3.91.69 \(\int \frac {1+2 e^{x^2} x}{e^{x^2}+x} \, dx\)

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

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Rubi [A]  time = 0.03, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {6684} \begin {gather*} \log \left (e^{x^2}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*E^x^2*x)/(E^x^2 + x),x]

[Out]

Log[E^x^2 + 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 (e^{x^2}+x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 8, normalized size = 1.00 \begin {gather*} \log \left (e^{x^2}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*E^x^2*x)/(E^x^2 + x),x]

[Out]

Log[E^x^2 + x]

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fricas [A]  time = 0.58, size = 7, normalized size = 0.88 \begin {gather*} \log \left (x + e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x^2)*x+1)/(exp(x^2)+x),x, algorithm="fricas")

[Out]

log(x + e^(x^2))

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giac [A]  time = 0.22, size = 7, normalized size = 0.88 \begin {gather*} \log \left (x + e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x^2)*x+1)/(exp(x^2)+x),x, algorithm="giac")

[Out]

log(x + e^(x^2))

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maple [A]  time = 0.03, size = 8, normalized size = 1.00

method	result	size
derivativedivides	\(\ln \left ({\mathrm e}^{x^{2}}+x \right )\)	\(8\)
default	\(\ln \left ({\mathrm e}^{x^{2}}+x \right )\)	\(8\)
norman	\(\ln \left ({\mathrm e}^{x^{2}}+x \right )\)	\(8\)
risch	\(\ln \left ({\mathrm e}^{x^{2}}+x \right )\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(x^2)*x+1)/(exp(x^2)+x),x,method=_RETURNVERBOSE)

[Out]

ln(exp(x^2)+x)

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maxima [A]  time = 0.35, size = 7, normalized size = 0.88 \begin {gather*} \log \left (x + e^{\left (x^{2}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x^2)*x+1)/(exp(x^2)+x),x, algorithm="maxima")

[Out]

log(x + e^(x^2))

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mupad [B]  time = 0.11, size = 7, normalized size = 0.88 \begin {gather*} \ln \left (x+{\mathrm {e}}^{x^2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(x^2) + 1)/(x + exp(x^2)),x)

[Out]

log(x + exp(x^2))

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sympy [A]  time = 0.10, size = 7, normalized size = 0.88 \begin {gather*} \log {\left (x + e^{x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*exp(x**2)*x+1)/(exp(x**2)+x),x)

[Out]

log(x + exp(x**2))

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