3.6.73 \(\int \frac {e^x (144-12 e-432 x+144 x^2)}{144 e^{2 x} \log (2)+e^x (-24 e-288 x+288 x^2) \log (2)+(e^2+144 x^2-288 x^3+144 x^4+e (24 x-24 x^2)) \log (2)} \, dx\)

Optimal. Leaf size=25 \[ \frac {e^x}{\left (-\frac {e}{12}+e^x-x+x^2\right ) \log (2)} \]

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Rubi [F]  time = 1.41, 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 (144-12 e-432 x+144 x^2\right )}{144 e^{2 x} \log (2)+e^x \left (-24 e-288 x+288 x^2\right ) \log (2)+\left (e^2+144 x^2-288 x^3+144 x^4+e \left (24 x-24 x^2\right )\right ) \log (2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(144 - 12*E - 432*x + 144*x^2))/(144*E^(2*x)*Log[2] + E^x*(-24*E - 288*x + 288*x^2)*Log[2] + (E^2 + 1
44*x^2 - 288*x^3 + 144*x^4 + E*(24*x - 24*x^2))*Log[2]),x]

[Out]

(12*(12 - E)*Defer[Int][E^(1 + x)/(E - 12*E^x + 12*x - 12*x^2)^2, x])/(E*Log[2]) - (432*Defer[Int][(E^x*x)/(-E
+ 12*E^x - 12*x + 12*x^2)^2, x])/Log[2] + (144*Defer[Int][(E^x*x^2)/(-E + 12*E^x - 12*x + 12*x^2)^2, x])/Log[
2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 e^x \left (12-e-36 x+12 x^2\right )}{\left (e-12 e^x-12 (-1+x) x\right )^2 \log (2)} \, dx\\ &=\frac {12 \int \frac {e^x \left (12-e-36 x+12 x^2\right )}{\left (e-12 e^x-12 (-1+x) x\right )^2} \, dx}{\log (2)}\\ &=\frac {12 \int \left (-\frac {\left (1-\frac {12}{e}\right ) e^{1+x}}{\left (e-12 e^x+12 x-12 x^2\right )^2}-\frac {36 e^x x}{\left (-e+12 e^x-12 x+12 x^2\right )^2}+\frac {12 e^x x^2}{\left (-e+12 e^x-12 x+12 x^2\right )^2}\right ) \, dx}{\log (2)}\\ &=\frac {144 \int \frac {e^x x^2}{\left (-e+12 e^x-12 x+12 x^2\right )^2} \, dx}{\log (2)}-\frac {432 \int \frac {e^x x}{\left (-e+12 e^x-12 x+12 x^2\right )^2} \, dx}{\log (2)}+\frac {\left (12 \left (-1+\frac {12}{e}\right )\right ) \int \frac {e^{1+x}}{\left (e-12 e^x+12 x-12 x^2\right )^2} \, dx}{\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 29, normalized size = 1.16 \begin {gather*} -\frac {e-12 (-1+x) x}{\left (e-12 e^x-12 (-1+x) x\right ) \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(144 - 12*E - 432*x + 144*x^2))/(144*E^(2*x)*Log[2] + E^x*(-24*E - 288*x + 288*x^2)*Log[2] + (E
^2 + 144*x^2 - 288*x^3 + 144*x^4 + E*(24*x - 24*x^2))*Log[2]),x]

[Out]

-((E - 12*(-1 + x)*x)/((E - 12*E^x - 12*(-1 + x)*x)*Log[2]))

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fricas [A]  time = 0.69, size = 40, normalized size = 1.60 \begin {gather*} -\frac {12 \, x^{2} - 12 \, x - e}{{\left (12 \, x^{2} - 12 \, x - e\right )} \log \relax (2) + 12 \, e^{x} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*log(2)*exp(x)^2+(-24*exp(1)+288*x^2-288*x)*log(2)*exp(x)+
(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^4-288*x^3+144*x^2)*log(2)),x, algorithm="fricas")

[Out]

-(12*x^2 - 12*x - e)/((12*x^2 - 12*x - e)*log(2) + 12*e^x*log(2))

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giac [A]  time = 0.29, size = 42, normalized size = 1.68 \begin {gather*} -\frac {12 \, x^{2} - 12 \, x - e}{12 \, x^{2} \log \relax (2) - 12 \, x \log \relax (2) - e \log \relax (2) + 12 \, e^{x} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*log(2)*exp(x)^2+(-24*exp(1)+288*x^2-288*x)*log(2)*exp(x)+
(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^4-288*x^3+144*x^2)*log(2)),x, algorithm="giac")

[Out]

-(12*x^2 - 12*x - e)/(12*x^2*log(2) - 12*x*log(2) - e*log(2) + 12*e^x*log(2))

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maple [A]  time = 0.42, size = 26, normalized size = 1.04

method	result	size
norman	\(-\frac {12 \,{\mathrm e}^{x}}{\ln \relax (2) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\)	\(26\)
risch	\(-\frac {-12 x^{2}+{\mathrm e}+12 x}{\ln \relax (2) \left (-12 x^{2}+{\mathrm e}-12 \,{\mathrm e}^{x}+12 x \right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*ln(2)*exp(x)^2+(-24*exp(1)+288*x^2-288*x)*ln(2)*exp(x)+(exp(1)^
2+(-24*x^2+24*x)*exp(1)+144*x^4-288*x^3+144*x^2)*ln(2)),x,method=_RETURNVERBOSE)

[Out]

-12*exp(x)/ln(2)/(-12*x^2+exp(1)-12*exp(x)+12*x)

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maxima [A]  time = 0.55, size = 42, normalized size = 1.68 \begin {gather*} -\frac {12 \, x^{2} - 12 \, x - e}{12 \, x^{2} \log \relax (2) - 12 \, x \log \relax (2) - e \log \relax (2) + 12 \, e^{x} \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(1)+144*x^2-432*x+144)*exp(x)/(144*log(2)*exp(x)^2+(-24*exp(1)+288*x^2-288*x)*log(2)*exp(x)+
(exp(1)^2+(-24*x^2+24*x)*exp(1)+144*x^4-288*x^3+144*x^2)*log(2)),x, algorithm="maxima")

[Out]

-(12*x^2 - 12*x - e)/(12*x^2*log(2) - 12*x*log(2) - e*log(2) + 12*e^x*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {{\mathrm {e}}^x\,\left (-144\,x^2+432\,x+12\,\mathrm {e}-144\right )}{144\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)+\ln \relax (2)\,\left ({\mathrm {e}}^2+\mathrm {e}\,\left (24\,x-24\,x^2\right )+144\,x^2-288\,x^3+144\,x^4\right )-{\mathrm {e}}^x\,\ln \relax (2)\,\left (-288\,x^2+288\,x+24\,\mathrm {e}\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(432*x + 12*exp(1) - 144*x^2 - 144))/(144*exp(2*x)*log(2) + log(2)*(exp(2) + exp(1)*(24*x - 24*x^
2) + 144*x^2 - 288*x^3 + 144*x^4) - exp(x)*log(2)*(288*x + 24*exp(1) - 288*x^2)),x)

[Out]

-int((exp(x)*(432*x + 12*exp(1) - 144*x^2 - 144))/(144*exp(2*x)*log(2) + log(2)*(exp(2) + exp(1)*(24*x - 24*x^
2) + 144*x^2 - 288*x^3 + 144*x^4) - exp(x)*log(2)*(288*x + 24*exp(1) - 288*x^2)), x)

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sympy [B]  time = 0.15, size = 41, normalized size = 1.64 \begin {gather*} \frac {- 12 x^{2} + 12 x + e}{12 x^{2} \log {\relax (2 )} - 12 x \log {\relax (2 )} + 12 e^{x} \log {\relax (2 )} - e \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12*exp(1)+144*x**2-432*x+144)*exp(x)/(144*ln(2)*exp(x)**2+(-24*exp(1)+288*x**2-288*x)*ln(2)*exp(x)
+(exp(1)**2+(-24*x**2+24*x)*exp(1)+144*x**4-288*x**3+144*x**2)*ln(2)),x)

[Out]

(-12*x**2 + 12*x + E)/(12*x**2*log(2) - 12*x*log(2) + 12*exp(x)*log(2) - E*log(2))

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