3.8.28 \(\int \frac {2^{\frac {1}{x}} (-120 x^2+(24+120 x) \log (2))}{x^2} \, dx\)

Optimal. Leaf size=16 \[ 3-2^{3+\frac {1}{x}} (3+15 x) \]

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Rubi [A]  time = 0.06, antiderivative size = 14, normalized size of antiderivative = 0.88, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2288} \begin {gather*} -3 2^{\frac {1}{x}+3} (5 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2^x^(-1)*(-120*x^2 + (24 + 120*x)*Log[2]))/x^2,x]

[Out]

-3*2^(3 + x^(-1))*(1 + 5*x)

Rule 2288

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 {gather*} \begin {aligned} \text {integral} &=-3 2^{3+\frac {1}{x}} (1+5 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 20, normalized size = 1.25 \begin {gather*} -\frac {3\ 2^{3+\frac {1}{x}} (\log (2)+x \log (32))}{\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2^x^(-1)*(-120*x^2 + (24 + 120*x)*Log[2]))/x^2,x]

[Out]

(-3*2^(3 + x^(-1))*(Log[2] + x*Log[32]))/Log[2]

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fricas [A]  time = 0.68, size = 12, normalized size = 0.75 \begin {gather*} -24 \cdot 2^{\left (\frac {1}{x}\right )} {\left (5 \, x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((120*x+24)*log(2)-120*x^2)*exp(log(2)/x)/x^2,x, algorithm="fricas")

[Out]

-24*2^(1/x)*(5*x + 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {24 \, {\left (5 \, x^{2} - {\left (5 \, x + 1\right )} \log \relax (2)\right )} 2^{\left (\frac {1}{x}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((120*x+24)*log(2)-120*x^2)*exp(log(2)/x)/x^2,x, algorithm="giac")

[Out]

integrate(-24*(5*x^2 - (5*x + 1)*log(2))*2^(1/x)/x^2, x)

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maple [A]  time = 0.14, size = 12, normalized size = 0.75

method	result	size
risch	\(\left (-24-120 x \right ) 2^{\frac {1}{x}}\)	\(12\)
gosper	\(-24 \,{\mathrm e}^{\frac {\ln \relax (2)}{x}} \left (1+5 x \right )\)	\(15\)
norman	\(\frac {-24 x \,{\mathrm e}^{\frac {\ln \relax (2)}{x}}-120 x^{2} {\mathrm e}^{\frac {\ln \relax (2)}{x}}}{x}\)	\(28\)
derivativedivides	\(-\frac {24 \ln \relax (2) {\mathrm e}^{\frac {\ln \relax (2)}{x}}+120 \,{\mathrm e}^{\frac {\ln \relax (2)}{x}} x \ln \relax (2)}{\ln \relax (2)}\)	\(31\)
default	\(-\frac {24 \ln \relax (2) {\mathrm e}^{\frac {\ln \relax (2)}{x}}+120 \,{\mathrm e}^{\frac {\ln \relax (2)}{x}} x \ln \relax (2)}{\ln \relax (2)}\)	\(31\)
meijerg	\(-120 \ln \relax (2) \left (-\ln \left (-\frac {\ln \relax (2)}{x}\right )-\expIntegralEi \left (1, -\frac {\ln \relax (2)}{x}\right )-\ln \relax (x )+\ln \left (\ln \relax (2)\right )+i \pi \right )+24-24 \,{\mathrm e}^{\frac {\ln \relax (2)}{x}}-120 \ln \relax (2) \left (-\frac {x \left (2+\frac {2 \ln \relax (2)}{x}\right )}{2 \ln \relax (2)}+\frac {x \,{\mathrm e}^{\frac {\ln \relax (2)}{x}}}{\ln \relax (2)}+\ln \left (-\frac {\ln \relax (2)}{x}\right )+\expIntegralEi \left (1, -\frac {\ln \relax (2)}{x}\right )+1+\ln \relax (x )-\ln \left (\ln \relax (2)\right )-i \pi +\frac {x}{\ln \relax (2)}\right )\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((120*x+24)*ln(2)-120*x^2)*exp(ln(2)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

(-24-120*x)*2^(1/x)

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maxima [C]  time = 0.58, size = 34, normalized size = 2.12 \begin {gather*} -120 \, {\rm Ei}\left (\frac {\log \relax (2)}{x}\right ) \log \relax (2) + 120 \, \Gamma \left (-1, -\frac {\log \relax (2)}{x}\right ) \log \relax (2) - 3 \cdot 2^{\frac {1}{x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((120*x+24)*log(2)-120*x^2)*exp(log(2)/x)/x^2,x, algorithm="maxima")

[Out]

-120*Ei(log(2)/x)*log(2) + 120*gamma(-1, -log(2)/x)*log(2) - 3*2^(1/x + 3)

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mupad [B]  time = 0.53, size = 12, normalized size = 0.75 \begin {gather*} -24\,2^{1/x}\,\left (5\,x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(log(2)/x)*(log(2)*(120*x + 24) - 120*x^2))/x^2,x)

[Out]

-24*2^(1/x)*(5*x + 1)

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sympy [A]  time = 0.16, size = 12, normalized size = 0.75 \begin {gather*} \left (- 120 x - 24\right ) e^{\frac {\log {\relax (2 )}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((120*x+24)*ln(2)-120*x**2)*exp(ln(2)/x)/x**2,x)

[Out]

(-120*x - 24)*exp(log(2)/x)

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