3.37.61 \(\int \frac {-4+160 x-416 x^2-96 x^3}{x \log ^2(e^{-80 x+104 x^2+16 x^3} x^2)} \, dx\)

Optimal. Leaf size=26 \[ \frac {2}{\log \left (e^{8 x \left (x+2 \left (-5+6 x+x^2\right )\right )} x^2\right )} \]

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Rubi [A]  time = 0.06, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6686} \begin {gather*} \frac {2}{\log \left (e^{16 x^3+104 x^2-80 x} x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 160*x - 416*x^2 - 96*x^3)/(x*Log[E^(-80*x + 104*x^2 + 16*x^3)*x^2]^2),x]

[Out]

2/Log[E^(-80*x + 104*x^2 + 16*x^3)*x^2]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {2}{\log \left (e^{-80 x+104 x^2+16 x^3} x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 24, normalized size = 0.92 \begin {gather*} \frac {2}{\log \left (e^{8 x \left (-10+13 x+2 x^2\right )} x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 160*x - 416*x^2 - 96*x^3)/(x*Log[E^(-80*x + 104*x^2 + 16*x^3)*x^2]^2),x]

[Out]

2/Log[E^(8*x*(-10 + 13*x + 2*x^2))*x^2]

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fricas [A]  time = 0.61, size = 24, normalized size = 0.92 \begin {gather*} \frac {2}{\log \left (x^{2} e^{\left (16 \, x^{3} + 104 \, x^{2} - 80 \, x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^3-416*x^2+160*x-4)/x/log(x^2*exp(8*x^3+52*x^2-40*x)^2)^2,x, algorithm="fricas")

[Out]

2/log(x^2*e^(16*x^3 + 104*x^2 - 80*x))

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giac [A]  time = 0.88, size = 22, normalized size = 0.85 \begin {gather*} \frac {2}{16 \, x^{3} + 104 \, x^{2} - 80 \, x + \log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^3-416*x^2+160*x-4)/x/log(x^2*exp(8*x^3+52*x^2-40*x)^2)^2,x, algorithm="giac")

[Out]

2/(16*x^3 + 104*x^2 - 80*x + log(x^2))

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maple [A]  time = 0.17, size = 27, normalized size = 1.04

method	result	size
default	\(\frac {2}{\ln \left (x^{2} {\mathrm e}^{16 x^{3}+104 x^{2}-80 x}\right )}\)	\(27\)
risch	\(\frac {4 i}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{4 \left (2 x^{2}+13 x -10\right ) x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )-2 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{4 \left (2 x^{2}+13 x -10\right ) x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )^{3}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{8 \left (2 x^{2}+13 x -10\right ) x}\right )^{3}+4 i \ln \relax (x )+4 i \ln \left ({\mathrm e}^{4 \left (2 x^{2}+13 x -10\right ) x}\right )}\)	\(327\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-96*x^3-416*x^2+160*x-4)/x/ln(x^2*exp(8*x^3+52*x^2-40*x)^2)^2,x,method=_RETURNVERBOSE)

[Out]

2/ln(x^2*exp(8*x^3+52*x^2-40*x)^2)

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maxima [A]  time = 0.76, size = 18, normalized size = 0.69 \begin {gather*} \frac {1}{8 \, x^{3} + 52 \, x^{2} - 40 \, x + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^3-416*x^2+160*x-4)/x/log(x^2*exp(8*x^3+52*x^2-40*x)^2)^2,x, algorithm="maxima")

[Out]

1/(8*x^3 + 52*x^2 - 40*x + log(x))

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mupad [B]  time = 2.51, size = 22, normalized size = 0.85 \begin {gather*} \frac {2}{\ln \left (x^2\right )-80\,x+104\,x^2+16\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(416*x^2 - 160*x + 96*x^3 + 4)/(x*log(x^2*exp(104*x^2 - 80*x + 16*x^3))^2),x)

[Out]

2/(log(x^2) - 80*x + 104*x^2 + 16*x^3)

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sympy [A]  time = 0.35, size = 20, normalized size = 0.77 \begin {gather*} \frac {2}{\log {\left (x^{2} e^{16 x^{3} + 104 x^{2} - 80 x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x**3-416*x**2+160*x-4)/x/ln(x**2*exp(8*x**3+52*x**2-40*x)**2)**2,x)

[Out]

2/log(x**2*exp(16*x**3 + 104*x**2 - 80*x))

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