3.21.79 \(\int \frac {92 e^{\frac {30 x^2}{23}}+e^{\frac {30 x^2}{23}} (-92+240 x^2) \log (4 x)}{23 x^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x} \]

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Rubi [A]  time = 0.23, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {12, 14, 2214, 2204, 2554, 6360} \begin {gather*} \frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(92*E^((30*x^2)/23) + E^((30*x^2)/23)*(-92 + 240*x^2)*Log[4*x])/(23*x^2),x]

[Out]

(4*E^((30*x^2)/23)*Log[4*x])/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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2214

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*F^(a + b*(c + d*x)^n))/(d*(m + 1)), x] - Dist[(b*n*Log[F])/(m + 1), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2554

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[(w*D[u, x]
)/u, x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 6360

Int[Erfi[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(2*b*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2}, b^2*x^2])/Sqrt[P
i], x] /; FreeQ[b, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{23} \int \frac {92 e^{\frac {30 x^2}{23}}+e^{\frac {30 x^2}{23}} \left (-92+240 x^2\right ) \log (4 x)}{x^2} \, dx\\ &=\frac {1}{23} \int \left (\frac {92 e^{\frac {30 x^2}{23}}}{x^2}+240 e^{\frac {30 x^2}{23}} \log (4 x)-\frac {92 e^{\frac {30 x^2}{23}} \log (4 x)}{x^2}\right ) \, dx\\ &=4 \int \frac {e^{\frac {30 x^2}{23}}}{x^2} \, dx-4 \int \frac {e^{\frac {30 x^2}{23}} \log (4 x)}{x^2} \, dx+\frac {240}{23} \int e^{\frac {30 x^2}{23}} \log (4 x) \, dx\\ &=-\frac {4 e^{\frac {30 x^2}{23}}}{x}+\frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x}+4 \int \left (-\frac {e^{\frac {30 x^2}{23}}}{x^2}+\frac {\sqrt {\frac {30 \pi }{23}} \text {erfi}\left (\sqrt {\frac {30}{23}} x\right )}{x}\right ) \, dx+\frac {240}{23} \int e^{\frac {30 x^2}{23}} \, dx-\frac {240}{23} \int \frac {\sqrt {\frac {23 \pi }{30}} \text {erfi}\left (\sqrt {\frac {30}{23}} x\right )}{2 x} \, dx\\ &=-\frac {4 e^{\frac {30 x^2}{23}}}{x}+4 \sqrt {\frac {30 \pi }{23}} \text {erfi}\left (\sqrt {\frac {30}{23}} x\right )+\frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x}-4 \int \frac {e^{\frac {30 x^2}{23}}}{x^2} \, dx\\ &=4 \sqrt {\frac {30 \pi }{23}} \text {erfi}\left (\sqrt {\frac {30}{23}} x\right )+\frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x}-\frac {240}{23} \int e^{\frac {30 x^2}{23}} \, dx\\ &=\frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 e^{\frac {30 x^2}{23}} \log (4 x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(92*E^((30*x^2)/23) + E^((30*x^2)/23)*(-92 + 240*x^2)*Log[4*x])/(23*x^2),x]

[Out]

(4*E^((30*x^2)/23)*Log[4*x])/x

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fricas [A]  time = 0.59, size = 15, normalized size = 0.83 \begin {gather*} \frac {4 \, e^{\left (\frac {30}{23} \, x^{2}\right )} \log \left (4 \, x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/23*((240*x^2-92)*exp(30/23*x^2)*log(4*x)+92*exp(30/23*x^2))/x^2,x, algorithm="fricas")

[Out]

4*e^(30/23*x^2)*log(4*x)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left ({\left (60 \, x^{2} - 23\right )} e^{\left (\frac {30}{23} \, x^{2}\right )} \log \left (4 \, x\right ) + 23 \, e^{\left (\frac {30}{23} \, x^{2}\right )}\right )}}{23 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/23*((240*x^2-92)*exp(30/23*x^2)*log(4*x)+92*exp(30/23*x^2))/x^2,x, algorithm="giac")

[Out]

integrate(4/23*((60*x^2 - 23)*e^(30/23*x^2)*log(4*x) + 23*e^(30/23*x^2))/x^2, x)

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maple [A]  time = 0.13, size = 16, normalized size = 0.89

method	result	size
norman	\(\frac {4 \,{\mathrm e}^{\frac {30 x^{2}}{23}} \ln \left (4 x \right )}{x}\)	\(16\)
risch	\(\frac {4 \,{\mathrm e}^{\frac {30 x^{2}}{23}} \ln \left (4 x \right )}{x}\)	\(16\)
meijerg	\(\left (\frac {2 i \sqrt {690}\, \ln \left (30\right ) \left (\frac {i \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}\right )}{23}-\frac {2 i \sqrt {690}\, \ln \left (23\right ) \left (\frac {i \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}\right )}{23}-\frac {2 \pi \sqrt {30}\, \sqrt {23}\, \left (\frac {i \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}\right )}{23}-\frac {4 i \sqrt {690}\, \left (\frac {i \left (-\gamma -2 \ln \relax (2)\right ) \sqrt {\pi }}{2 \sqrt {-x^{2}}}+\frac {15 i \left (-\frac {23 \ln \left (-\frac {30 x^{2}}{23}\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{30}+\frac {23 \left (\gamma +3 \ln \relax (2)+\ln \relax (3)+\ln \relax (5)-\ln \left (23\right )+\ln \left (-x^{2}\right )\right ) \sqrt {\pi }}{30}\right )}{23 \sqrt {-x^{2}}}+\frac {i \ln \left (-\frac {30 x^{2}}{23}\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{2 \sqrt {-x^{2}}}+\frac {i \ln \left (23\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{2 \sqrt {-x^{2}}}-\frac {\pi ^{\frac {3}{2}}}{2 \sqrt {-x^{2}}}+\frac {i \ln \relax (x ) \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {i \ln \relax (x ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}-\frac {i \sqrt {\pi }\, \ln \left (-\frac {30 x^{2}}{23}\right )}{2 \sqrt {-x^{2}}}-\frac {i \ln \left (30\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{2 \sqrt {-x^{2}}}-\frac {i \ln \left (23\right ) \sqrt {\pi }}{2 \sqrt {-x^{2}}}+\frac {i \ln \left (30\right ) \sqrt {\pi }}{2 \sqrt {-x^{2}}}+\frac {\pi ^{\frac {3}{2}} \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{2 \sqrt {-x^{2}}}\right )}{23}\right ) x +\frac {8 \ln \relax (2) \sqrt {30}\, \sqrt {23}\, \sqrt {\pi }\, \erfi \left (\frac {x \sqrt {30}\, \sqrt {23}}{23}\right )}{23}+\left (\frac {i \sqrt {690}\, \ln \left (30\right ) \left (-\frac {2 i \sqrt {\pi }}{\sqrt {-x^{2}}}+\frac {2 i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}\right )}{23}-\frac {i \sqrt {690}\, \ln \left (23\right ) \left (-\frac {2 i \sqrt {\pi }}{\sqrt {-x^{2}}}+\frac {2 i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}\right )}{23}-\frac {\pi \sqrt {30}\, \sqrt {23}\, \left (-\frac {2 i \sqrt {\pi }}{\sqrt {-x^{2}}}+\frac {2 i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}\right )}{23}-\frac {2 i \sqrt {690}\, \left (-\frac {i \left (-\gamma -2 \ln \relax (2)\right ) \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {i \ln \left (-\frac {30 x^{2}}{23}\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}+\frac {2 i \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}+\frac {\pi ^{\frac {3}{2}}}{\sqrt {-x^{2}}}-\frac {i \ln \left (30\right ) \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {i \ln \left (23\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}-\frac {2 i \ln \relax (x ) \sqrt {\pi }}{\sqrt {-x^{2}}}+\frac {i \ln \left (30\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}+\frac {i \sqrt {\pi }\, \ln \left (-\frac {30 x^{2}}{23}\right )}{\sqrt {-x^{2}}}-\frac {30 i \left (-\frac {23 \ln \left (-\frac {30 x^{2}}{23}\right ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{30}+\frac {23 \left (\gamma +3 \ln \relax (2)+\ln \relax (3)+\ln \relax (5)-\ln \left (23\right )+\ln \left (-x^{2}\right )\right ) \sqrt {\pi }}{30}\right )}{23 \sqrt {-x^{2}}}+\frac {2 i \ln \relax (x ) \sqrt {\pi }\, \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}-\frac {\pi ^{\frac {3}{2}} \mathrm {erfc}\left (\frac {\sqrt {30}\, \sqrt {23}\, \sqrt {-x^{2}}}{23}\right )}{\sqrt {-x^{2}}}+\frac {i \ln \left (23\right ) \sqrt {\pi }}{\sqrt {-x^{2}}}-\frac {2 i \sqrt {\pi }}{\sqrt {-x^{2}}}\right )}{23}\right ) x +\frac {-2 \ln \left (30\right ) {\mathrm e}^{\frac {30 x^{2}}{23}}+2 \ln \left (23\right ) {\mathrm e}^{\frac {30 x^{2}}{23}}-2 i \pi \,{\mathrm e}^{\frac {30 x^{2}}{23}}-\frac {2 i \sqrt {690}\, \left (\frac {i \sqrt {690}\, \ln \relax (x ) {\mathrm e}^{\frac {30 x^{2}}{23}}}{15}+\frac {i \sqrt {690}\, \ln \left (30\right ) {\mathrm e}^{\frac {30 x^{2}}{23}}}{30}-\frac {i \sqrt {690}\, \ln \left (23\right ) {\mathrm e}^{\frac {30 x^{2}}{23}}}{30}-\frac {\pi \sqrt {30}\, \sqrt {23}\, {\mathrm e}^{\frac {30 x^{2}}{23}}}{30}+\frac {i \sqrt {690}\, {\mathrm e}^{\frac {30 x^{2}}{23}}}{15}\right )}{23}}{x}-\frac {i \sqrt {690}\, \left (\frac {240 \ln \relax (2)}{23}-\frac {120}{23}\right ) \left (\frac {i \sqrt {690}\, {\mathrm e}^{\frac {30 x^{2}}{23}}}{15 x}-2 i \sqrt {\pi }\, \erfi \left (\frac {x \sqrt {30}\, \sqrt {23}}{23}\right )\right )}{60}\)	\(1181\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/23*((240*x^2-92)*exp(30/23*x^2)*ln(4*x)+92*exp(30/23*x^2))/x^2,x,method=_RETURNVERBOSE)

[Out]

4*exp(30/23*x^2)/x*ln(4*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, \sqrt {\frac {30}{23}} \sqrt {-x^{2}} \Gamma \left (-\frac {1}{2}, -\frac {30}{23} \, x^{2}\right )}{x} + \frac {4 \, e^{\left (\frac {30}{23} \, x^{2}\right )} \log \relax (x)}{x} + \frac {4}{23} \, \int \frac {{\left (120 \, x^{2} \log \relax (2) - 46 \, \log \relax (2) - 23\right )} e^{\left (\frac {30}{23} \, x^{2}\right )}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/23*((240*x^2-92)*exp(30/23*x^2)*log(4*x)+92*exp(30/23*x^2))/x^2,x, algorithm="maxima")

[Out]

-2*sqrt(30/23)*sqrt(-x^2)*gamma(-1/2, -30/23*x^2)/x + 4*e^(30/23*x^2)*log(x)/x + 4/23*integrate((120*x^2*log(2
) - 46*log(2) - 23)*e^(30/23*x^2)/x^2, x)

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mupad [B]  time = 1.44, size = 15, normalized size = 0.83 \begin {gather*} \frac {4\,\ln \left (4\,x\right )\,{\mathrm {e}}^{\frac {30\,x^2}{23}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp((30*x^2)/23) + (log(4*x)*exp((30*x^2)/23)*(240*x^2 - 92))/23)/x^2,x)

[Out]

(4*log(4*x)*exp((30*x^2)/23))/x

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sympy [A]  time = 0.26, size = 15, normalized size = 0.83 \begin {gather*} \frac {4 e^{\frac {30 x^{2}}{23}} \log {\left (4 x \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/23*((240*x**2-92)*exp(30/23*x**2)*ln(4*x)+92*exp(30/23*x**2))/x**2,x)

[Out]

4*exp(30*x**2/23)*log(4*x)/x

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