3.19.98 \(\int e^{-x^4} (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+(-3+2 e^{x^4} x+12 x^4) \log (x)) \, dx\)

Optimal. Leaf size=23 \[ -\left (\left (3 e^{-x^4}-x\right ) (4+x (25+\log (x)))\right ) \]

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Rubi [C]  time = 0.63, antiderivative size = 188, normalized size of antiderivative = 8.17, number of steps used = 13, number of rules used = 7, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6742, 2208, 2209, 2218, 2554, 15, 6561} \begin {gather*} 3 x \, _2F_2\left (\frac {1}{4},\frac {1}{4};\frac {5}{4},\frac {5}{4};-x^4\right )-\frac {12}{25} x^5 \, _2F_2\left (\frac {5}{4},\frac {5}{4};\frac {9}{4},\frac {9}{4};-x^4\right )-12 e^{-x^4}+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}+\frac {3 x \Gamma \left (\frac {5}{4}\right ) \log (x)}{\sqrt [4]{x^4}}-\frac {3 x \Gamma \left (\frac {1}{4}\right ) \log (x)}{4 \sqrt [4]{x^4}}+\frac {3 x \log (x) \Gamma \left (\frac {1}{4},x^4\right )}{4 \sqrt [4]{x^4}}+25 x^2+x^2 \log (x)-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}-\frac {3 x^5 \log (x) \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-78 + 48*x^3 + 300*x^4 + E^x^4*(4 + 51*x) + (-3 + 2*E^x^4*x + 12*x^4)*Log[x])/E^x^4,x]

[Out]

-12/E^x^4 + 4*x + 25*x^2 + (39*x*Gamma[1/4, x^4])/(2*(x^4)^(1/4)) - (75*x^5*Gamma[5/4, x^4])/(x^4)^(5/4) + 3*x
*HypergeometricPFQ[{1/4, 1/4}, {5/4, 5/4}, -x^4] - (12*x^5*HypergeometricPFQ[{5/4, 5/4}, {9/4, 9/4}, -x^4])/25
+ x^2*Log[x] - (3*x*Gamma[1/4]*Log[x])/(4*(x^4)^(1/4)) + (3*x*Gamma[5/4]*Log[x])/(x^4)^(1/4) + (3*x*Gamma[1/4
, x^4]*Log[x])/(4*(x^4)^(1/4)) - (3*x^5*Gamma[5/4, x^4]*Log[x])/(x^4)^(5/4)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

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 6561

Int[Gamma[n_, (b_.)*(x_)]/(x_), x_Symbol] :> Simp[Gamma[n]*Log[x], x] - Simp[((b*x)^n*HypergeometricPFQ[{n, n}
, {1 + n, 1 + n}, -(b*x)])/n^2, x] /; FreeQ[{b, n}, x] &&  !IntegerQ[n]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4-78 e^{-x^4}+51 x+48 e^{-x^4} x^3+300 e^{-x^4} x^4+e^{-x^4} \left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx\\ &=4 x+\frac {51 x^2}{2}+48 \int e^{-x^4} x^3 \, dx-78 \int e^{-x^4} \, dx+300 \int e^{-x^4} x^4 \, dx+\int e^{-x^4} \left (-3+2 e^{x^4} x+12 x^4\right ) \log (x) \, dx\\ &=-12 e^{-x^4}+4 x+\frac {51 x^2}{2}+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\int \left (x+\frac {3 \Gamma \left (\frac {1}{4},x^4\right )}{4 \sqrt [4]{x^4}}-\frac {3 \Gamma \left (\frac {5}{4},x^4\right )}{\sqrt [4]{x^4}}\right ) \, dx\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\frac {3}{4} \int \frac {\Gamma \left (\frac {1}{4},x^4\right )}{\sqrt [4]{x^4}} \, dx+3 \int \frac {\Gamma \left (\frac {5}{4},x^4\right )}{\sqrt [4]{x^4}} \, dx\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\frac {(3 x) \int \frac {\Gamma \left (\frac {1}{4},x^4\right )}{x} \, dx}{4 \sqrt [4]{x^4}}+\frac {(3 x) \int \frac {\Gamma \left (\frac {5}{4},x^4\right )}{x} \, dx}{\sqrt [4]{x^4}}\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+x^2 \log (x)+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}-\frac {(3 x) \operatorname {Subst}\left (\int \frac {\Gamma \left (\frac {1}{4},x\right )}{x} \, dx,x,x^4\right )}{16 \sqrt [4]{x^4}}+\frac {(3 x) \operatorname {Subst}\left (\int \frac {\Gamma \left (\frac {5}{4},x\right )}{x} \, dx,x,x^4\right )}{4 \sqrt [4]{x^4}}\\ &=-12 e^{-x^4}+4 x+25 x^2+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+3 x \, _2F_2\left (\frac {1}{4},\frac {1}{4};\frac {5}{4},\frac {5}{4};-x^4\right )-\frac {12}{25} x^5 \, _2F_2\left (\frac {5}{4},\frac {5}{4};\frac {9}{4},\frac {9}{4};-x^4\right )+x^2 \log (x)-\frac {3 x \Gamma \left (\frac {1}{4}\right ) \log (x)}{4 \sqrt [4]{x^4}}+\frac {3 x \Gamma \left (\frac {5}{4}\right ) \log (x)}{\sqrt [4]{x^4}}+\frac {3 x \Gamma \left (\frac {1}{4},x^4\right ) \log (x)}{4 \sqrt [4]{x^4}}-\frac {3 x^5 \Gamma \left (\frac {5}{4},x^4\right ) \log (x)}{\left (x^4\right )^{5/4}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 26, normalized size = 1.13 \begin {gather*} e^{-x^4} \left (-3+e^{x^4} x\right ) (4+25 x+x \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-78 + 48*x^3 + 300*x^4 + E^x^4*(4 + 51*x) + (-3 + 2*E^x^4*x + 12*x^4)*Log[x])/E^x^4,x]

[Out]

((-3 + E^x^4*x)*(4 + 25*x + x*Log[x]))/E^x^4

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fricas [B]  time = 0.83, size = 41, normalized size = 1.78 \begin {gather*} {\left ({\left (25 \, x^{2} + 4 \, x\right )} e^{\left (x^{4}\right )} + {\left (x^{2} e^{\left (x^{4}\right )} - 3 \, x\right )} \log \relax (x) - 75 \, x - 12\right )} e^{\left (-x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(x^4)+12*x^4-3)*log(x)+(51*x+4)*exp(x^4)+300*x^4+48*x^3-78)/exp(x^4),x, algorithm="fricas")

[Out]

((25*x^2 + 4*x)*e^(x^4) + (x^2*e^(x^4) - 3*x)*log(x) - 75*x - 12)*e^(-x^4)

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giac [B]  time = 0.19, size = 43, normalized size = 1.87 \begin {gather*} x^{2} \log \relax (x) - 3 \, x e^{\left (-x^{4}\right )} \log \relax (x) + 25 \, x^{2} - 75 \, x e^{\left (-x^{4}\right )} + 4 \, x - 12 \, e^{\left (-x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(x^4)+12*x^4-3)*log(x)+(51*x+4)*exp(x^4)+300*x^4+48*x^3-78)/exp(x^4),x, algorithm="giac")

[Out]

x^2*log(x) - 3*x*e^(-x^4)*log(x) + 25*x^2 - 75*x*e^(-x^4) + 4*x - 12*e^(-x^4)

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maple [A]  time = 0.05, size = 33, normalized size = 1.43

method	result	size
default	\(4 x +\left (-12-75 x -3 x \ln \relax (x )\right ) {\mathrm e}^{-x^{4}}+25 x^{2}+x^{2} \ln \relax (x )\)	\(33\)
norman	\(\left (-12+x^{2} {\mathrm e}^{x^{4}} \ln \relax (x )-75 x +4 x \,{\mathrm e}^{x^{4}}-3 x \ln \relax (x )+25 x^{2} {\mathrm e}^{x^{4}}\right ) {\mathrm e}^{-x^{4}}\)	\(44\)
risch	\(x \left (x \,{\mathrm e}^{x^{4}}-3\right ) {\mathrm e}^{-x^{4}} \ln \relax (x )+\left (25 x^{2} {\mathrm e}^{x^{4}}+4 x \,{\mathrm e}^{x^{4}}-75 x -12\right ) {\mathrm e}^{-x^{4}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*exp(x^4)+12*x^4-3)*ln(x)+(51*x+4)*exp(x^4)+300*x^4+48*x^3-78)/exp(x^4),x,method=_RETURNVERBOSE)

[Out]

4*x+(-12-75*x-3*x*ln(x))/exp(x^4)+25*x^2+x^2*ln(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {75 \, x^{5} \Gamma \left (\frac {5}{4}, x^{4}\right )}{{\left (x^{4}\right )}^{\frac {5}{4}}} + x^{2} \log \relax (x) - 3 \, x e^{\left (-x^{4}\right )} \log \relax (x) + 25 \, x^{2} + \frac {39 \, x \Gamma \left (\frac {1}{4}, x^{4}\right )}{2 \, {\left (x^{4}\right )}^{\frac {1}{4}}} + 4 \, x - 12 \, e^{\left (-x^{4}\right )} + 3 \, \int e^{\left (-x^{4}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(x^4)+12*x^4-3)*log(x)+(51*x+4)*exp(x^4)+300*x^4+48*x^3-78)/exp(x^4),x, algorithm="maxima")

[Out]

-75*x^5*gamma(5/4, x^4)/(x^4)^(5/4) + x^2*log(x) - 3*x*e^(-x^4)*log(x) + 25*x^2 + 39/2*x*gamma(1/4, x^4)/(x^4)
^(1/4) + 4*x - 12*e^(-x^4) + 3*integrate(e^(-x^4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\mathrm {e}}^{-x^4}\,\left ({\mathrm {e}}^{x^4}\,\left (51\,x+4\right )+48\,x^3+300\,x^4+\ln \relax (x)\,\left (2\,x\,{\mathrm {e}}^{x^4}+12\,x^4-3\right )-78\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x^4)*(exp(x^4)*(51*x + 4) + 48*x^3 + 300*x^4 + log(x)*(2*x*exp(x^4) + 12*x^4 - 3) - 78),x)

[Out]

int(exp(-x^4)*(exp(x^4)*(51*x + 4) + 48*x^3 + 300*x^4 + log(x)*(2*x*exp(x^4) + 12*x^4 - 3) - 78), x)

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sympy [B]  time = 0.33, size = 32, normalized size = 1.39 \begin {gather*} x^{2} \log {\relax (x )} + 25 x^{2} + 4 x + \left (- 3 x \log {\relax (x )} - 75 x - 12\right ) e^{- x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*exp(x**4)+12*x**4-3)*ln(x)+(51*x+4)*exp(x**4)+300*x**4+48*x**3-78)/exp(x**4),x)

[Out]

x**2*log(x) + 25*x**2 + 4*x + (-3*x*log(x) - 75*x - 12)*exp(-x**4)

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