3.99.42 \(\int \frac {e^x (-784-170 x-34 x^2-2 x^3+(196+28 x+x^2) \log (4))+e^x (112+12 x+2 x^2+(-28-2 x) \log (4)) \log (3 x)+e^x (-4+\log (4)) \log ^2(3 x)}{196+28 x+x^2+(-28-2 x) \log (3 x)+\log ^2(3 x)} \, dx\)

Optimal. Leaf size=24 \[ e^x \left (-4+\log (4)+\frac {2 x^2}{-14-x+\log (3 x)}\right ) \]

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Rubi [F]  time = 3.66, 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 (-784-170 x-34 x^2-2 x^3+\left (196+28 x+x^2\right ) \log (4)\right )+e^x \left (112+12 x+2 x^2+(-28-2 x) \log (4)\right ) \log (3 x)+e^x (-4+\log (4)) \log ^2(3 x)}{196+28 x+x^2+(-28-2 x) \log (3 x)+\log ^2(3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-784 - 170*x - 34*x^2 - 2*x^3 + (196 + 28*x + x^2)*Log[4]) + E^x*(112 + 12*x + 2*x^2 + (-28 - 2*x)*L
og[4])*Log[3*x] + E^x*(-4 + Log[4])*Log[3*x]^2)/(196 + 28*x + x^2 + (-28 - 2*x)*Log[3*x] + Log[3*x]^2),x]

[Out]

-(E^x*(4 - Log[4])) + 392*(2 - Log[2])*Defer[Int][E^x/(14 + x - Log[3*x])^2, x] - 196*(4 - Log[4])*Defer[Int][
E^x/(14 + x - Log[3*x])^2, x] - 2*(85 - 28*Log[2])*Defer[Int][(E^x*x)/(14 + x - Log[3*x])^2, x] + 56*(2 - Log[
2])*Defer[Int][(E^x*x)/(14 + x - Log[3*x])^2, x] + 56*(3 - Log[2])*Defer[Int][(E^x*x)/(14 + x - Log[3*x])^2, x
] - 28*(4 - Log[4])*Defer[Int][(E^x*x)/(14 + x - Log[3*x])^2, x] + 28*Defer[Int][(E^x*x^2)/(14 + x - Log[3*x])
^2, x] + 4*(3 - Log[2])*Defer[Int][(E^x*x^2)/(14 + x - Log[3*x])^2, x] - 2*(17 - Log[2])*Defer[Int][(E^x*x^2)/
(14 + x - Log[3*x])^2, x] - (4 - Log[4])*Defer[Int][(E^x*x^2)/(14 + x - Log[3*x])^2, x] + 28*(4 - Log[4])*Defe
r[Int][E^x/(14 + x - Log[3*x]), x] - 4*(3 - Log[2])*Defer[Int][(E^x*x)/(14 + x - Log[3*x]), x] + 2*(4 - Log[4]
)*Defer[Int][(E^x*x)/(14 + x - Log[3*x]), x] - 2*Defer[Int][(E^x*x^2)/(14 + x - Log[3*x]), x] + 56*(2 - Log[2]
)*Defer[Int][E^x/(-14 - x + Log[3*x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-784-170 x-34 x^2-2 x^3+\left (196+28 x+x^2\right ) \log (4)\right )+e^x \left (112+12 x+2 x^2+(-28-2 x) \log (4)\right ) \log (3 x)+e^x (-4+\log (4)) \log ^2(3 x)}{(14+x-\log (3 x))^2} \, dx\\ &=\int \left (-\frac {2 e^x x^3}{(14+x-\log (3 x))^2}-\frac {784 e^x \left (1-\frac {\log (2)}{2}\right )}{(14+x-\log (3 x))^2}-\frac {170 e^x x \left (1-\frac {28 \log (2)}{85}\right )}{(14+x-\log (3 x))^2}-\frac {34 e^x x^2 \left (1-\frac {\log (2)}{17}\right )}{(14+x-\log (3 x))^2}+\frac {2 e^x x^2 \log (3 x)}{(14+x-\log (3 x))^2}+\frac {112 e^x \left (1-\frac {\log (2)}{2}\right ) \log (3 x)}{(14+x-\log (3 x))^2}+\frac {12 e^x x \left (1-\frac {\log (2)}{3}\right ) \log (3 x)}{(14+x-\log (3 x))^2}+\frac {e^x (-4+\log (4)) \log ^2(3 x)}{(14+x-\log (3 x))^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^x x^3}{(14+x-\log (3 x))^2} \, dx\right )+2 \int \frac {e^x x^2 \log (3 x)}{(14+x-\log (3 x))^2} \, dx-(2 (85-28 \log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \frac {e^x \log (3 x)}{(14+x-\log (3 x))^2} \, dx-(392 (2-\log (2))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(4 (3-\log (2))) \int \frac {e^x x \log (3 x)}{(14+x-\log (3 x))^2} \, dx-\left (34 \left (1-\frac {\log (2)}{17}\right )\right ) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx+(-4+\log (4)) \int \frac {e^x \log ^2(3 x)}{(14+x-\log (3 x))^2} \, dx\\ &=2 \int \left (\frac {e^x x^2 (14+x)}{(14+x-\log (3 x))^2}-\frac {e^x x^2}{14+x-\log (3 x)}\right ) \, dx-2 \int \frac {e^x x^3}{(14+x-\log (3 x))^2} \, dx-(2 (85-28 \log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \left (\frac {e^x (14+x)}{(14+x-\log (3 x))^2}+\frac {e^x}{-14-x+\log (3 x)}\right ) \, dx-(392 (2-\log (2))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(4 (3-\log (2))) \int \left (\frac {e^x x (14+x)}{(14+x-\log (3 x))^2}-\frac {e^x x}{14+x-\log (3 x)}\right ) \, dx-\left (34 \left (1-\frac {\log (2)}{17}\right )\right ) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx+(-4+\log (4)) \int \left (e^x+\frac {e^x (14+x)^2}{(14+x-\log (3 x))^2}-\frac {2 e^x (14+x)}{14+x-\log (3 x)}\right ) \, dx\\ &=-\left (2 \int \frac {e^x x^3}{(14+x-\log (3 x))^2} \, dx\right )+2 \int \frac {e^x x^2 (14+x)}{(14+x-\log (3 x))^2} \, dx-2 \int \frac {e^x x^2}{14+x-\log (3 x)} \, dx-(2 (85-28 \log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \frac {e^x (14+x)}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \frac {e^x}{-14-x+\log (3 x)} \, dx-(392 (2-\log (2))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(4 (3-\log (2))) \int \frac {e^x x (14+x)}{(14+x-\log (3 x))^2} \, dx-(4 (3-\log (2))) \int \frac {e^x x}{14+x-\log (3 x)} \, dx-\left (34 \left (1-\frac {\log (2)}{17}\right )\right ) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx+(2 (4-\log (4))) \int \frac {e^x (14+x)}{14+x-\log (3 x)} \, dx+(-4+\log (4)) \int e^x \, dx+(-4+\log (4)) \int \frac {e^x (14+x)^2}{(14+x-\log (3 x))^2} \, dx\\ &=-e^x (4-\log (4))+2 \int \left (\frac {14 e^x x^2}{(14+x-\log (3 x))^2}+\frac {e^x x^3}{(14+x-\log (3 x))^2}\right ) \, dx-2 \int \frac {e^x x^3}{(14+x-\log (3 x))^2} \, dx-2 \int \frac {e^x x^2}{14+x-\log (3 x)} \, dx-(2 (85-28 \log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \left (\frac {14 e^x}{(14+x-\log (3 x))^2}+\frac {e^x x}{(14+x-\log (3 x))^2}\right ) \, dx+(56 (2-\log (2))) \int \frac {e^x}{-14-x+\log (3 x)} \, dx-(392 (2-\log (2))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(4 (3-\log (2))) \int \left (\frac {14 e^x x}{(14+x-\log (3 x))^2}+\frac {e^x x^2}{(14+x-\log (3 x))^2}\right ) \, dx-(4 (3-\log (2))) \int \frac {e^x x}{14+x-\log (3 x)} \, dx-\left (34 \left (1-\frac {\log (2)}{17}\right )\right ) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx+(2 (4-\log (4))) \int \left (\frac {14 e^x}{14+x-\log (3 x)}+\frac {e^x x}{14+x-\log (3 x)}\right ) \, dx+(-4+\log (4)) \int \left (\frac {196 e^x}{(14+x-\log (3 x))^2}+\frac {28 e^x x}{(14+x-\log (3 x))^2}+\frac {e^x x^2}{(14+x-\log (3 x))^2}\right ) \, dx\\ &=-e^x (4-\log (4))-2 \int \frac {e^x x^2}{14+x-\log (3 x)} \, dx+28 \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx-(2 (85-28 \log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(56 (2-\log (2))) \int \frac {e^x}{-14-x+\log (3 x)} \, dx-(392 (2-\log (2))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(784 (2-\log (2))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(4 (3-\log (2))) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx-(4 (3-\log (2))) \int \frac {e^x x}{14+x-\log (3 x)} \, dx+(56 (3-\log (2))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx-\left (34 \left (1-\frac {\log (2)}{17}\right )\right ) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx+(2 (4-\log (4))) \int \frac {e^x x}{14+x-\log (3 x)} \, dx-(28 (4-\log (4))) \int \frac {e^x x}{(14+x-\log (3 x))^2} \, dx+(28 (4-\log (4))) \int \frac {e^x}{14+x-\log (3 x)} \, dx-(196 (4-\log (4))) \int \frac {e^x}{(14+x-\log (3 x))^2} \, dx+(-4+\log (4)) \int \frac {e^x x^2}{(14+x-\log (3 x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.92, size = 24, normalized size = 1.00 \begin {gather*} e^x \left (-4+\log (4)-\frac {2 x^2}{14+x-\log (3 x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-784 - 170*x - 34*x^2 - 2*x^3 + (196 + 28*x + x^2)*Log[4]) + E^x*(112 + 12*x + 2*x^2 + (-28 -
2*x)*Log[4])*Log[3*x] + E^x*(-4 + Log[4])*Log[3*x]^2)/(196 + 28*x + x^2 + (-28 - 2*x)*Log[3*x] + Log[3*x]^2),x
]

[Out]

E^x*(-4 + Log[4] - (2*x^2)/(14 + x - Log[3*x]))

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fricas [A]  time = 0.70, size = 43, normalized size = 1.79 \begin {gather*} -\frac {2 \, {\left ({\left (\log \relax (2) - 2\right )} e^{x} \log \left (3 \, x\right ) + {\left (x^{2} - {\left (x + 14\right )} \log \relax (2) + 2 \, x + 28\right )} e^{x}\right )}}{x - \log \left (3 \, x\right ) + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)-4)*exp(x)*log(3*x)^2+(2*(-2*x-28)*log(2)+2*x^2+12*x+112)*exp(x)*log(3*x)+(2*(x^2+28*x+196
)*log(2)-2*x^3-34*x^2-170*x-784)*exp(x))/(log(3*x)^2+(-2*x-28)*log(3*x)+x^2+28*x+196),x, algorithm="fricas")

[Out]

-2*((log(2) - 2)*e^x*log(3*x) + (x^2 - (x + 14)*log(2) + 2*x + 28)*e^x)/(x - log(3*x) + 14)

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giac [B]  time = 0.18, size = 59, normalized size = 2.46 \begin {gather*} -\frac {2 \, {\left (x^{2} e^{x} - x e^{x} \log \relax (2) + e^{x} \log \relax (2) \log \left (3 \, x\right ) + 2 \, x e^{x} - 14 \, e^{x} \log \relax (2) - 2 \, e^{x} \log \left (3 \, x\right ) + 28 \, e^{x}\right )}}{x - \log \left (3 \, x\right ) + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)-4)*exp(x)*log(3*x)^2+(2*(-2*x-28)*log(2)+2*x^2+12*x+112)*exp(x)*log(3*x)+(2*(x^2+28*x+196
)*log(2)-2*x^3-34*x^2-170*x-784)*exp(x))/(log(3*x)^2+(-2*x-28)*log(3*x)+x^2+28*x+196),x, algorithm="giac")

[Out]

-2*(x^2*e^x - x*e^x*log(2) + e^x*log(2)*log(3*x) + 2*x*e^x - 14*e^x*log(2) - 2*e^x*log(3*x) + 28*e^x)/(x - log
(3*x) + 14)

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maple [A]  time = 0.03, size = 30, normalized size = 1.25

method	result	size
risch	\(2 \,{\mathrm e}^{x} \ln \relax (2)-4 \,{\mathrm e}^{x}-\frac {2 x^{2} {\mathrm e}^{x}}{x -\ln \left (3 x \right )+14}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(2)-4)*exp(x)*ln(3*x)^2+(2*(-2*x-28)*ln(2)+2*x^2+12*x+112)*exp(x)*ln(3*x)+(2*(x^2+28*x+196)*ln(2)-2*
x^3-34*x^2-170*x-784)*exp(x))/(ln(3*x)^2+(-2*x-28)*ln(3*x)+x^2+28*x+196),x,method=_RETURNVERBOSE)

[Out]

2*exp(x)*ln(2)-4*exp(x)-2*x^2*exp(x)/(x-ln(3*x)+14)

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maxima [A]  time = 0.49, size = 47, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left (x^{2} - x {\left (\log \relax (2) - 2\right )} + {\left (\log \relax (3) - 14\right )} \log \relax (2) + {\left (\log \relax (2) - 2\right )} \log \relax (x) - 2 \, \log \relax (3) + 28\right )} e^{x}}{x - \log \relax (3) - \log \relax (x) + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)-4)*exp(x)*log(3*x)^2+(2*(-2*x-28)*log(2)+2*x^2+12*x+112)*exp(x)*log(3*x)+(2*(x^2+28*x+196
)*log(2)-2*x^3-34*x^2-170*x-784)*exp(x))/(log(3*x)^2+(-2*x-28)*log(3*x)+x^2+28*x+196),x, algorithm="maxima")

[Out]

-2*(x^2 - x*(log(2) - 2) + (log(3) - 14)*log(2) + (log(2) - 2)*log(x) - 2*log(3) + 28)*e^x/(x - log(3) - log(x
) + 14)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^x\,\left (2\,\ln \relax (2)-4\right )\,{\ln \left (3\,x\right )}^2+{\mathrm {e}}^x\,\left (12\,x-2\,\ln \relax (2)\,\left (2\,x+28\right )+2\,x^2+112\right )\,\ln \left (3\,x\right )-{\mathrm {e}}^x\,\left (170\,x+34\,x^2+2\,x^3-2\,\ln \relax (2)\,\left (x^2+28\,x+196\right )+784\right )}{28\,x+{\ln \left (3\,x\right )}^2+x^2-\ln \left (3\,x\right )\,\left (2\,x+28\right )+196} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3*x)^2*exp(x)*(2*log(2) - 4) - exp(x)*(170*x + 34*x^2 + 2*x^3 - 2*log(2)*(28*x + x^2 + 196) + 784) +
log(3*x)*exp(x)*(12*x - 2*log(2)*(2*x + 28) + 2*x^2 + 112))/(28*x + log(3*x)^2 + x^2 - log(3*x)*(2*x + 28) + 1
96),x)

[Out]

int((log(3*x)^2*exp(x)*(2*log(2) - 4) - exp(x)*(170*x + 34*x^2 + 2*x^3 - 2*log(2)*(28*x + x^2 + 196) + 784) +
log(3*x)*exp(x)*(12*x - 2*log(2)*(2*x + 28) + 2*x^2 + 112))/(28*x + log(3*x)^2 + x^2 - log(3*x)*(2*x + 28) + 1
96), x)

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sympy [A]  time = 0.43, size = 49, normalized size = 2.04 \begin {gather*} \frac {\left (- 2 x^{2} - 4 x + 2 x \log {\relax (2 )} - 2 \log {\relax (2 )} \log {\left (3 x \right )} + 4 \log {\left (3 x \right )} - 56 + 28 \log {\relax (2 )}\right ) e^{x}}{x - \log {\left (3 x \right )} + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(2)-4)*exp(x)*ln(3*x)**2+(2*(-2*x-28)*ln(2)+2*x**2+12*x+112)*exp(x)*ln(3*x)+(2*(x**2+28*x+196)
*ln(2)-2*x**3-34*x**2-170*x-784)*exp(x))/(ln(3*x)**2+(-2*x-28)*ln(3*x)+x**2+28*x+196),x)

[Out]

(-2*x**2 - 4*x + 2*x*log(2) - 2*log(2)*log(3*x) + 4*log(3*x) - 56 + 28*log(2))*exp(x)/(x - log(3*x) + 14)

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