3.8.70 \(\int \frac {-10 x^2-11 x^3+120 x^6+120 x^7+(3 x^2+3 x^3-40 x^6-40 x^7) \log (x+x^2)}{e^{-20+10 x^4} (9+9 x)+e^{-20+10 x^4} (-6-6 x) \log (x+x^2)+e^{-20+10 x^4} (1+x) \log ^2(x+x^2)} \, dx\)

Optimal. Leaf size=23 \[ \frac {e^{-10 \left (-2+x^4\right )} x^3}{-3+\log (x (1+x))} \]

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Rubi [F]  time = 3.32, 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 {-10 x^2-11 x^3+120 x^6+120 x^7+\left (3 x^2+3 x^3-40 x^6-40 x^7\right ) \log \left (x+x^2\right )}{e^{-20+10 x^4} (9+9 x)+e^{-20+10 x^4} (-6-6 x) \log \left (x+x^2\right )+e^{-20+10 x^4} (1+x) \log ^2\left (x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-10*x^2 - 11*x^3 + 120*x^6 + 120*x^7 + (3*x^2 + 3*x^3 - 40*x^6 - 40*x^7)*Log[x + x^2])/(E^(-20 + 10*x^4)*
(9 + 9*x) + E^(-20 + 10*x^4)*(-6 - 6*x)*Log[x + x^2] + E^(-20 + 10*x^4)*(1 + x)*Log[x + x^2]^2),x]

[Out]

-Defer[Int][E^(20 - 10*x^4)/(-3 + Log[x*(1 + x)])^2, x] + Defer[Int][(E^(20 - 10*x^4)*x)/(-3 + Log[x*(1 + x)])
^2, x] - 2*Defer[Int][(E^(20 - 10*x^4)*x^2)/(-3 + Log[x*(1 + x)])^2, x] + Defer[Int][E^(20 - 10*x^4)/((1 + x)*
(-3 + Log[x*(1 + x)])^2), x] + 3*Defer[Int][(E^(20 - 10*x^4)*x^2)/(-3 + Log[x*(1 + x)]), x] - 40*Defer[Int][(E
^(20 - 10*x^4)*x^6)/(-3 + Log[x*(1 + x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{20-10 x^4} x^2 \left (-10-11 x+120 x^4+120 x^5-\left (-3-3 x+40 x^4+40 x^5\right ) \log (x (1+x))\right )}{(1+x) (3-\log (x (1+x)))^2} \, dx\\ &=\int \left (\frac {e^{20-10 x^4} x^2 \left (-3+40 x^4\right )}{3-\log (x (1+x))}-\frac {e^{20-10 x^4} x^2 (1+2 x)}{(1+x) (-3+\log (x (1+x)))^2}\right ) \, dx\\ &=\int \frac {e^{20-10 x^4} x^2 \left (-3+40 x^4\right )}{3-\log (x (1+x))} \, dx-\int \frac {e^{20-10 x^4} x^2 (1+2 x)}{(1+x) (-3+\log (x (1+x)))^2} \, dx\\ &=-\int \left (\frac {e^{20-10 x^4}}{(-3+\log (x (1+x)))^2}-\frac {e^{20-10 x^4} x}{(-3+\log (x (1+x)))^2}+\frac {2 e^{20-10 x^4} x^2}{(-3+\log (x (1+x)))^2}-\frac {e^{20-10 x^4}}{(1+x) (-3+\log (x (1+x)))^2}\right ) \, dx+\int \left (\frac {3 e^{20-10 x^4} x^2}{-3+\log (x (1+x))}-\frac {40 e^{20-10 x^4} x^6}{-3+\log (x (1+x))}\right ) \, dx\\ &=-\left (2 \int \frac {e^{20-10 x^4} x^2}{(-3+\log (x (1+x)))^2} \, dx\right )+3 \int \frac {e^{20-10 x^4} x^2}{-3+\log (x (1+x))} \, dx-40 \int \frac {e^{20-10 x^4} x^6}{-3+\log (x (1+x))} \, dx-\int \frac {e^{20-10 x^4}}{(-3+\log (x (1+x)))^2} \, dx+\int \frac {e^{20-10 x^4} x}{(-3+\log (x (1+x)))^2} \, dx+\int \frac {e^{20-10 x^4}}{(1+x) (-3+\log (x (1+x)))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 23, normalized size = 1.00 \begin {gather*} \frac {e^{20-10 x^4} x^3}{-3+\log (x (1+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10*x^2 - 11*x^3 + 120*x^6 + 120*x^7 + (3*x^2 + 3*x^3 - 40*x^6 - 40*x^7)*Log[x + x^2])/(E^(-20 + 10
*x^4)*(9 + 9*x) + E^(-20 + 10*x^4)*(-6 - 6*x)*Log[x + x^2] + E^(-20 + 10*x^4)*(1 + x)*Log[x + x^2]^2),x]

[Out]

(E^(20 - 10*x^4)*x^3)/(-3 + Log[x*(1 + x)])

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fricas [A]  time = 0.93, size = 32, normalized size = 1.39 \begin {gather*} \frac {x^{3}}{e^{\left (10 \, x^{4} - 20\right )} \log \left (x^{2} + x\right ) - 3 \, e^{\left (10 \, x^{4} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^7-40*x^6+3*x^3+3*x^2)*log(x^2+x)+120*x^7+120*x^6-11*x^3-10*x^2)/((x+1)*exp(5*x^4-10)^2*log(x
^2+x)^2+(-6*x-6)*exp(5*x^4-10)^2*log(x^2+x)+(9*x+9)*exp(5*x^4-10)^2),x, algorithm="fricas")

[Out]

x^3/(e^(10*x^4 - 20)*log(x^2 + x) - 3*e^(10*x^4 - 20))

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giac [A]  time = 0.52, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^{3} e^{\left (-10 \, x^{4} + 20\right )}}{\log \left (x^{2} + x\right ) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^7-40*x^6+3*x^3+3*x^2)*log(x^2+x)+120*x^7+120*x^6-11*x^3-10*x^2)/((x+1)*exp(5*x^4-10)^2*log(x
^2+x)^2+(-6*x-6)*exp(5*x^4-10)^2*log(x^2+x)+(9*x+9)*exp(5*x^4-10)^2),x, algorithm="giac")

[Out]

x^3*e^(-10*x^4 + 20)/(log(x^2 + x) - 3)

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maple [C]  time = 0.09, size = 104, normalized size = 4.52

method	result	size
risch	\(\frac {2 i x^{3} {\mathrm e}^{-10 x^{4}+20}}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x +1\right )\right ) \mathrm {csgn}\left (i x \left (x +1\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x +1\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x +1\right )\right ) \mathrm {csgn}\left (i x \left (x +1\right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (x +1\right )\right )^{3}+2 i \ln \relax (x )+2 i \ln \left (x +1\right )-6 i}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*x^7-40*x^6+3*x^3+3*x^2)*ln(x^2+x)+120*x^7+120*x^6-11*x^3-10*x^2)/((x+1)*exp(5*x^4-10)^2*ln(x^2+x)^2+
(-6*x-6)*exp(5*x^4-10)^2*ln(x^2+x)+(9*x+9)*exp(5*x^4-10)^2),x,method=_RETURNVERBOSE)

[Out]

2*I*x^3/(Pi*csgn(I*x)*csgn(I*(x+1))*csgn(I*x*(x+1))-Pi*csgn(I*x)*csgn(I*x*(x+1))^2-Pi*csgn(I*(x+1))*csgn(I*x*(
x+1))^2+Pi*csgn(I*x*(x+1))^3+2*I*ln(x)+2*I*ln(x+1)-6*I)*exp(-10*x^4+20)

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maxima [A]  time = 0.68, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^{3} e^{\left (-10 \, x^{4} + 20\right )}}{\log \left (x + 1\right ) + \log \relax (x) - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^7-40*x^6+3*x^3+3*x^2)*log(x^2+x)+120*x^7+120*x^6-11*x^3-10*x^2)/((x+1)*exp(5*x^4-10)^2*log(x
^2+x)^2+(-6*x-6)*exp(5*x^4-10)^2*log(x^2+x)+(9*x+9)*exp(5*x^4-10)^2),x, algorithm="maxima")

[Out]

x^3*e^(-10*x^4 + 20)/(log(x + 1) + log(x) - 3)

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mupad [B]  time = 0.96, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^3\,{\mathrm {e}}^{20}\,{\mathrm {e}}^{-10\,x^4}}{\ln \left (x^2+x\right )-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x + x^2)*(3*x^2 + 3*x^3 - 40*x^6 - 40*x^7) - 10*x^2 - 11*x^3 + 120*x^6 + 120*x^7)/(exp(10*x^4 - 20)*(
9*x + 9) - log(x + x^2)*exp(10*x^4 - 20)*(6*x + 6) + log(x + x^2)^2*exp(10*x^4 - 20)*(x + 1)),x)

[Out]

(x^3*exp(20)*exp(-10*x^4))/(log(x + x^2) - 3)

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sympy [A]  time = 0.40, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^{3} e^{20 - 10 x^{4}}}{\log {\left (x^{2} + x \right )} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x**7-40*x**6+3*x**3+3*x**2)*ln(x**2+x)+120*x**7+120*x**6-11*x**3-10*x**2)/((x+1)*exp(5*x**4-10
)**2*ln(x**2+x)**2+(-6*x-6)*exp(5*x**4-10)**2*ln(x**2+x)+(9*x+9)*exp(5*x**4-10)**2),x)

[Out]

x**3*exp(20 - 10*x**4)/(log(x**2 + x) - 3)

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