3.58.60 \(\int \frac {15-15 x+e^{x^2} (-3+3 x+6 x^2)+e^{x+x^2} (-3+6 x+6 x^2)}{-5 x+e^{x^2} x+e^{x+x^2} x} \, dx\)

Optimal. Leaf size=27 \[ 3 \left (x+\log \left (\frac {5-e^{x^2}-e^{x+x^2}}{x}\right )\right ) \]

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Rubi [F]  time = 1.31, 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 {15-15 x+e^{x^2} \left (-3+3 x+6 x^2\right )+e^{x+x^2} \left (-3+6 x+6 x^2\right )}{-5 x+e^{x^2} x+e^{x+x^2} x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(15 - 15*x + E^x^2*(-3 + 3*x + 6*x^2) + E^(x + x^2)*(-3 + 6*x + 6*x^2))/(-5*x + E^x^2*x + E^(x + x^2)*x),x
]

[Out]

3*x + 3*x^2 + 3*Log[1 + E^x] - 3*Log[x] + 15*Defer[Int][E^x/((1 + E^x)*(-5 + E^x^2 + E^(x + x^2))), x] + 30*De
fer[Int][x/((1 + E^x)*(-5 + E^x^2 + E^(x + x^2))), x] + 30*Defer[Int][(E^x*x)/((1 + E^x)*(-5 + E^x^2 + E^(x +
x^2))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {15 \left (e^x+2 x+2 e^x x\right )}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )}+\frac {3 \left (-1-e^x+x+2 e^x x+2 x^2+2 e^x x^2\right )}{\left (1+e^x\right ) x}\right ) \, dx\\ &=3 \int \frac {-1-e^x+x+2 e^x x+2 x^2+2 e^x x^2}{\left (1+e^x\right ) x} \, dx+15 \int \frac {e^x+2 x+2 e^x x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx\\ &=3 \int \left (-\frac {1}{1+e^x}+\frac {-1+2 x+2 x^2}{x}\right ) \, dx+15 \int \left (\frac {e^x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )}+\frac {2 x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )}+\frac {2 e^x x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )}\right ) \, dx\\ &=-\left (3 \int \frac {1}{1+e^x} \, dx\right )+3 \int \frac {-1+2 x+2 x^2}{x} \, dx+15 \int \frac {e^x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {e^x x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx\\ &=3 \int \left (2-\frac {1}{x}+2 x\right ) \, dx-3 \operatorname {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right )+15 \int \frac {e^x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {e^x x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx\\ &=6 x+3 x^2-3 \log (x)-3 \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+3 \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )+15 \int \frac {e^x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {e^x x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx\\ &=3 x+3 x^2+3 \log \left (1+e^x\right )-3 \log (x)+15 \int \frac {e^x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx+30 \int \frac {e^x x}{\left (1+e^x\right ) \left (-5+e^{x^2}+e^{x+x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 27, normalized size = 1.00 \begin {gather*} 3 \left (x+\log \left (5-e^{x^2}-e^{x+x^2}\right )-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 - 15*x + E^x^2*(-3 + 3*x + 6*x^2) + E^(x + x^2)*(-3 + 6*x + 6*x^2))/(-5*x + E^x^2*x + E^(x + x^2
)*x),x]

[Out]

3*(x + Log[5 - E^x^2 - E^(x + x^2)] - Log[x])

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fricas [A]  time = 0.84, size = 23, normalized size = 0.85 \begin {gather*} 3 \, x - 3 \, \log \relax (x) + 3 \, \log \left (e^{\left (x^{2} + x\right )} + e^{\left (x^{2}\right )} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+6*x-3)*exp(x^2+x)+(6*x^2+3*x-3)*exp(x^2)-15*x+15)/(x*exp(x^2+x)+exp(x^2)*x-5*x),x, algorithm
="fricas")

[Out]

3*x - 3*log(x) + 3*log(e^(x^2 + x) + e^(x^2) - 5)

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giac [A]  time = 0.22, size = 23, normalized size = 0.85 \begin {gather*} 3 \, x - 3 \, \log \relax (x) + 3 \, \log \left (e^{\left (x^{2} + x\right )} + e^{\left (x^{2}\right )} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+6*x-3)*exp(x^2+x)+(6*x^2+3*x-3)*exp(x^2)-15*x+15)/(x*exp(x^2+x)+exp(x^2)*x-5*x),x, algorithm
="giac")

[Out]

3*x - 3*log(x) + 3*log(e^(x^2 + x) + e^(x^2) - 5)

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maple [A]  time = 0.05, size = 24, normalized size = 0.89

method	result	size
norman	\(3 x -3 \ln \relax (x )+3 \ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{x^{2}+x}-5\right )\)	\(24\)
risch	\(3 x -3 \ln \relax (x )+3 \ln \left ({\mathrm e}^{x^{2}}+{\mathrm e}^{\left (x +1\right ) x}-5\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^2+6*x-3)*exp(x^2+x)+(6*x^2+3*x-3)*exp(x^2)-15*x+15)/(x*exp(x^2+x)+exp(x^2)*x-5*x),x,method=_RETURNVE
RBOSE)

[Out]

3*x-3*ln(x)+3*ln(exp(x^2)+exp(x^2+x)-5)

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maxima [A]  time = 0.39, size = 36, normalized size = 1.33 \begin {gather*} 3 \, x - 3 \, \log \relax (x) + 3 \, \log \left (\frac {{\left (e^{x} + 1\right )} e^{\left (x^{2}\right )} - 5}{e^{x} + 1}\right ) + 3 \, \log \left (e^{x} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+6*x-3)*exp(x^2+x)+(6*x^2+3*x-3)*exp(x^2)-15*x+15)/(x*exp(x^2+x)+exp(x^2)*x-5*x),x, algorithm
="maxima")

[Out]

3*x - 3*log(x) + 3*log(((e^x + 1)*e^(x^2) - 5)/(e^x + 1)) + 3*log(e^x + 1)

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mupad [B]  time = 0.15, size = 24, normalized size = 0.89 \begin {gather*} 3\,x+3\,\ln \left ({\mathrm {e}}^{x^2}+{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x-5\right )-3\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)*(3*x + 6*x^2 - 3) - 15*x + exp(x + x^2)*(6*x + 6*x^2 - 3) + 15)/(x*exp(x^2) - 5*x + x*exp(x + x^
2)),x)

[Out]

3*x + 3*log(exp(x^2) + exp(x^2)*exp(x) - 5) - 3*log(x)

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sympy [A]  time = 0.29, size = 24, normalized size = 0.89 \begin {gather*} 3 x - 3 \log {\relax (x )} + 3 \log {\left (e^{x^{2}} + e^{x^{2} + x} - 5 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**2+6*x-3)*exp(x**2+x)+(6*x**2+3*x-3)*exp(x**2)-15*x+15)/(x*exp(x**2+x)+exp(x**2)*x-5*x),x)

[Out]

3*x - 3*log(x) + 3*log(exp(x**2) + exp(x**2 + x) - 5)

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