3.66.2 \(\int \frac {-25-25 e^{\frac {6}{e^{20}}}-26 x^2}{25 x+25 e^{\frac {6}{e^{20}}} x-26 x^3} \, dx\)

Optimal. Leaf size=25 \[ \log \left (-x+\frac {1+e^{\frac {6}{e^{20}}}-\frac {x^2}{25}}{x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6, 1593, 446, 72} \begin {gather*} \log \left (25 \left (1+e^{\frac {6}{e^{20}}}\right )-26 x^2\right )-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25 - 25*E^(6/E^20) - 26*x^2)/(25*x + 25*E^(6/E^20)*x - 26*x^3),x]

[Out]

-Log[x] + Log[25*(1 + E^(6/E^20)) - 26*x^2]

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-25-25 e^{\frac {6}{e^{20}}}-26 x^2}{\left (25+25 e^{\frac {6}{e^{20}}}\right ) x-26 x^3} \, dx\\ &=\int \frac {-25-25 e^{\frac {6}{e^{20}}}-26 x^2}{x \left (25+25 e^{\frac {6}{e^{20}}}-26 x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-25-25 e^{\frac {6}{e^{20}}}-26 x}{\left (25+25 e^{\frac {6}{e^{20}}}-26 x\right ) x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {52}{25+25 e^{\frac {6}{e^{20}}}-26 x}-\frac {1}{x}\right ) \, dx,x,x^2\right )\\ &=-\log (x)+\log \left (25 \left (1+e^{\frac {6}{e^{20}}}\right )-26 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 22, normalized size = 0.88 \begin {gather*} -\log (x)+\log \left (25+25 e^{\frac {6}{e^{20}}}-26 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 - 25*E^(6/E^20) - 26*x^2)/(25*x + 25*E^(6/E^20)*x - 26*x^3),x]

[Out]

-Log[x] + Log[25 + 25*E^(6/E^20) - 26*x^2]

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fricas [A]  time = 0.76, size = 20, normalized size = 0.80 \begin {gather*} \log \left (26 \, x^{2} - 25 \, e^{\left (6 \, e^{\left (-20\right )}\right )} - 25\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(6/exp(5)^4)-26*x^2-25)/(25*x*exp(6/exp(5)^4)-26*x^3+25*x),x, algorithm="fricas")

[Out]

log(26*x^2 - 25*e^(6*e^(-20)) - 25) - log(x)

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giac [A]  time = 0.14, size = 23, normalized size = 0.92 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2}\right ) + \log \left ({\left | 26 \, x^{2} - 25 \, e^{\left (6 \, e^{\left (-20\right )}\right )} - 25 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(6/exp(5)^4)-26*x^2-25)/(25*x*exp(6/exp(5)^4)-26*x^3+25*x),x, algorithm="giac")

[Out]

-1/2*log(x^2) + log(abs(26*x^2 - 25*e^(6*e^(-20)) - 25))

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maple [A]  time = 0.47, size = 21, normalized size = 0.84

method	result	size
default	\(-\ln \relax (x )+\ln \left (26 x^{2}-25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}}-25\right )\)	\(21\)
risch	\(-\ln \relax (x )+\ln \left (26 x^{2}-25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}}-25\right )\)	\(21\)
norman	\(-\ln \relax (x )+\ln \left (-26 x^{2}+25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}}+25\right )\)	\(23\)
meijerg	\(-\frac {25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}} \left (-\ln \left (1-\frac {26 x^{2}}{25 \left (1+{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )}\right )+2 \ln \relax (x )+\ln \relax (2)+\ln \left (13\right )-2 \ln \relax (5)-\ln \left (1+{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )+i \pi \right )}{2 \left (25+25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )}+\frac {25 \left (1+{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right ) \ln \left (1-\frac {26 x^{2}}{25 \left (1+{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )}\right )}{2 \left (25+25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )}-\frac {25 \left (-\ln \left (1-\frac {26 x^{2}}{25 \left (1+{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )}\right )+2 \ln \relax (x )+\ln \relax (2)+\ln \left (13\right )-2 \ln \relax (5)-\ln \left (1+{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )+i \pi \right )}{2 \left (25+25 \,{\mathrm e}^{6 \,{\mathrm e}^{-20}}\right )}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-25*exp(6/exp(5)^4)-26*x^2-25)/(25*x*exp(6/exp(5)^4)-26*x^3+25*x),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+ln(26*x^2-25*exp(6*exp(-20))-25)

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maxima [A]  time = 0.38, size = 20, normalized size = 0.80 \begin {gather*} \log \left (26 \, x^{2} - 25 \, e^{\left (6 \, e^{\left (-20\right )}\right )} - 25\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(6/exp(5)^4)-26*x^2-25)/(25*x*exp(6/exp(5)^4)-26*x^3+25*x),x, algorithm="maxima")

[Out]

log(26*x^2 - 25*e^(6*e^(-20)) - 25) - log(x)

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mupad [B]  time = 0.25, size = 20, normalized size = 0.80 \begin {gather*} \ln \left (-52\,x^2+50\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-20}}+50\right )-\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(25*exp(6*exp(-20)) + 26*x^2 + 25)/(25*x + 25*x*exp(6*exp(-20)) - 26*x^3),x)

[Out]

log(50*exp(6*exp(-20)) - 52*x^2 + 50) - log(x)

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sympy [A]  time = 0.28, size = 20, normalized size = 0.80 \begin {gather*} - \log {\relax (x )} + \log {\left (x^{2} - \frac {25 e^{\frac {6}{e^{20}}}}{26} - \frac {25}{26} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-25*exp(6/exp(5)**4)-26*x**2-25)/(25*x*exp(6/exp(5)**4)-26*x**3+25*x),x)

[Out]

-log(x) + log(x**2 - 25*exp(6*exp(-20))/26 - 25/26)

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