3.1.73 \(\int \frac {-1+x^7}{\log (x)} \, dx\) [73]

Optimal. Leaf size=10 \[ \operatorname {ExpIntegralEi}(8 \log (x))-\operatorname {LogIntegral}(x) \]

[Out]

Ei(8*ln(x))-Li(x)

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Rubi [A]
time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2367, 2335, 2346, 2209} \begin {gather*} \operatorname {ExpIntegralEi}(8 \log (x))-\operatorname {LogIntegral}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + x^7)/Log[x],x]

[Out]

ExpIntegralEi[8*Log[x]] - LogIntegral[x]

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]

Rule 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2367

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = Expand
Integrand[(a + b*Log[c*x^n])^p, (d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}
, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left (-\frac {1}{\log (x)}+\frac {x^7}{\log (x)}\right ) \, dx\\ &=-\int \frac {1}{\log (x)} \, dx+\int \frac {x^7}{\log (x)} \, dx\\ &=-\operatorname {LogIntegral}(x)+\text {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (x)\right )\\ &=\operatorname {ExpIntegralEi}(8 \log (x))-\operatorname {LogIntegral}(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 10, normalized size = 1.00 \begin {gather*} \operatorname {ExpIntegralEi}(8 \log (x))-\operatorname {LogIntegral}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^7)/Log[x],x]

[Out]

ExpIntegralEi[8*Log[x]] - LogIntegral[x]

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Maple [A]
time = 0.02, size = 16, normalized size = 1.60

method	result	size
default	\(-\expIntegral _{1}\left (-8 \ln \left (x \right )\right )+\expIntegral _{1}\left (-\ln \left (x \right )\right )\)	\(16\)
risch	\(-\expIntegral _{1}\left (-8 \ln \left (x \right )\right )+\expIntegral _{1}\left (-\ln \left (x \right )\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7-1)/ln(x),x,method=_RETURNVERBOSE)

[Out]

-Ei(1,-8*ln(x))+Ei(1,-ln(x))

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Maxima [A]
time = 0.44, size = 11, normalized size = 1.10 \begin {gather*} {\rm Ei}\left (8 \, \log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7-1)/log(x),x, algorithm="maxima")

[Out]

Ei(8*log(x)) - Ei(log(x))

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Fricas [A]
time = 0.58, size = 9, normalized size = 0.90 \begin {gather*} \operatorname {log\_integral}\left (x^{8}\right ) - \operatorname {log\_integral}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7-1)/log(x),x, algorithm="fricas")

[Out]

log_integral(x^8) - log_integral(x)

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Sympy [A]
time = 0.70, size = 10, normalized size = 1.00 \begin {gather*} - \operatorname {Ei}{\left (\log {\left (x \right )} \right )} + \operatorname {Ei}{\left (8 \log {\left (x \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**7-1)/ln(x),x)

[Out]

-Ei(log(x)) + Ei(8*log(x))

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Giac [A]
time = 0.59, size = 11, normalized size = 1.10 \begin {gather*} {\rm Ei}\left (8 \, \log \left (x\right )\right ) - {\rm Ei}\left (\log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^7-1)/log(x),x, algorithm="giac")

[Out]

Ei(8*log(x)) - Ei(log(x))

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Mupad [B]
time = 0.12, size = 10, normalized size = 1.00 \begin {gather*} \mathrm {ei}\left (8\,\ln \left (x\right )\right )-\mathrm {logint}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7 - 1)/log(x),x)

[Out]

ei(8*log(x)) - logint(x)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int((x^7-1)/ln(x),x)

[Out]

not solved

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