3.49.95 \(\int \frac {450-141855 x+93650 x^2-15476 x^3+(-11250 x+7470 x^2-1240 x^3) \log (\frac {e^{2 x}}{x})+(-225 x+150 x^2-25 x^3) \log ^2(\frac {e^{2 x}}{x})}{562500 x-372000 x^2+61504 x^3+(45000 x-29880 x^2+4960 x^3) \log (\frac {e^{2 x}}{x})+(900 x-600 x^2+100 x^3) \log ^2(\frac {e^{2 x}}{x})} \, dx\)

Optimal. Leaf size=36 \[ \frac {1}{4} \left (-x+\frac {2}{25+\frac {x}{5 (3-x)}+\log \left (\frac {e^{2 x}}{x}\right )}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.08, 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 {450-141855 x+93650 x^2-15476 x^3+\left (-11250 x+7470 x^2-1240 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (-225 x+150 x^2-25 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )}{562500 x-372000 x^2+61504 x^3+\left (45000 x-29880 x^2+4960 x^3\right ) \log \left (\frac {e^{2 x}}{x}\right )+\left (900 x-600 x^2+100 x^3\right ) \log ^2\left (\frac {e^{2 x}}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(450 - 141855*x + 93650*x^2 - 15476*x^3 + (-11250*x + 7470*x^2 - 1240*x^3)*Log[E^(2*x)/x] + (-225*x + 150*
x^2 - 25*x^3)*Log[E^(2*x)/x]^2)/(562500*x - 372000*x^2 + 61504*x^3 + (45000*x - 29880*x^2 + 4960*x^3)*Log[E^(2
*x)/x] + (900*x - 600*x^2 + 100*x^3)*Log[E^(2*x)/x]^2),x]

[Out]

-1/4*x - (615*Defer[Int][(375 - 124*x - 5*(-3 + x)*Log[E^(2*x)/x])^(-2), x])/2 + (225*Defer[Int][1/(x*(375 - 1
24*x - 5*(-3 + x)*Log[E^(2*x)/x])^2), x])/2 + (325*Defer[Int][x/(375 - 124*x - 5*(-3 + x)*Log[E^(2*x)/x])^2, x
])/2 - 25*Defer[Int][x^2/(375 - 124*x - 5*(-3 + x)*Log[E^(2*x)/x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {450-141855 x+93650 x^2-15476 x^3-10 x \left (1125-747 x+124 x^2\right ) \log \left (\frac {e^{2 x}}{x}\right )-25 (-3+x)^2 x \log ^2\left (\frac {e^{2 x}}{x}\right )}{4 x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {450-141855 x+93650 x^2-15476 x^3-10 x \left (1125-747 x+124 x^2\right ) \log \left (\frac {e^{2 x}}{x}\right )-25 (-3+x)^2 x \log ^2\left (\frac {e^{2 x}}{x}\right )}{x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx\\ &=\frac {1}{4} \int \left (-1-\frac {10 \left (-45+123 x-65 x^2+10 x^3\right )}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}\right ) \, dx\\ &=-\frac {x}{4}-\frac {5}{2} \int \frac {-45+123 x-65 x^2+10 x^3}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx\\ &=-\frac {x}{4}-\frac {5}{2} \int \frac {-45+123 x-65 x^2+10 x^3}{x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx\\ &=-\frac {x}{4}-\frac {5}{2} \int \left (\frac {123}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}-\frac {45}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}-\frac {65 x}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}+\frac {10 x^2}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2}\right ) \, dx\\ &=-\frac {x}{4}-25 \int \frac {x^2}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {225}{2} \int \frac {1}{x \left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {325}{2} \int \frac {x}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx-\frac {615}{2} \int \frac {1}{\left (-375+124 x-15 \log \left (\frac {e^{2 x}}{x}\right )+5 x \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx\\ &=-\frac {x}{4}-25 \int \frac {x^2}{\left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {225}{2} \int \frac {1}{x \left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx+\frac {325}{2} \int \frac {x}{\left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx-\frac {615}{2} \int \frac {1}{\left (375-124 x-5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 52, normalized size = 1.44 \begin {gather*} -\frac {30-385 x+124 x^2+5 (-3+x) x \log \left (\frac {e^{2 x}}{x}\right )}{4 \left (-375+124 x+5 (-3+x) \log \left (\frac {e^{2 x}}{x}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(450 - 141855*x + 93650*x^2 - 15476*x^3 + (-11250*x + 7470*x^2 - 1240*x^3)*Log[E^(2*x)/x] + (-225*x
+ 150*x^2 - 25*x^3)*Log[E^(2*x)/x]^2)/(562500*x - 372000*x^2 + 61504*x^3 + (45000*x - 29880*x^2 + 4960*x^3)*Lo
g[E^(2*x)/x] + (900*x - 600*x^2 + 100*x^3)*Log[E^(2*x)/x]^2),x]

[Out]

-1/4*(30 - 385*x + 124*x^2 + 5*(-3 + x)*x*Log[E^(2*x)/x])/(-375 + 124*x + 5*(-3 + x)*Log[E^(2*x)/x])

________________________________________________________________________________________

fricas [A]  time = 0.98, size = 51, normalized size = 1.42 \begin {gather*} -\frac {124 \, x^{2} + 5 \, {\left (x^{2} - 3 \, x\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right ) - 385 \, x + 30}{4 \, {\left (5 \, {\left (x - 3\right )} \log \left (\frac {e^{\left (2 \, x\right )}}{x}\right ) + 124 \, x - 375\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3+150*x^2-225*x)*log(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-11250*x)*log(exp(x)^2/x)-15476*x^3+93
650*x^2-141855*x+450)/((100*x^3-600*x^2+900*x)*log(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*log(exp(x)^2/x)+
61504*x^3-372000*x^2+562500*x),x, algorithm="fricas")

[Out]

-1/4*(124*x^2 + 5*(x^2 - 3*x)*log(e^(2*x)/x) - 385*x + 30)/(5*(x - 3)*log(e^(2*x)/x) + 124*x - 375)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 30, normalized size = 0.83 \begin {gather*} -\frac {1}{4} \, x + \frac {5 \, {\left (x - 3\right )}}{2 \, {\left (10 \, x^{2} - 5 \, x \log \relax (x) + 94 \, x + 15 \, \log \relax (x) - 375\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3+150*x^2-225*x)*log(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-11250*x)*log(exp(x)^2/x)-15476*x^3+93
650*x^2-141855*x+450)/((100*x^3-600*x^2+900*x)*log(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*log(exp(x)^2/x)+
61504*x^3-372000*x^2+562500*x),x, algorithm="giac")

[Out]

-1/4*x + 5/2*(x - 3)/(10*x^2 - 5*x*log(x) + 94*x + 15*log(x) - 375)

________________________________________________________________________________________

maple [B]  time = 0.44, size = 112, normalized size = 3.11

method	result	size
default	\(\frac {\left (-31-\frac {5 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )}{4}-\frac {5 \ln \relax (x )}{4}+\frac {5 x}{2}\right ) x^{2}+\frac {13 x}{4}+\frac {5 x^{2} \ln \relax (x )}{4}-\frac {5 x^{3}}{2}+\frac {1095}{4}+\frac {45 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )}{4}}{10 x^{2}-5 x \ln \relax (x )+5 x \left (\ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )-2 \ln \left ({\mathrm e}^{x}\right )+\ln \relax (x )\right )+10 x \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+124 x -15 \ln \left (\frac {{\mathrm e}^{2 x}}{x}\right )-375}\)	\(112\)
risch	\(-\frac {x}{4}+\frac {5 i \left (x -3\right )}{5 \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )-5 \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}+5 \pi x \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )-10 \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+5 \pi x \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-5 \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}+5 \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}-15 \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )+15 \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}-15 \pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+30 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}-15 \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+15 \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{2}-15 \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{2 x}}{x}\right )^{3}+30 i \ln \relax (x )-60 i \ln \left ({\mathrm e}^{x}\right )+248 i x +20 i x \ln \left ({\mathrm e}^{x}\right )-10 i x \ln \relax (x )-750 i}\)	\(337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*x^3+150*x^2-225*x)*ln(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-11250*x)*ln(exp(x)^2/x)-15476*x^3+93650*x^2-
141855*x+450)/((100*x^3-600*x^2+900*x)*ln(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*ln(exp(x)^2/x)+61504*x^3-
372000*x^2+562500*x),x,method=_RETURNVERBOSE)

[Out]

((-31-5/4*ln(exp(x)^2/x)-5/4*ln(x)+5/2*x)*x^2+13/4*x+5/4*x^2*ln(x)-5/2*x^3+1095/4+45/4*ln(exp(x)^2/x))/(10*x^2
-5*x*ln(x)+5*x*(ln(exp(x)^2/x)-2*ln(exp(x))+ln(x))+10*x*(ln(exp(x))-x)+124*x-15*ln(exp(x)^2/x)-375)

________________________________________________________________________________________

maxima [A]  time = 0.40, size = 47, normalized size = 1.31 \begin {gather*} -\frac {10 \, x^{3} + 94 \, x^{2} - 5 \, {\left (x^{2} - 3 \, x\right )} \log \relax (x) - 385 \, x + 30}{4 \, {\left (10 \, x^{2} - 5 \, {\left (x - 3\right )} \log \relax (x) + 94 \, x - 375\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^3+150*x^2-225*x)*log(exp(x)^2/x)^2+(-1240*x^3+7470*x^2-11250*x)*log(exp(x)^2/x)-15476*x^3+93
650*x^2-141855*x+450)/((100*x^3-600*x^2+900*x)*log(exp(x)^2/x)^2+(4960*x^3-29880*x^2+45000*x)*log(exp(x)^2/x)+
61504*x^3-372000*x^2+562500*x),x, algorithm="maxima")

[Out]

-1/4*(10*x^3 + 94*x^2 - 5*(x^2 - 3*x)*log(x) - 385*x + 30)/(10*x^2 - 5*(x - 3)*log(x) + 94*x - 375)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {141855\,x+{\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )}^2\,\left (25\,x^3-150\,x^2+225\,x\right )-93650\,x^2+15476\,x^3+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )\,\left (1240\,x^3-7470\,x^2+11250\,x\right )-450}{562500\,x+{\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )}^2\,\left (100\,x^3-600\,x^2+900\,x\right )-372000\,x^2+61504\,x^3+\ln \left (\frac {{\mathrm {e}}^{2\,x}}{x}\right )\,\left (4960\,x^3-29880\,x^2+45000\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(141855*x + log(exp(2*x)/x)^2*(225*x - 150*x^2 + 25*x^3) - 93650*x^2 + 15476*x^3 + log(exp(2*x)/x)*(11250
*x - 7470*x^2 + 1240*x^3) - 450)/(562500*x + log(exp(2*x)/x)^2*(900*x - 600*x^2 + 100*x^3) - 372000*x^2 + 6150
4*x^3 + log(exp(2*x)/x)*(45000*x - 29880*x^2 + 4960*x^3)),x)

[Out]

int(-(141855*x + log(exp(2*x)/x)^2*(225*x - 150*x^2 + 25*x^3) - 93650*x^2 + 15476*x^3 + log(exp(2*x)/x)*(11250
*x - 7470*x^2 + 1240*x^3) - 450)/(562500*x + log(exp(2*x)/x)^2*(900*x - 600*x^2 + 100*x^3) - 372000*x^2 + 6150
4*x^3 + log(exp(2*x)/x)*(45000*x - 29880*x^2 + 4960*x^3)), x)

________________________________________________________________________________________

sympy [A]  time = 0.31, size = 26, normalized size = 0.72 \begin {gather*} - \frac {x}{4} + \frac {5 x - 15}{248 x + \left (10 x - 30\right ) \log {\left (\frac {e^{2 x}}{x} \right )} - 750} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x**3+150*x**2-225*x)*ln(exp(x)**2/x)**2+(-1240*x**3+7470*x**2-11250*x)*ln(exp(x)**2/x)-15476*x
**3+93650*x**2-141855*x+450)/((100*x**3-600*x**2+900*x)*ln(exp(x)**2/x)**2+(4960*x**3-29880*x**2+45000*x)*ln(e
xp(x)**2/x)+61504*x**3-372000*x**2+562500*x),x)

[Out]

-x/4 + (5*x - 15)/(248*x + (10*x - 30)*log(exp(2*x)/x) - 750)

________________________________________________________________________________________