\(\int \frac {-6 x^2-30 x^3+(-3-8 x+6 x^3) \log (x)+(-3-4 x+2 x^3) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx\) [7661]

Optimal result

Integrand size = 50, antiderivative size = 29 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {\frac {3}{4 x^2}+\frac {2}{x}+x+\frac {1+5 x}{\log (x)}}{\log ^2(x)} \]

[Out]

(x+2/x+3/4/x^2+(1+5*x)/ln(x))/ln(x)^2

Rubi [F]

\[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx \]

[In]

Int[(-6*x^2 - 30*x^3 + (-3 - 8*x + 6*x^3)*Log[x] + (-3 - 4*x + 2*x^3)*Log[x]^2)/(2*x^3*Log[x]^4),x]

[Out]

Log[x]^(-3) + (5*x)/Log[x]^3 + (5*x)/(2*Log[x]^2) + (5*x)/(2*Log[x]) - (5*LogIntegral[x])/2 + Defer[Int][(-3 -
8*x + 6*x^3)/(x^3*Log[x]^3), x]/2 + Defer[Int][(-3 - 4*x + 2*x^3)/(x^3*Log[x]^2), x]/2

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{x^3 \log ^4(x)} \, dx \\ & = \frac {1}{2} \int \left (-\frac {6 (1+5 x)}{x \log ^4(x)}+\frac {-3-8 x+6 x^3}{x^3 \log ^3(x)}+\frac {-3-4 x+2 x^3}{x^3 \log ^2(x)}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx-3 \int \frac {1+5 x}{x \log ^4(x)} \, dx \\ & = \frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx-3 \int \left (\frac {5}{\log ^4(x)}+\frac {1}{x \log ^4(x)}\right ) \, dx \\ & = \frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx-3 \int \frac {1}{x \log ^4(x)} \, dx-15 \int \frac {1}{\log ^4(x)} \, dx \\ & = \frac {5 x}{\log ^3(x)}+\frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx-3 \text {Subst}\left (\int \frac {1}{x^4} \, dx,x,\log (x)\right )-5 \int \frac {1}{\log ^3(x)} \, dx \\ & = \frac {1}{\log ^3(x)}+\frac {5 x}{\log ^3(x)}+\frac {5 x}{2 \log ^2(x)}+\frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx-\frac {5}{2} \int \frac {1}{\log ^2(x)} \, dx \\ & = \frac {1}{\log ^3(x)}+\frac {5 x}{\log ^3(x)}+\frac {5 x}{2 \log ^2(x)}+\frac {5 x}{2 \log (x)}+\frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx-\frac {5}{2} \int \frac {1}{\log (x)} \, dx \\ & = \frac {1}{\log ^3(x)}+\frac {5 x}{\log ^3(x)}+\frac {5 x}{2 \log ^2(x)}+\frac {5 x}{2 \log (x)}-\frac {5 \operatorname {LogIntegral}(x)}{2}+\frac {1}{2} \int \frac {-3-8 x+6 x^3}{x^3 \log ^3(x)} \, dx+\frac {1}{2} \int \frac {-3-4 x+2 x^3}{x^3 \log ^2(x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {1}{\log ^3(x)}+\frac {5 x}{\log ^3(x)}+\frac {3}{4 x^2 \log ^2(x)}+\frac {2}{x \log ^2(x)}+\frac {x}{\log ^2(x)} \]

[In]

Integrate[(-6*x^2 - 30*x^3 + (-3 - 8*x + 6*x^3)*Log[x] + (-3 - 4*x + 2*x^3)*Log[x]^2)/(2*x^3*Log[x]^4),x]

[Out]

Log[x]^(-3) + (5*x)/Log[x]^3 + 3/(4*x^2*Log[x]^2) + 2/(x*Log[x]^2) + x/Log[x]^2

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14

method	result	size
norman	\(\frac {x^{2}+x^{3} \ln \left (x \right )+5 x^{3}+2 x \ln \left (x \right )+\frac {3 \ln \left (x \right )}{4}}{x^{2} \ln \left (x \right )^{3}}\)	\(33\)
default	\(\frac {x}{\ln \left (x \right )^{2}}+\frac {5 x}{\ln \left (x \right )^{3}}+\frac {2}{x \ln \left (x \right )^{2}}+\frac {1}{\ln \left (x \right )^{3}}+\frac {3}{4 x^{2} \ln \left (x \right )^{2}}\)	\(37\)
risch	\(\frac {4 x^{3} \ln \left (x \right )+20 x^{3}+4 x^{2}+8 x \ln \left (x \right )+3 \ln \left (x \right )}{4 x^{2} \ln \left (x \right )^{3}}\)	\(37\)
parallelrisch	\(-\frac {-4 x^{3} \ln \left (x \right )-20 x^{3}-4 x^{2}-8 x \ln \left (x \right )-3 \ln \left (x \right )}{4 x^{2} \ln \left (x \right )^{3}}\)	\(37\)
parts	\(\frac {x}{\ln \left (x \right )^{2}}+\frac {5 x}{\ln \left (x \right )^{3}}+\frac {2}{x \ln \left (x \right )^{2}}+\frac {1}{\ln \left (x \right )^{3}}+\frac {3}{4 x^{2} \ln \left (x \right )^{2}}\)	\(37\)

[In]

int(1/2*((2*x^3-4*x-3)*ln(x)^2+(6*x^3-8*x-3)*ln(x)-30*x^3-6*x^2)/x^3/ln(x)^4,x,method=_RETURNVERBOSE)

[Out]

(x^2+x^3*ln(x)+5*x^3+2*x*ln(x)+3/4*ln(x))/x^2/ln(x)^3

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {20 \, x^{3} + 4 \, x^{2} + {\left (4 \, x^{3} + 8 \, x + 3\right )} \log \left (x\right )}{4 \, x^{2} \log \left (x\right )^{3}} \]

[In]

integrate(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x^3/log(x)^4,x, algorithm="fricas")

[Out]

1/4*(20*x^3 + 4*x^2 + (4*x^3 + 8*x + 3)*log(x))/(x^2*log(x)^3)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {20 x^{3} + 4 x^{2} + \left (4 x^{3} + 8 x + 3\right ) \log {\left (x \right )}}{4 x^{2} \log {\left (x \right )}^{3}} \]

[In]

integrate(1/2*((2*x**3-4*x-3)*ln(x)**2+(6*x**3-8*x-3)*ln(x)-30*x**3-6*x**2)/x**3/ln(x)**4,x)

[Out]

(20*x**3 + 4*x**2 + (4*x**3 + 8*x + 3)*log(x))/(4*x**2*log(x)**3)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {1}{\log \left (x\right )^{3}} + 3 \, \Gamma \left (-1, 2 \, \log \left (x\right )\right ) + \Gamma \left (-1, -\log \left (x\right )\right ) + 2 \, \Gamma \left (-1, \log \left (x\right )\right ) + 6 \, \Gamma \left (-2, 2 \, \log \left (x\right )\right ) - 3 \, \Gamma \left (-2, -\log \left (x\right )\right ) + 4 \, \Gamma \left (-2, \log \left (x\right )\right ) - 15 \, \Gamma \left (-3, -\log \left (x\right )\right ) \]

[In]

integrate(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x^3/log(x)^4,x, algorithm="maxima")

[Out]

1/log(x)^3 + 3*gamma(-1, 2*log(x)) + gamma(-1, -log(x)) + 2*gamma(-1, log(x)) + 6*gamma(-2, 2*log(x)) - 3*gamm
a(-2, -log(x)) + 4*gamma(-2, log(x)) - 15*gamma(-3, -log(x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {4 \, x^{3} \log \left (x\right ) + 20 \, x^{3} + 4 \, x^{2} + 8 \, x \log \left (x\right ) + 3 \, \log \left (x\right )}{4 \, x^{2} \log \left (x\right )^{3}} \]

[In]

integrate(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x^3/log(x)^4,x, algorithm="giac")

[Out]

1/4*(4*x^3*log(x) + 20*x^3 + 4*x^2 + 8*x*log(x) + 3*log(x))/(x^2*log(x)^3)

Mupad [B] (verification not implemented)

Time = 12.90 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {\ln \left (x\right )\,\left (x^3+2\,x+\frac {3}{4}\right )+x^2+5\,x^3}{x^2\,{\ln \left (x\right )}^3} \]

[In]

int(-((log(x)^2*(4*x - 2*x^3 + 3))/2 + (log(x)*(8*x - 6*x^3 + 3))/2 + 3*x^2 + 15*x^3)/(x^3*log(x)^4),x)

[Out]

(log(x)*(2*x + x^3 + 3/4) + x^2 + 5*x^3)/(x^2*log(x)^3)