3.48.53 \(\int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} (-162-240 x-32 x^2+24 x \log (\log (4)))+e^{\frac {12-3 \log (\log (4))}{9+4 x}} (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4)))}{81 x^3+72 x^4+16 x^5} \, dx\)

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

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Rubi [F]  time = 7.80, 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^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{81 x^3+72 x^4+16 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(12 - 3*Log[Log[4]]))/(9 + 4*x))*(-162 - 240*x - 32*x^2 + 24*x*Log[Log[4]]) + E^((12 - 3*Log[Log[4]
])/(9 + 4*x))*(-486*x - 720*x^2 - 96*x^3 + 72*x^2*Log[Log[4]]))/(81*x^3 + 72*x^4 + 16*x^5),x]

[Out]

(16*E^(24/(9 + 4*x)))/(81*Log[4]^(6/(9 + 4*x))) - (8*E^(12/(9 + 4*x)))/(3*Log[4]^(3/(9 + 4*x))) - (8*ExpIntegr
alEi[(3*(4 - Log[Log[4]]))/(9 + 4*x)]*(4 - Log[Log[4]]))/9 + (64*ExpIntegralEi[(6*(4 - Log[Log[4]]))/(9 + 4*x)
]*(4 - Log[Log[4]]))/243 - 2*Defer[Int][E^(24/(9 + 4*x))/(x^3*Log[4]^(6/(9 + 4*x))), x] - (8*(4 - Log[Log[4]])
*Defer[Int][E^(24/(9 + 4*x))/(x^2*Log[4]^(6/(9 + 4*x))), x])/27 + (64*(4 - Log[Log[4]])*Defer[Int][E^(24/(9 +
4*x))/(x*Log[4]^(6/(9 + 4*x))), x])/243 - 6*Defer[Int][E^(12/(9 + 4*x))/(x^2*Log[4]^(3/(9 + 4*x))), x] - (8*(4
- Log[Log[4]])*Defer[Int][E^(12/(9 + 4*x))/(x*Log[4]^(3/(9 + 4*x))), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{x^3 \left (81+72 x+16 x^2\right )} \, dx\\ &=\int \frac {e^{\frac {2 (12-3 \log (\log (4)))}{9+4 x}} \left (-162-240 x-32 x^2+24 x \log (\log (4))\right )+e^{\frac {12-3 \log (\log (4))}{9+4 x}} \left (-486 x-720 x^2-96 x^3+72 x^2 \log (\log (4))\right )}{x^3 (9+4 x)^2} \, dx\\ &=\int \frac {2 e^{\frac {12}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) \left (e^{\frac {12}{9+4 x}}+3 x \log ^{\frac {3}{9+4 x}}(4)\right ) \left (-81-16 x^2-12 x (10-\log (\log (4)))\right )}{x^3 (9+4 x)^2} \, dx\\ &=2 \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) \left (e^{\frac {12}{9+4 x}}+3 x \log ^{\frac {3}{9+4 x}}(4)\right ) \left (-81-16 x^2-12 x (10-\log (\log (4)))\right )}{x^3 (9+4 x)^2} \, dx\\ &=2 \int \left (\frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) \left (-81-16 x^2-12 x (10-\log (\log (4)))\right )}{x^3 (9+4 x)^2}+\frac {3 e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4) \left (-81-16 x^2-12 x (10-\log (\log (4)))\right )}{x^2 (9+4 x)^2}\right ) \, dx\\ &=2 \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) \left (-81-16 x^2-12 x (10-\log (\log (4)))\right )}{x^3 (9+4 x)^2} \, dx+6 \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4) \left (-81-16 x^2-12 x (10-\log (\log (4)))\right )}{x^2 (9+4 x)^2} \, dx\\ &=2 \int \left (-\frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^3}+\frac {4 e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) (-4+\log (\log (4)))}{27 x^2}-\frac {32 e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) (-4+\log (\log (4)))}{243 x}+\frac {64 e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) (-4+\log (\log (4)))}{27 (9+4 x)^2}+\frac {128 e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4) (-4+\log (\log (4)))}{243 (9+4 x)}\right ) \, dx+6 \int \left (-\frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x^2}+\frac {4 e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4) (-4+\log (\log (4)))}{27 x}-\frac {16 e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4) (-4+\log (\log (4)))}{3 (9+4 x)^2}-\frac {16 e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4) (-4+\log (\log (4)))}{27 (9+4 x)}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^3} \, dx\right )-6 \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x^2} \, dx+\frac {1}{243} (64 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x} \, dx-\frac {1}{27} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^2} \, dx-\frac {1}{9} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x} \, dx-\frac {1}{243} (256 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{9+4 x} \, dx+\frac {1}{9} (32 (4-\log (\log (4)))) \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{9+4 x} \, dx-\frac {1}{27} (128 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{(9+4 x)^2} \, dx+(32 (4-\log (\log (4)))) \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{(9+4 x)^2} \, dx\\ &=-\left (2 \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^3} \, dx\right )-6 \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x^2} \, dx+\frac {1}{243} (64 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x} \, dx-\frac {1}{243} (64 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{24/x} \log ^{-\frac {6}{x}}(4)}{x} \, dx,x,9+4 x\right )-\frac {1}{27} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^2} \, dx-\frac {1}{9} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x} \, dx+\frac {1}{9} (8 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{12/x} \log ^{-\frac {3}{x}}(4)}{x} \, dx,x,9+4 x\right )-\frac {1}{27} (32 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{24/x} \log ^{-\frac {6}{x}}(4)}{x^2} \, dx,x,9+4 x\right )+(8 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{12/x} \log ^{-\frac {3}{x}}(4)}{x^2} \, dx,x,9+4 x\right )\\ &=-\left (2 \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^3} \, dx\right )-6 \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x^2} \, dx+\frac {1}{243} (64 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x} \, dx-\frac {1}{243} (64 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{\frac {24-6 \log (\log (4))}{x}}}{x} \, dx,x,9+4 x\right )-\frac {1}{27} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^2} \, dx-\frac {1}{9} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x} \, dx+\frac {1}{9} (8 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{\frac {12-3 \log (\log (4))}{x}}}{x} \, dx,x,9+4 x\right )-\frac {1}{27} (32 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{\frac {24-6 \log (\log (4))}{x}}}{x^2} \, dx,x,9+4 x\right )+(8 (4-\log (\log (4)))) \operatorname {Subst}\left (\int \frac {e^{\frac {12-3 \log (\log (4))}{x}}}{x^2} \, dx,x,9+4 x\right )\\ &=\frac {16}{81} e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)-\frac {8}{3} e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)-\frac {8}{9} \text {Ei}\left (\frac {3 (4-\log (\log (4)))}{9+4 x}\right ) (4-\log (\log (4)))+\frac {64}{243} \text {Ei}\left (\frac {6 (4-\log (\log (4)))}{9+4 x}\right ) (4-\log (\log (4)))-2 \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^3} \, dx-6 \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x^2} \, dx+\frac {1}{243} (64 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x} \, dx-\frac {1}{27} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{x^2} \, dx-\frac {1}{9} (8 (4-\log (\log (4)))) \int \frac {e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.74, size = 61, normalized size = 2.26 \begin {gather*} -2 \left (-\frac {e^{\frac {24}{9+4 x}} \log ^{-\frac {6}{9+4 x}}(4)}{2 x^2}-\frac {3 e^{\frac {12}{9+4 x}} \log ^{-\frac {3}{9+4 x}}(4)}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(12 - 3*Log[Log[4]]))/(9 + 4*x))*(-162 - 240*x - 32*x^2 + 24*x*Log[Log[4]]) + E^((12 - 3*Log[
Log[4]])/(9 + 4*x))*(-486*x - 720*x^2 - 96*x^3 + 72*x^2*Log[Log[4]]))/(81*x^3 + 72*x^4 + 16*x^5),x]

[Out]

-2*(-1/2*E^(24/(9 + 4*x))/(x^2*Log[4]^(6/(9 + 4*x))) - (3*E^(12/(9 + 4*x)))/(x*Log[4]^(3/(9 + 4*x))))

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fricas [A]  time = 0.63, size = 42, normalized size = 1.56 \begin {gather*} \frac {6 \, x e^{\left (-\frac {3 \, {\left (\log \left (2 \, \log \relax (2)\right ) - 4\right )}}{4 \, x + 9}\right )} + e^{\left (-\frac {6 \, {\left (\log \left (2 \, \log \relax (2)\right ) - 4\right )}}{4 \, x + 9}\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x*log(2*log(2))-32*x^2-240*x-162)*exp((-3*log(2*log(2))+12)/(4*x+9))^2+(72*x^2*log(2*log(2))-96
*x^3-720*x^2-486*x)*exp((-3*log(2*log(2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x, algorithm="fricas")

[Out]

(6*x*e^(-3*(log(2*log(2)) - 4)/(4*x + 9)) + e^(-6*(log(2*log(2)) - 4)/(4*x + 9)))/x^2

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giac [B]  time = 4.40, size = 74, normalized size = 2.74 \begin {gather*} \frac {6 \, x e^{\left (\frac {4 \, {\left (x \log \relax (2) + x \log \left (\log \relax (2)\right ) - 4 \, x\right )}}{3 \, {\left (4 \, x + 9\right )}} - \frac {1}{3} \, \log \relax (2) + \frac {4}{3}\right )} \log \relax (2)^{\frac {1}{3}} + e^{\left (\frac {8 \, {\left (x \log \relax (2) + x \log \left (\log \relax (2)\right ) - 4 \, x\right )}}{3 \, {\left (4 \, x + 9\right )}} - \frac {2}{3} \, \log \relax (2) + \frac {8}{3}\right )}}{x^{2} \log \relax (2)^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x*log(2*log(2))-32*x^2-240*x-162)*exp((-3*log(2*log(2))+12)/(4*x+9))^2+(72*x^2*log(2*log(2))-96
*x^3-720*x^2-486*x)*exp((-3*log(2*log(2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x, algorithm="giac")

[Out]

(6*x*e^(4/3*(x*log(2) + x*log(log(2)) - 4*x)/(4*x + 9) - 1/3*log(2) + 4/3)*log(2)^(1/3) + e^(8/3*(x*log(2) + x
*log(log(2)) - 4*x)/(4*x + 9) - 2/3*log(2) + 8/3))/(x^2*log(2)^(2/3))

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maple [A]  time = 0.38, size = 45, normalized size = 1.67

method	result	size
risch	\(\frac {{\mathrm e}^{-\frac {6 \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )-4\right )}{4 x +9}}}{x^{2}}+\frac {6 \,{\mathrm e}^{-\frac {3 \left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )-4\right )}{4 x +9}}}{x}\)	\(45\)
norman	\(\frac {9 \,{\mathrm e}^{\frac {-6 \ln \left (2 \ln \relax (2)\right )+24}{4 x +9}}+54 x \,{\mathrm e}^{\frac {-3 \ln \left (2 \ln \relax (2)\right )+12}{4 x +9}}+4 x \,{\mathrm e}^{\frac {-6 \ln \left (2 \ln \relax (2)\right )+24}{4 x +9}}+24 x^{2} {\mathrm e}^{\frac {-3 \ln \left (2 \ln \relax (2)\right )+12}{4 x +9}}}{x^{2} \left (4 x +9\right )}\)	\(102\)
derivativedivides	\(\text {Expression too large to display}\)	\(8703\)
default	\(\text {Expression too large to display}\)	\(8703\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x*ln(2*ln(2))-32*x^2-240*x-162)*exp((-3*ln(2*ln(2))+12)/(4*x+9))^2+(72*x^2*ln(2*ln(2))-96*x^3-720*x^2
-486*x)*exp((-3*ln(2*ln(2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x,method=_RETURNVERBOSE)

[Out]

1/x^2*exp(-6*(ln(2)+ln(ln(2))-4)/(4*x+9))+6/x*exp(-3*(ln(2)+ln(ln(2))-4)/(4*x+9))

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maxima [B]  time = 0.56, size = 78, normalized size = 2.89 \begin {gather*} \frac {6 \, x e^{\left (\frac {3 \, \log \relax (2)}{4 \, x + 9} - \frac {3 \, \log \left (\log \relax (2)\right )}{4 \, x + 9} + \frac {12}{4 \, x + 9}\right )} + e^{\left (-\frac {6 \, \log \left (\log \relax (2)\right )}{4 \, x + 9} + \frac {24}{4 \, x + 9}\right )}}{2^{\frac {6}{4 \, x + 9}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x*log(2*log(2))-32*x^2-240*x-162)*exp((-3*log(2*log(2))+12)/(4*x+9))^2+(72*x^2*log(2*log(2))-96
*x^3-720*x^2-486*x)*exp((-3*log(2*log(2))+12)/(4*x+9)))/(16*x^5+72*x^4+81*x^3),x, algorithm="maxima")

[Out]

(6*x*e^(3*log(2)/(4*x + 9) - 3*log(log(2))/(4*x + 9) + 12/(4*x + 9)) + e^(-6*log(log(2))/(4*x + 9) + 24/(4*x +
9)))/(2^(6/(4*x + 9))*x^2)

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mupad [B]  time = 0.73, size = 67, normalized size = 2.48 \begin {gather*} \frac {{\mathrm {e}}^{\frac {12}{4\,x+9}}\,\left ({\mathrm {e}}^{\frac {12}{4\,x+9}}+3\,2^{\frac {3}{4\,x+9}+1}\,x\,{\ln \relax (2)}^{\frac {3}{4\,x+9}}\right )\,{\left (\frac {1}{64\,{\ln \relax (2)}^6}\right )}^{\frac {1}{4\,x+9}}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(2*(3*log(2*log(2)) - 12))/(4*x + 9))*(240*x - 24*x*log(2*log(2)) + 32*x^2 + 162) + exp(-(3*log(2*l
og(2)) - 12)/(4*x + 9))*(486*x - 72*x^2*log(2*log(2)) + 720*x^2 + 96*x^3))/(81*x^3 + 72*x^4 + 16*x^5),x)

[Out]

(exp(12/(4*x + 9))*(exp(12/(4*x + 9)) + 3*2^(3/(4*x + 9) + 1)*x*log(2)^(3/(4*x + 9)))*(1/(64*log(2)^6))^(1/(4*
x + 9)))/x^2

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sympy [B]  time = 0.38, size = 42, normalized size = 1.56 \begin {gather*} \frac {6 x^{2} e^{\frac {12 - 3 \log {\left (2 \log {\relax (2 )} \right )}}{4 x + 9}} + x e^{\frac {2 \left (12 - 3 \log {\left (2 \log {\relax (2 )} \right )}\right )}{4 x + 9}}}{x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x*ln(2*ln(2))-32*x**2-240*x-162)*exp((-3*ln(2*ln(2))+12)/(4*x+9))**2+(72*x**2*ln(2*ln(2))-96*x*
*3-720*x**2-486*x)*exp((-3*ln(2*ln(2))+12)/(4*x+9)))/(16*x**5+72*x**4+81*x**3),x)

[Out]

(6*x**2*exp((12 - 3*log(2*log(2)))/(4*x + 9)) + x*exp(2*(12 - 3*log(2*log(2)))/(4*x + 9)))/x**3

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