3.93.92 \(\int \frac {-6-4 e^{50-2 x}+8 x^3+e^{25-x} (8 x-4 x^2)}{-3+e^{50-2 x}-3 x+2 e^{25-x} x^2+x^4} \, dx\)

Optimal. Leaf size=23 \[ \log \left (\left (3+3 x-\left (e^{25-x}+x^2\right )^2\right )^2\right ) \]

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Rubi [A]  time = 0.09, antiderivative size = 34, normalized size of antiderivative = 1.48, number of steps used = 1, number of rules used = 1, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6684} \begin {gather*} 2 \log \left (-x^4-2 e^{25-x} x^2+3 x-e^{50-2 x}+3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6 - 4*E^(50 - 2*x) + 8*x^3 + E^(25 - x)*(8*x - 4*x^2))/(-3 + E^(50 - 2*x) - 3*x + 2*E^(25 - x)*x^2 + x^4
),x]

[Out]

2*Log[3 - E^(50 - 2*x) + 3*x - 2*E^(25 - x)*x^2 - x^4]

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 \log \left (3-e^{50-2 x}+3 x-2 e^{25-x} x^2-x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.29, size = 45, normalized size = 1.96 \begin {gather*} -4 x+2 \log \left (e^{50}-3 e^{2 x}-3 e^{2 x} x+2 e^{25+x} x^2+e^{2 x} x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 - 4*E^(50 - 2*x) + 8*x^3 + E^(25 - x)*(8*x - 4*x^2))/(-3 + E^(50 - 2*x) - 3*x + 2*E^(25 - x)*x^2
+ x^4),x]

[Out]

-4*x + 2*Log[E^50 - 3*E^(2*x) - 3*E^(2*x)*x + 2*E^(25 + x)*x^2 + E^(2*x)*x^4]

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fricas [A]  time = 0.71, size = 28, normalized size = 1.22 \begin {gather*} 2 \, \log \left (x^{4} + 2 \, x^{2} e^{\left (-x + 25\right )} - 3 \, x + e^{\left (-2 \, x + 50\right )} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+25)^2+(-4*x^2+8*x)*exp(-x+25)+8*x^3-6)/(exp(-x+25)^2+2*x^2*exp(-x+25)+x^4-3*x-3),x, algor
ithm="fricas")

[Out]

2*log(x^4 + 2*x^2*e^(-x + 25) - 3*x + e^(-2*x + 50) - 3)

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giac [A]  time = 0.27, size = 40, normalized size = 1.74 \begin {gather*} -4 \, x + 2 \, \log \left (x^{4} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{\left (x + 25\right )} - 3 \, x e^{\left (2 \, x\right )} + e^{50} - 3 \, e^{\left (2 \, x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+25)^2+(-4*x^2+8*x)*exp(-x+25)+8*x^3-6)/(exp(-x+25)^2+2*x^2*exp(-x+25)+x^4-3*x-3),x, algor
ithm="giac")

[Out]

-4*x + 2*log(x^4*e^(2*x) + 2*x^2*e^(x + 25) - 3*x*e^(2*x) + e^50 - 3*e^(2*x))

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maple [A]  time = 0.14, size = 31, normalized size = 1.35

method	result	size
norman	\(2 \ln \left ({\mathrm e}^{-2 x +50}+2 x^{2} {\mathrm e}^{-x +25}+x^{4}-3 x -3\right )\)	\(31\)
risch	\(-100+2 \ln \left ({\mathrm e}^{-2 x +50}+2 x^{2} {\mathrm e}^{-x +25}+x^{4}-3 x -3\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-4*exp(-x+25)^2+(-4*x^2+8*x)*exp(-x+25)+8*x^3-6)/(exp(-x+25)^2+2*x^2*exp(-x+25)+x^4-3*x-3),x,method=_RETU
RNVERBOSE)

[Out]

2*ln(exp(-x+25)^2+2*x^2*exp(-x+25)+x^4-3*x-3)

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maxima [B]  time = 0.48, size = 54, normalized size = 2.35 \begin {gather*} -4 \, x + 2 \, \log \left (x^{4} - 3 \, x - 3\right ) + 2 \, \log \left (\frac {2 \, x^{2} e^{\left (x + 25\right )} + {\left (x^{4} - 3 \, x - 3\right )} e^{\left (2 \, x\right )} + e^{50}}{x^{4} - 3 \, x - 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+25)^2+(-4*x^2+8*x)*exp(-x+25)+8*x^3-6)/(exp(-x+25)^2+2*x^2*exp(-x+25)+x^4-3*x-3),x, algor
ithm="maxima")

[Out]

-4*x + 2*log(x^4 - 3*x - 3) + 2*log((2*x^2*e^(x + 25) + (x^4 - 3*x - 3)*e^(2*x) + e^50)/(x^4 - 3*x - 3))

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mupad [B]  time = 0.26, size = 28, normalized size = 1.22 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^{50-2\,x}-3\,x+2\,x^2\,{\mathrm {e}}^{25-x}+x^4-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*exp(50 - 2*x) - exp(25 - x)*(8*x - 4*x^2) - 8*x^3 + 6)/(exp(50 - 2*x) - 3*x + 2*x^2*exp(25 - x) + x^4
- 3),x)

[Out]

2*log(exp(50 - 2*x) - 3*x + 2*x^2*exp(25 - x) + x^4 - 3)

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sympy [A]  time = 0.19, size = 27, normalized size = 1.17 \begin {gather*} 2 \log {\left (x^{4} + 2 x^{2} e^{25 - x} - 3 x + e^{50 - 2 x} - 3 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-4*exp(-x+25)**2+(-4*x**2+8*x)*exp(-x+25)+8*x**3-6)/(exp(-x+25)**2+2*x**2*exp(-x+25)+x**4-3*x-3),x)

[Out]

2*log(x**4 + 2*x**2*exp(25 - x) - 3*x + exp(50 - 2*x) - 3)

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