3.21.1 \(\int \frac {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} (-6+2 x+x^2)}{25+e^{e^{-1+4 x}} (-6+x^2)} \, dx\)

Optimal. Leaf size=21 \[ x+\log \left (-6+25 e^{-e^{-1+4 x}}+x^2\right ) \]

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Rubi [F]  time = 1.98, 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 {25-100 e^{-1+4 x}+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25 - 100*E^(-1 + 4*x) + E^E^(-1 + 4*x)*(-6 + 2*x + x^2))/(25 + E^E^(-1 + 4*x)*(-6 + x^2)),x]

[Out]

x + Log[6 - x^2] - 100*Defer[Int][E^(-1 + 4*x)/(25 + E^E^(-1 + 4*x)*(-6 + x^2)), x] + 25*Defer[Int][1/((Sqrt[6
] - x)*(25 + E^E^(-1 + 4*x)*(-6 + x^2))), x] - 25*Defer[Int][1/((Sqrt[6] + x)*(25 + E^E^(-1 + 4*x)*(-6 + x^2))
), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {100 e^{-1+4 x}}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2}+\frac {25-6 e^{e^{-1+4 x}}+2 e^{e^{-1+4 x}} x+e^{e^{-1+4 x}} x^2}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2}\right ) \, dx\\ &=-\left (100 \int \frac {e^{-1+4 x}}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2} \, dx\right )+\int \frac {25-6 e^{e^{-1+4 x}}+2 e^{e^{-1+4 x}} x+e^{e^{-1+4 x}} x^2}{25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2} \, dx\\ &=-\left (100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\right )+\int \frac {25+e^{e^{-1+4 x}} \left (-6+2 x+x^2\right )}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\\ &=-\left (100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\right )+\int \left (\frac {-6+2 x+x^2}{-6+x^2}-\frac {50 x}{\left (-6+x^2\right ) \left (25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2\right )}\right ) \, dx\\ &=-\left (50 \int \frac {x}{\left (-6+x^2\right ) \left (25-6 e^{e^{-1+4 x}}+e^{e^{-1+4 x}} x^2\right )} \, dx\right )-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx+\int \frac {-6+2 x+x^2}{-6+x^2} \, dx\\ &=-\left (50 \int \frac {x}{\left (-6+x^2\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )} \, dx\right )-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx+\int \left (1+\frac {2 x}{-6+x^2}\right ) \, dx\\ &=x+2 \int \frac {x}{-6+x^2} \, dx-50 \int \left (-\frac {1}{2 \left (\sqrt {6}-x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )}+\frac {1}{2 \left (\sqrt {6}+x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )}\right ) \, dx-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\\ &=x+\log \left (6-x^2\right )+25 \int \frac {1}{\left (\sqrt {6}-x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )} \, dx-25 \int \frac {1}{\left (\sqrt {6}+x\right ) \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right )} \, dx-100 \int \frac {e^{-1+4 x}}{25+e^{e^{-1+4 x}} \left (-6+x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 29, normalized size = 1.38 \begin {gather*} -e^{-1+4 x}+x+\log \left (25+e^{e^{-1+4 x}} \left (-6+x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 - 100*E^(-1 + 4*x) + E^E^(-1 + 4*x)*(-6 + 2*x + x^2))/(25 + E^E^(-1 + 4*x)*(-6 + x^2)),x]

[Out]

-E^(-1 + 4*x) + x + Log[25 + E^E^(-1 + 4*x)*(-6 + x^2)]

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fricas [B]  time = 0.64, size = 40, normalized size = 1.90 \begin {gather*} x - e^{\left (4 \, x - 1\right )} + \log \left (x^{2} - 6\right ) + \log \left (\frac {{\left (x^{2} - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} + 25}{x^{2} - 6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x-6)*exp(exp(4*x-1))-100*exp(4*x-1)+25)/((x^2-6)*exp(exp(4*x-1))+25),x, algorithm="fricas")

[Out]

x - e^(4*x - 1) + log(x^2 - 6) + log(((x^2 - 6)*e^(e^(4*x - 1)) + 25)/(x^2 - 6))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{2} + 2 \, x - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} - 100 \, e^{\left (4 \, x - 1\right )} + 25}{{\left (x^{2} - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} + 25}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x-6)*exp(exp(4*x-1))-100*exp(4*x-1)+25)/((x^2-6)*exp(exp(4*x-1))+25),x, algorithm="giac")

[Out]

integrate(((x^2 + 2*x - 6)*e^(e^(4*x - 1)) - 100*e^(4*x - 1) + 25)/((x^2 - 6)*e^(e^(4*x - 1)) + 25), x)

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maple [A]  time = 0.08, size = 34, normalized size = 1.62

method	result	size
norman	\(x -{\mathrm e}^{4 x -1}+\ln \left ({\mathrm e}^{{\mathrm e}^{4 x -1}} x^{2}-6 \,{\mathrm e}^{{\mathrm e}^{4 x -1}}+25\right )\)	\(34\)
risch	\(x +\ln \left (x^{2}-6\right )-{\mathrm e}^{4 x -1}+\ln \left ({\mathrm e}^{{\mathrm e}^{4 x -1}}+\frac {25}{x^{2}-6}\right )\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+2*x-6)*exp(exp(4*x-1))-100*exp(4*x-1)+25)/((x^2-6)*exp(exp(4*x-1))+25),x,method=_RETURNVERBOSE)

[Out]

x-exp(4*x-1)+ln(exp(exp(4*x-1))*x^2-6*exp(exp(4*x-1))+25)

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maxima [B]  time = 0.55, size = 45, normalized size = 2.14 \begin {gather*} {\left (x e - e^{\left (4 \, x\right )}\right )} e^{\left (-1\right )} + \log \left (x^{2} - 6\right ) + \log \left (\frac {{\left (x^{2} - 6\right )} e^{\left (e^{\left (4 \, x - 1\right )}\right )} + 25}{x^{2} - 6}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+2*x-6)*exp(exp(4*x-1))-100*exp(4*x-1)+25)/((x^2-6)*exp(exp(4*x-1))+25),x, algorithm="maxima")

[Out]

(x*e - e^(4*x))*e^(-1) + log(x^2 - 6) + log(((x^2 - 6)*e^(e^(4*x - 1)) + 25)/(x^2 - 6))

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mupad [B]  time = 1.21, size = 33, normalized size = 1.57 \begin {gather*} x+\ln \left (x^2\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x-1}}-6\,{\mathrm {e}}^{{\mathrm {e}}^{4\,x-1}}+25\right )-{\mathrm {e}}^{4\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(4*x - 1))*(2*x + x^2 - 6) - 100*exp(4*x - 1) + 25)/(exp(exp(4*x - 1))*(x^2 - 6) + 25),x)

[Out]

x + log(x^2*exp(exp(4*x - 1)) - 6*exp(exp(4*x - 1)) + 25) - exp(4*x - 1)

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sympy [A]  time = 0.40, size = 31, normalized size = 1.48 \begin {gather*} x - e^{4 x - 1} + \log {\left (x^{2} - 6 \right )} + \log {\left (e^{e^{4 x - 1}} + \frac {25}{x^{2} - 6} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+2*x-6)*exp(exp(4*x-1))-100*exp(4*x-1)+25)/((x**2-6)*exp(exp(4*x-1))+25),x)

[Out]

x - exp(4*x - 1) + log(x**2 - 6) + log(exp(exp(4*x - 1)) + 25/(x**2 - 6))

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