3.2.62 \(\int \frac {a^x}{x^2} \, dx\) [162]

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

[Out]

-a^x/x+Ei(x*ln(a))*ln(a)

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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2208, 2209} \begin {gather*} \log (a) \text {ExpIntegralEi}(x \log (a))-\frac {a^x}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2208

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] &&  !TrueQ[$UseGamm
a]

Rule 2209

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

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*}

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} -\frac {a^x}{x}+\text {Ei}(x \log (a)) \log (a) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.03, size = 21, normalized size = 1.24

method	result	size
risch	\(-\frac {a^{x}}{x}-\ln \left (a \right ) \expIntegral \left (1, -x \ln \left (a \right )\right )\)	\(21\)
meijerg	\(-\ln \left (a \right ) \left (\frac {1}{x \ln \left (a \right )}+1-\ln \left (x \right )-i \pi -\ln \left (\ln \left (a \right )\right )-\frac {2+2 x \ln \left (a \right )}{2 x \ln \left (a \right )}+\frac {{\mathrm e}^{x \ln \left (a \right )}}{x \ln \left (a \right )}+\ln \left (-x \ln \left (a \right )\right )+\expIntegral \left (1, -x \ln \left (a \right )\right )\right )\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a^x/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 3.06, size = 10, normalized size = 0.59 \begin {gather*} \Gamma \left (-1, -x \log \left (a\right )\right ) \log \left (a\right ) \end {gather*}

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)

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Fricas [A]
time = 0.89, size = 19, normalized size = 1.12 \begin {gather*} \frac {x {\rm Ei}\left (x \log \left (a\right )\right ) \log \left (a\right ) - a^{x}}{x} \end {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{x}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(a**x/x**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.13, size = 19, normalized size = 1.12 \begin {gather*} -\ln \left (a\right )\,\mathrm {expint}\left (-x\,\ln \left (a\right )\right )-\frac {a^x}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- log(a)*expint(-x*log(a)) - a^x/x

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