3.162 \(\int \frac{a^x}{x^2} \, dx\)

Optimal. Leaf size=17 \[ \log (a) \text{ExpIntegralEi}(x \log (a))-\frac{a^x}{x} \]

[Out]

-(a^x/x) + ExpIntegralEi[x*Log[a]]*Log[a]

________________________________________________________________________________________

Rubi [A]  time = 0.0190059, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2177, 2178} \[ \log (a) \text{ExpIntegralEi}(x \log (a))-\frac{a^x}{x} \]

Antiderivative was successfully verified.

[In]

Int[a^x/x^2,x]

[Out]

-(a^x/x) + ExpIntegralEi[x*Log[a]]*Log[a]

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{a^x}{x^2} \, dx &=-\frac{a^x}{x}+\log (a) \int \frac{a^x}{x} \, dx\\ &=-\frac{a^x}{x}+\text{Ei}(x \log (a)) \log (a)\\ \end{align*}

Mathematica [A]  time = 0.0107686, size = 17, normalized size = 1. \[ \log (a) \text{ExpIntegralEi}(x \log (a))-\frac{a^x}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[a^x/x^2,x]

[Out]

-(a^x/x) + ExpIntegralEi[x*Log[a]]*Log[a]

________________________________________________________________________________________

Maple [A]  time = 0.02, size = 21, normalized size = 1.2 \begin{align*} -{\frac{{a}^{x}}{x}}-\ln \left ( a \right ){\it Ei} \left ( 1,-x\ln \left ( a \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x/x^2,x)

[Out]

-a^x/x-ln(a)*Ei(1,-x*ln(a))

________________________________________________________________________________________

Maxima [A]  time = 1.03775, size = 14, normalized size = 0.82 \begin{align*} \Gamma \left (-1, -x \log \left (a\right )\right ) \log \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x/x^2,x, algorithm="maxima")

[Out]

gamma(-1, -x*log(a))*log(a)

________________________________________________________________________________________

Fricas [A]  time = 1.67461, size = 45, normalized size = 2.65 \begin{align*} \frac{x{\rm Ei}\left (x \log \left (a\right )\right ) \log \left (a\right ) - a^{x}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x/x^2,x, algorithm="fricas")

[Out]

(x*Ei(x*log(a))*log(a) - a^x)/x

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a**x/x**2,x)

[Out]

Integral(a**x/x**2, x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a^{x}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a^x/x^2,x, algorithm="giac")

[Out]

integrate(a^x/x^2, x)