3.21.97 \(\int \frac {-4 x^2+x^2 (i \pi +\log (3))}{60 \log ^2(5)} \, dx\)

Optimal. Leaf size=20 \[ \frac {x^3 (-4+i \pi +\log (3))}{180 \log ^2(5)} \]

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Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6, 12, 30} \begin {gather*} -\frac {x^3 (4-i \pi -\log (3))}{180 \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^2 + x^2*(I*Pi + Log[3]))/(60*Log[5]^2),x]

[Out]

-1/180*(x^3*(4 - I*Pi - Log[3]))/Log[5]^2

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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} &=\int \frac {x^2 (-4+i \pi +\log (3))}{60 \log ^2(5)} \, dx\\ &=-\frac {(4-i \pi -\log (3)) \int x^2 \, dx}{60 \log ^2(5)}\\ &=-\frac {x^3 (4-i \pi -\log (3))}{180 \log ^2(5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} \frac {x^3 (-4+i \pi +\log (3))}{180 \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 + x^2*(I*Pi + Log[3]))/(60*Log[5]^2),x]

[Out]

(x^3*(-4 + I*Pi + Log[3]))/(180*Log[5]^2)

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fricas [A]  time = 0.60, size = 22, normalized size = 1.10 \begin {gather*} \frac {{\left (i \, \pi - 4\right )} x^{3} + x^{3} \log \relax (3)}{180 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/60*(x^2*(log(3)+I*pi)-4*x^2)/log(5)^2,x, algorithm="fricas")

[Out]

1/180*((I*pi - 4)*x^3 + x^3*log(3))/log(5)^2

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giac [A]  time = 0.14, size = 22, normalized size = 1.10 \begin {gather*} \frac {{\left (i \, \pi + \log \relax (3)\right )} x^{3} - 4 \, x^{3}}{180 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/60*(x^2*(log(3)+I*pi)-4*x^2)/log(5)^2,x, algorithm="giac")

[Out]

1/180*((I*pi + log(3))*x^3 - 4*x^3)/log(5)^2

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maple [A]  time = 0.04, size = 18, normalized size = 0.90

method	result	size
gosper	\(\frac {x^{3} \left (\ln \relax (3)+i \pi -4\right )}{180 \ln \relax (5)^{2}}\)	\(18\)
norman	\(\frac {x^{3} \left (\ln \relax (3)+i \pi -4\right )}{180 \ln \relax (5)^{2}}\)	\(18\)
default	\(\frac {\frac {x^{3} \left (\ln \relax (3)+i \pi \right )}{3}-\frac {4 x^{3}}{3}}{60 \ln \relax (5)^{2}}\)	\(25\)
risch	\(\frac {i x^{3} \pi }{180 \ln \relax (5)^{2}}+\frac {x^{3} \ln \relax (3)}{180 \ln \relax (5)^{2}}-\frac {x^{3}}{45 \ln \relax (5)^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/60*(x^2*(ln(3)+I*Pi)-4*x^2)/ln(5)^2,x,method=_RETURNVERBOSE)

[Out]

1/180/ln(5)^2*x^3*(ln(3)+I*Pi-4)

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maxima [A]  time = 0.40, size = 24, normalized size = 1.20 \begin {gather*} -\frac {{\left (-i \, \pi - \log \relax (3)\right )} x^{3} + 4 \, x^{3}}{180 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/60*(x^2*(log(3)+I*pi)-4*x^2)/log(5)^2,x, algorithm="maxima")

[Out]

-1/180*((-I*pi - log(3))*x^3 + 4*x^3)/log(5)^2

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mupad [B]  time = 1.17, size = 19, normalized size = 0.95 \begin {gather*} \frac {x^3\,\left (\Pi -\ln \relax (3)\,1{}\mathrm {i}+4{}\mathrm {i}\right )\,1{}\mathrm {i}}{180\,{\ln \relax (5)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2*(Pi*1i + log(3)))/60 - x^2/15)/log(5)^2,x)

[Out]

(x^3*(Pi - log(3)*1i + 4i)*1i)/(180*log(5)^2)

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sympy [A]  time = 0.06, size = 17, normalized size = 0.85 \begin {gather*} \frac {x^{3} \left (-4 + \log {\relax (3 )} + i \pi \right )}{180 \log {\relax (5 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/60*(x**2*(ln(3)+I*pi)-4*x**2)/ln(5)**2,x)

[Out]

x**3*(-4 + log(3) + I*pi)/(180*log(5)**2)

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