\(\int \frac {2 x^3+e^{.\frac {3}{2}/x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx\) [1353]

Optimal result

Integrand size = 35, antiderivative size = 27 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-2+x+\frac {\left (-e^{\left .\frac {3}{2}\right /x}+x\right ) \log (2) \log ^2(\log (4))}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 14, 2326} \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x-\frac {e^{\left .\frac {3}{2}\right /x} \log (2) \log ^2(\log (4))}{x} \]

[In]

Int[(2*x^3 + E^(3/(2*x))*(3 + 2*x)*Log[2]*Log[Log[4]]^2)/(2*x^3),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{x^3} \, dx \\ & = \frac {1}{2} \int \left (2+\frac {e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{x^3}\right ) \, dx \\ & = x+\frac {1}{2} \left (\log (2) \log ^2(\log (4))\right ) \int \frac {e^{\left .\frac {3}{2}\right /x} (3+2 x)}{x^3} \, dx \\ & = x-\frac {e^{\left .\frac {3}{2}\right /x} \log (2) \log ^2(\log (4))}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x-\frac {e^{\left .\frac {3}{2}\right /x} \log (2) \log ^2(\log (4))}{x} \]

[In]

Integrate[(2*x^3 + E^(3/(2*x))*(3 + 2*x)*Log[2]*Log[Log[4]]^2)/(2*x^3),x]

[Out]

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

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
parts	\(x -\frac {\ln \left (2\right ) \ln \left (2 \ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {3}{2 x}}}{x}\)	\(23\)
parallelrisch	\(-\frac {2 \ln \left (2\right ) {\mathrm e}^{\frac {3}{2 x}} \ln \left (2 \ln \left (2\right )\right )^{2}-2 x^{2}}{2 x}\)	\(29\)
risch	\(x -\frac {\ln \left (2\right ) \left (\ln \left (2\right )^{2}+2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )+\ln \left (\ln \left (2\right )\right )^{2}\right ) {\mathrm e}^{\frac {3}{2 x}}}{x}\)	\(33\)
norman	\(\frac {x^{3}+\left (-\ln \left (2\right )^{3}-2 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right )-\ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2}\right ) x \,{\mathrm e}^{\frac {3}{2 x}}}{x^{2}}\)	\(42\)
derivativedivides	\(x -\frac {2 \ln \left (2\right )^{3} {\mathrm e}^{\frac {3}{2 x}}}{3}-\frac {2 \ln \left (2\right )^{3} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {2 \ln \left (2\right ) {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )^{2}}{3}-\frac {4 \ln \left (2\right )^{2} {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )}{3}-\frac {2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {4 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right ) \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}\)	\(129\)
default	\(x -\frac {2 \ln \left (2\right )^{3} {\mathrm e}^{\frac {3}{2 x}}}{3}-\frac {2 \ln \left (2\right )^{3} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {2 \ln \left (2\right ) {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )^{2}}{3}-\frac {4 \ln \left (2\right )^{2} {\mathrm e}^{\frac {3}{2 x}} \ln \left (\ln \left (2\right )\right )}{3}-\frac {2 \ln \left (2\right ) \ln \left (\ln \left (2\right )\right )^{2} \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}-\frac {4 \ln \left (2\right )^{2} \ln \left (\ln \left (2\right )\right ) \left (\frac {3 \,{\mathrm e}^{\frac {3}{2 x}}}{2 x}-{\mathrm e}^{\frac {3}{2 x}}\right )}{3}\)	\(129\)

[In]

int(1/2*((3+2*x)*ln(2)*exp(3/2/x)*ln(2*ln(2))^2+2*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

x-ln(2)*ln(2*ln(2))^2/x*exp(3/2/x)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-\frac {e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right ) \log \left (2 \, \log \left (2\right )\right )^{2} - x^{2}}{x} \]

[In]

integrate(1/2*((3+2*x)*log(2)*exp(3/2/x)*log(2*log(2))^2+2*x^3)/x^3,x, algorithm="fricas")

[Out]

-(e^(3/2/x)*log(2)*log(2*log(2))^2 - x^2)/x

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x + \frac {\left (- \log {\left (2 \right )}^{3} - \log {\left (2 \right )} \log {\left (\log {\left (2 \right )} \right )}^{2} - 2 \log {\left (2 \right )}^{2} \log {\left (\log {\left (2 \right )} \right )}\right ) e^{\frac {3}{2 x}}}{x} \]

[In]

integrate(1/2*((3+2*x)*ln(2)*exp(3/2/x)*ln(2*ln(2))**2+2*x**3)/x**3,x)

[Out]

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

Maxima [C] (verification not implemented)

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

Time = 0.21 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-\frac {2}{3} \, e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right ) \log \left (2 \, \log \left (2\right )\right )^{2} + \frac {2}{3} \, \Gamma \left (2, -\frac {3}{2 \, x}\right ) \log \left (2\right ) \log \left (2 \, \log \left (2\right )\right )^{2} + x \]

[In]

integrate(1/2*((3+2*x)*log(2)*exp(3/2/x)*log(2*log(2))^2+2*x^3)/x^3,x, algorithm="maxima")

[Out]

-2/3*e^(3/2/x)*log(2)*log(2*log(2))^2 + 2/3*gamma(2, -3/2/x)*log(2)*log(2*log(2))^2 + x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).

Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=-{\left (\frac {e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right )^{3}}{x^{2}} + \frac {2 \, e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right )^{2} \log \left (\log \left (2\right )\right )}{x^{2}} + \frac {e^{\left (\frac {3}{2 \, x}\right )} \log \left (2\right ) \log \left (\log \left (2\right )\right )^{2}}{x^{2}} - 1\right )} x \]

[In]

integrate(1/2*((3+2*x)*log(2)*exp(3/2/x)*log(2*log(2))^2+2*x^3)/x^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {2 x^3+e^{\left .\frac {3}{2}\right /x} (3+2 x) \log (2) \log ^2(\log (4))}{2 x^3} \, dx=x-\frac {{\mathrm {e}}^{\frac {3}{2\,x}}\,\ln \left (2\right )\,{\ln \left (\ln \left (4\right )\right )}^2}{x} \]

[In]

int((x^3 + (log(2*log(2))^2*exp(3/(2*x))*log(2)*(2*x + 3))/2)/x^3,x)

[Out]

x - (exp(3/(2*x))*log(2)*log(log(4))^2)/x