3.70.69 \(\int -\frac {\log (2)}{x^4 (i \pi +\log (5))} \, dx\)

Optimal. Leaf size=19 \[ \frac {\log (2)}{3 x^3 (i \pi +\log (5))} \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 30} \begin {gather*} \frac {\log (2)}{3 x^3 (\log (5)+i \pi )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-(Log[2]/(x^4*(I*Pi + Log[5]))),x]

[Out]

Log[2]/(3*x^3*(I*Pi + Log[5]))

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\log (2) \int \frac {1}{x^4} \, dx}{i \pi +\log (5)}\\ &=\frac {\log (2)}{3 x^3 (i \pi +\log (5))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log (2)}{3 x^3 (i \pi +\log (5))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-(Log[2]/(x^4*(I*Pi + Log[5]))),x]

[Out]

Log[2]/(3*x^3*(I*Pi + Log[5]))

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fricas [A]  time = 0.97, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log \relax (2)}{3 i \, \pi x^{3} + 3 \, x^{3} \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(2)/x^4/(log(5)+I*pi),x, algorithm="fricas")

[Out]

log(2)/(3*I*pi*x^3 + 3*x^3*log(5))

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giac [A]  time = 0.42, size = 15, normalized size = 0.79 \begin {gather*} \frac {\log \relax (2)}{3 \, {\left (i \, \pi + \log \relax (5)\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(2)/x^4/(log(5)+I*pi),x, algorithm="giac")

[Out]

1/3*log(2)/((I*pi + log(5))*x^3)

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maple [A]  time = 0.05, size = 17, normalized size = 0.89

method	result	size
gosper	\(\frac {\ln \relax (2)}{3 \left (\ln \relax (5)+i \pi \right ) x^{3}}\)	\(17\)
default	\(\frac {\ln \relax (2)}{3 \left (\ln \relax (5)+i \pi \right ) x^{3}}\)	\(17\)
risch	\(\frac {\ln \relax (2)}{3 \left (\ln \relax (5)+i \pi \right ) x^{3}}\)	\(17\)
norman	\(-\frac {\ln \relax (2) \left (i \pi -\ln \relax (5)\right )}{3 \left (\pi ^{2}+\ln \relax (5)^{2}\right ) x^{3}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-ln(2)/x^4/(ln(5)+I*Pi),x,method=_RETURNVERBOSE)

[Out]

1/3*ln(2)/(ln(5)+I*Pi)/x^3

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maxima [A]  time = 0.37, size = 15, normalized size = 0.79 \begin {gather*} \frac {\log \relax (2)}{3 \, {\left (i \, \pi + \log \relax (5)\right )} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(2)/x^4/(log(5)+I*pi),x, algorithm="maxima")

[Out]

1/3*log(2)/((I*pi + log(5))*x^3)

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mupad [B]  time = 0.08, size = 16, normalized size = 0.84 \begin {gather*} \frac {\ln \relax (2)}{3\,x^3\,\left (\ln \relax (5)+\Pi \,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-log(2)/(x^4*(Pi*1i + log(5))),x)

[Out]

log(2)/(3*x^3*(Pi*1i + log(5)))

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sympy [A]  time = 0.06, size = 17, normalized size = 0.89 \begin {gather*} - \frac {\log {\relax (2 )}}{3 x^{3} \left (- \log {\relax (5 )} - i \pi \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-ln(2)/x**4/(ln(5)+I*pi),x)

[Out]

-log(2)/(3*x**3*(-log(5) - I*pi))

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