+25)/ln(x)^3,x,method=_RETURNVERBOSE)

[Out]

x+27*x^3/ln(x)^2/(exp(x)+5)

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maxima [B]  time = 0.50, size = 38, normalized size = 2.00 \begin {gather*} \frac {x e^{x} \log \relax (x)^{2} + 27 \, x^{3} + 5 \, x \log \relax (x)^{2}}{e^{x} \log \relax (x)^{2} + 5 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)^2+10*exp(x)+25)*log(x)^3+((-27*x^3+81*x^2)*exp(x)+405*x^2)*log(x)-54*exp(x)*x^2-270*x^2)/(e
xp(x)^2+10*exp(x)+25)/log(x)^3,x, algorithm="maxima")

[Out]

(x*e^x*log(x)^2 + 27*x^3 + 5*x*log(x)^2)/(e^x*log(x)^2 + 5*log(x)^2)

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mupad [B]  time = 0.43, size = 212, normalized size = 11.16 \begin {gather*} x+\frac {675\,x^5}{15\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+75\,{\mathrm {e}}^x+125}+\frac {\frac {945\,x^4}{2}-\frac {405\,x^5}{2}}{{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^x+25}+\frac {\frac {27\,x\,\left (3\,x^2\,{\mathrm {e}}^x-x^3\,{\mathrm {e}}^x+15\,x^2\right )}{2\,{\left ({\mathrm {e}}^x+5\right )}^2}-\frac {27\,x\,\ln \relax (x)\,\left (90\,x^2\,{\mathrm {e}}^x-35\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+9\,x^2\,{\mathrm {e}}^{2\,x}-7\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}+225\,x^2\right )}{2\,{\left ({\mathrm {e}}^x+5\right )}^3}}{\ln \relax (x)}+\frac {\frac {27\,x^3}{{\mathrm {e}}^x+5}-\frac {27\,x^3\,\ln \relax (x)\,\left (3\,{\mathrm {e}}^x-x\,{\mathrm {e}}^x+15\right )}{2\,{\left ({\mathrm {e}}^x+5\right )}^2}}{{\ln \relax (x)}^2}+\frac {\frac {27\,x^5}{2}-\frac {189\,x^4}{2}+\frac {243\,x^3}{2}}{{\mathrm {e}}^x+5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(54*x^2*exp(x) - log(x)^3*(exp(2*x) + 10*exp(x) + 25) + 270*x^2 - log(x)*(exp(x)*(81*x^2 - 27*x^3) + 405*
x^2))/(log(x)^3*(exp(2*x) + 10*exp(x) + 25)),x)

[Out]

x + (675*x^5)/(15*exp(2*x) + exp(3*x) + 75*exp(x) + 125) + ((945*x^4)/2 - (405*x^5)/2)/(exp(2*x) + 10*exp(x) +
25) + ((27*x*(3*x^2*exp(x) - x^3*exp(x) + 15*x^2))/(2*(exp(x) + 5)^2) - (27*x*log(x)*(90*x^2*exp(x) - 35*x^3*
exp(x) - 5*x^4*exp(x) + 9*x^2*exp(2*x) - 7*x^3*exp(2*x) + x^4*exp(2*x) + 225*x^2))/(2*(exp(x) + 5)^3))/log(x)
+ ((27*x^3)/(exp(x) + 5) - (27*x^3*log(x)*(3*exp(x) - x*exp(x) + 15))/(2*(exp(x) + 5)^2))/log(x)^2 + ((243*x^3
)/2 - (189*x^4)/2 + (27*x^5)/2)/(exp(x) + 5)

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sympy [A]  time = 0.23, size = 20, normalized size = 1.05 \begin {gather*} \frac {27 x^{3}}{e^{x} \log {\relax (x )}^{2} + 5 \log {\relax (x )}^{2}} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)**2+10*exp(x)+25)*ln(x)**3+((-27*x**3+81*x**2)*exp(x)+405*x**2)*ln(x)-54*exp(x)*x**2-270*x**
2)/(exp(x)**2+10*exp(x)+25)/ln(x)**3,x)

[Out]

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

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