3.92.76 \(\int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {3}{11+4^{x (4+x)}-\frac {x}{2}} \]

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Rubi [F]  time = 0.90, 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 {6+4^{4 x+x^2} (-48-24 x) \log (4)}{484+4^{1+8 x+2 x^2}+4^{4 x+x^2} (88-4 x)-44 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(6 + 4^(4*x + x^2)*(-48 - 24*x)*Log[4])/(484 + 4^(1 + 8*x + 2*x^2) + 4^(4*x + x^2)*(88 - 4*x) - 44*x + x^2
),x]

[Out]

6*(1 + 88*Log[4])*Defer[Int][(22 + 2^(1 + 8*x + 2*x^2) - x)^(-2), x] - 24*Log[4]*Defer[Int][(22 + 2^(1 + 8*x +
2*x^2) - x)^(-1), x] + 240*Log[4]*Defer[Int][x/(22 + 2^(1 + 8*x + 2*x^2) - x)^2, x] - 12*Log[4]*Defer[Int][x/
(22 + 2^(1 + 8*x + 2*x^2) - x), x] - 12*Log[4]*Defer[Int][x^2/(22 + 2^(1 + 8*x + 2*x^2) - x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {6+4^{4 x+x^2} (-48-24 x) \log (4)}{\left (22+2^{1+8 x+2 x^2}-x\right )^2} \, dx\\ &=\int \left (-\frac {12 (2+x) \log (4)}{22+2^{1+8 x+2 x^2}-x}-\frac {6 \left (-1-88 \log (4)-40 x \log (4)+2 x^2 \log (4)\right )}{\left (22+2^{1+8 x+2 x^2}-x\right )^2}\right ) \, dx\\ &=-\left (6 \int \frac {-1-88 \log (4)-40 x \log (4)+2 x^2 \log (4)}{\left (22+2^{1+8 x+2 x^2}-x\right )^2} \, dx\right )-(12 \log (4)) \int \frac {2+x}{22+2^{1+8 x+2 x^2}-x} \, dx\\ &=-\left (6 \int \left (-\frac {40 x \log (4)}{\left (22+2^{1+8 x+2 x^2}-x\right )^2}+\frac {2 x^2 \log (4)}{\left (22+2^{1+8 x+2 x^2}-x\right )^2}-\frac {1+88 \log (4)}{\left (22+2^{1+8 x+2 x^2}-x\right )^2}\right ) \, dx\right )-(12 \log (4)) \int \left (\frac {2}{22+2^{1+8 x+2 x^2}-x}+\frac {x}{22+2^{1+8 x+2 x^2}-x}\right ) \, dx\\ &=-\left ((12 \log (4)) \int \frac {x}{22+2^{1+8 x+2 x^2}-x} \, dx\right )-(12 \log (4)) \int \frac {x^2}{\left (22+2^{1+8 x+2 x^2}-x\right )^2} \, dx-(24 \log (4)) \int \frac {1}{22+2^{1+8 x+2 x^2}-x} \, dx+(240 \log (4)) \int \frac {x}{\left (22+2^{1+8 x+2 x^2}-x\right )^2} \, dx+(6 (1+88 \log (4))) \int \frac {1}{\left (22+2^{1+8 x+2 x^2}-x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.42, size = 21, normalized size = 1.17 \begin {gather*} \frac {6}{22+2^{1+8 x+2 x^2}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + 4^(4*x + x^2)*(-48 - 24*x)*Log[4])/(484 + 4^(1 + 8*x + 2*x^2) + 4^(4*x + x^2)*(88 - 4*x) - 44*x
+ x^2),x]

[Out]

6/(22 + 2^(1 + 8*x + 2*x^2) - x)

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fricas [A]  time = 0.51, size = 22, normalized size = 1.22 \begin {gather*} \frac {6}{2 \cdot 2^{2 \, x^{2} + 8 \, x} - x + 22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4*x)*log(2))^2+(-4*x+88)*exp(2*(x^2+4*
x)*log(2))+x^2-44*x+484),x, algorithm="fricas")

[Out]

6/(2*2^(2*x^2 + 8*x) - x + 22)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {6 \, {\left (8 \cdot 2^{2 \, x^{2} + 8 \, x} {\left (x + 2\right )} \log \relax (2) - 1\right )}}{4 \cdot 2^{2 \, x^{2} + 8 \, x} {\left (x - 22\right )} - x^{2} - 4 \cdot 2^{4 \, x^{2} + 16 \, x} + 44 \, x - 484}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4*x)*log(2))^2+(-4*x+88)*exp(2*(x^2+4*
x)*log(2))+x^2-44*x+484),x, algorithm="giac")

[Out]

integrate(6*(8*2^(2*x^2 + 8*x)*(x + 2)*log(2) - 1)/(4*2^(2*x^2 + 8*x)*(x - 22) - x^2 - 4*2^(4*x^2 + 16*x) + 44
*x - 484), x)

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maple [A]  time = 0.10, size = 17, normalized size = 0.94

method	result	size
risch	\(-\frac {6}{x -2 \,4^{\left (4+x \right ) x}-22}\)	\(17\)
norman	\(-\frac {6}{x -2 \,{\mathrm e}^{2 \left (x^{2}+4 x \right ) \ln \relax (2)}-22}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(-24*x-48)*ln(2)*exp(2*(x^2+4*x)*ln(2))+6)/(4*exp(2*(x^2+4*x)*ln(2))^2+(-4*x+88)*exp(2*(x^2+4*x)*ln(2))
+x^2-44*x+484),x,method=_RETURNVERBOSE)

[Out]

-6/(x-2*4^((4+x)*x)-22)

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maxima [A]  time = 0.48, size = 23, normalized size = 1.28 \begin {gather*} -\frac {6}{x - 2 \, e^{\left (2 \, x^{2} \log \relax (2) + 8 \, x \log \relax (2)\right )} - 22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-24*x-48)*log(2)*exp(2*(x^2+4*x)*log(2))+6)/(4*exp(2*(x^2+4*x)*log(2))^2+(-4*x+88)*exp(2*(x^2+4*
x)*log(2))+x^2-44*x+484),x, algorithm="maxima")

[Out]

-6/(x - 2*e^(2*x^2*log(2) + 8*x*log(2)) - 22)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {2\,{\mathrm {e}}^{2\,\ln \relax (2)\,\left (x^2+4\,x\right )}\,\ln \relax (2)\,\left (24\,x+48\right )-6}{4\,{\mathrm {e}}^{4\,\ln \relax (2)\,\left (x^2+4\,x\right )}-44\,x-{\mathrm {e}}^{2\,\ln \relax (2)\,\left (x^2+4\,x\right )}\,\left (4\,x-88\right )+x^2+484} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*exp(2*log(2)*(4*x + x^2))*log(2)*(24*x + 48) - 6)/(4*exp(4*log(2)*(4*x + x^2)) - 44*x - exp(2*log(2)*(
4*x + x^2))*(4*x - 88) + x^2 + 484),x)

[Out]

int(-(2*exp(2*log(2)*(4*x + x^2))*log(2)*(24*x + 48) - 6)/(4*exp(4*log(2)*(4*x + x^2)) - 44*x - exp(2*log(2)*(
4*x + x^2))*(4*x - 88) + x^2 + 484), x)

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sympy [A]  time = 0.16, size = 19, normalized size = 1.06 \begin {gather*} \frac {6}{- x + 2 e^{\left (2 x^{2} + 8 x\right ) \log {\relax (2 )}} + 22} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-24*x-48)*ln(2)*exp(2*(x**2+4*x)*ln(2))+6)/(4*exp(2*(x**2+4*x)*ln(2))**2+(-4*x+88)*exp(2*(x**2+4
*x)*ln(2))+x**2-44*x+484),x)

[Out]

6/(-x + 2*exp((2*x**2 + 8*x)*log(2)) + 22)

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