3.30.17 \(\int \frac {-1-10 e^{\frac {x^3}{3}} x^2+2 e^{\frac {2 x^3}{3}} x^2}{25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x} \, dx\)

Optimal. Leaf size=18 \[ \log \left (\left (-5+e^{\frac {x^3}{3}}\right )^2-x\right ) \]

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Rubi [A]  time = 0.07, antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 1, number of rules used = 1, integrand size = 58, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {6684} \begin {gather*} \log \left (-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x+25\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 10*E^(x^3/3)*x^2 + 2*E^((2*x^3)/3)*x^2)/(25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x),x]

[Out]

Log[25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x]

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} &=\log \left (25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.20, size = 26, normalized size = 1.44 \begin {gather*} \log \left (25-10 e^{\frac {x^3}{3}}+e^{\frac {2 x^3}{3}}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 10*E^(x^3/3)*x^2 + 2*E^((2*x^3)/3)*x^2)/(25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x),x]

[Out]

Log[25 - 10*E^(x^3/3) + E^((2*x^3)/3) - x]

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fricas [A]  time = 0.69, size = 20, normalized size = 1.11 \begin {gather*} \log \left (-x + e^{\left (\frac {2}{3} \, x^{3}\right )} - 10 \, e^{\left (\frac {1}{3} \, x^{3}\right )} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x, algorithm="fri
cas")

[Out]

log(-x + e^(2/3*x^3) - 10*e^(1/3*x^3) + 25)

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giac [A]  time = 0.27, size = 20, normalized size = 1.11 \begin {gather*} \log \left (x - e^{\left (\frac {2}{3} \, x^{3}\right )} + 10 \, e^{\left (\frac {1}{3} \, x^{3}\right )} - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x, algorithm="gia
c")

[Out]

log(x - e^(2/3*x^3) + 10*e^(1/3*x^3) - 25)

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maple [A]  time = 0.04, size = 21, normalized size = 1.17

method	result	size
risch	\(\ln \left ({\mathrm e}^{\frac {2 x^{3}}{3}}-10 \,{\mathrm e}^{\frac {x^{3}}{3}}-x +25\right )\)	\(21\)
derivativedivides	\(\ln \left ({\mathrm e}^{\frac {2 x^{3}}{3}}-10 \,{\mathrm e}^{\frac {x^{3}}{3}}-x +25\right )\)	\(23\)
default	\(\ln \left ({\mathrm e}^{\frac {2 x^{3}}{3}}-10 \,{\mathrm e}^{\frac {x^{3}}{3}}-x +25\right )\)	\(23\)
norman	\(\ln \left (-{\mathrm e}^{\frac {2 x^{3}}{3}}+x +10 \,{\mathrm e}^{\frac {x^{3}}{3}}-25\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x,method=_RETURNVERBOSE
)

[Out]

ln(exp(2/3*x^3)-10*exp(1/3*x^3)-x+25)

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maxima [A]  time = 0.56, size = 20, normalized size = 1.11 \begin {gather*} \log \left (x - e^{\left (\frac {2}{3} \, x^{3}\right )} + 10 \, e^{\left (\frac {1}{3} \, x^{3}\right )} - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2*exp(1/3*x^3)^2-10*x^2*exp(1/3*x^3)-1)/(exp(1/3*x^3)^2-10*exp(1/3*x^3)-x+25),x, algorithm="max
ima")

[Out]

log(x - e^(2/3*x^3) + 10*e^(1/3*x^3) - 25)

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mupad [B]  time = 0.14, size = 20, normalized size = 1.11 \begin {gather*} \ln \left (x+10\,{\mathrm {e}}^{\frac {x^3}{3}}-{\mathrm {e}}^{\frac {2\,x^3}{3}}-25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^2*exp(x^3/3) - 2*x^2*exp((2*x^3)/3) + 1)/(x + 10*exp(x^3/3) - exp((2*x^3)/3) - 25),x)

[Out]

log(x + 10*exp(x^3/3) - exp((2*x^3)/3) - 25)

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sympy [A]  time = 0.14, size = 20, normalized size = 1.11 \begin {gather*} \log {\left (- x + e^{\frac {2 x^{3}}{3}} - 10 e^{\frac {x^{3}}{3}} + 25 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2*exp(1/3*x**3)**2-10*x**2*exp(1/3*x**3)-1)/(exp(1/3*x**3)**2-10*exp(1/3*x**3)-x+25),x)

[Out]

log(-x + exp(2*x**3/3) - 10*exp(x**3/3) + 25)

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