\(\int e^{-x^4} (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+(-3+2 e^{x^4} x+12 x^4) \log (x)) \, dx\) [1898]

Optimal result

Integrand size = 49, antiderivative size = 23 \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=-\left (\left (3 e^{-x^4}-x\right ) (4+x (25+\log (x)))\right ) \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.37 (sec) , antiderivative size = 188, normalized size of antiderivative = 8.17, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6874, 2239, 2240, 2250, 2634, 15, 6696} \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=3 x \, _2F_2\left (\frac {1}{4},\frac {1}{4};\frac {5}{4},\frac {5}{4};-x^4\right )-\frac {12}{25} x^5 \, _2F_2\left (\frac {5}{4},\frac {5}{4};\frac {9}{4},\frac {9}{4};-x^4\right )-12 e^{-x^4}+\frac {39 x \Gamma \left (\frac {1}{4},x^4\right )}{2 \sqrt [4]{x^4}}+\frac {3 x \operatorname {Gamma}\left (\frac {5}{4}\right ) \log (x)}{\sqrt [4]{x^4}}-\frac {3 x \operatorname {Gamma}\left (\frac {1}{4}\right ) \log (x)}{4 \sqrt [4]{x^4}}+\frac {3 x \log (x) \Gamma \left (\frac {1}{4},x^4\right )}{4 \sqrt [4]{x^4}}+25 x^2+x^2 \log (x)-\frac {75 x^5 \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}-\frac {3 x^5 \log (x) \Gamma \left (\frac {5}{4},x^4\right )}{\left (x^4\right )^{5/4}}+4 x \]

[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 2239

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 2240

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 2250

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

Rule 2634

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 6696

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

Rule 6874

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

Rubi steps \begin{align*} \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) \text {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) \text {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 \operatorname {Gamma}\left (\frac {1}{4}\right ) \log (x)}{4 \sqrt [4]{x^4}}+\frac {3 x \operatorname {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{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=e^{-x^4} \left (-3+e^{x^4} x\right ) (4+25 x+x \log (x)) \]

[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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43

method	result	size
default	\(4 x +\left (-12-75 x -3 x \ln \left (x \right )\right ) {\mathrm e}^{-x^{4}}+25 x^{2}+x^{2} \ln \left (x \right )\)	\(33\)
parts	\(4 x +\left (-12-75 x -3 x \ln \left (x \right )\right ) {\mathrm e}^{-x^{4}}+25 x^{2}+x^{2} \ln \left (x \right )\)	\(33\)
norman	\(\left (-12+x^{2} {\mathrm e}^{x^{4}} \ln \left (x \right )-75 x +4 x \,{\mathrm e}^{x^{4}}-3 x \ln \left (x \right )+25 x^{2} {\mathrm e}^{x^{4}}\right ) {\mathrm e}^{-x^{4}}\)	\(44\)
risch	\(x^{2} \ln \left (x \right )+25 x^{2}-3 x \,{\mathrm e}^{-x^{4}} \ln \left (x \right )+4 x -75 \,{\mathrm e}^{-x^{4}} x -12 \,{\mathrm e}^{-x^{4}}\)	\(44\)
parallelrisch	\(-\left (12-x^{2} {\mathrm e}^{x^{4}} \ln \left (x \right )-25 x^{2} {\mathrm e}^{x^{4}}+3 x \ln \left (x \right )-4 x \,{\mathrm e}^{x^{4}}+75 x \right ) {\mathrm e}^{-x^{4}}\)	\(46\)

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78 \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx={\left ({\left (25 \, x^{2} + 4 \, x\right )} e^{\left (x^{4}\right )} + {\left (x^{2} e^{\left (x^{4}\right )} - 3 \, x\right )} \log \left (x\right ) - 75 \, x - 12\right )} e^{\left (-x^{4}\right )} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).

Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=x^{2} \log {\left (x \right )} + 25 x^{2} + 4 x + \left (- 3 x \log {\left (x \right )} - 75 x - 12\right ) e^{- x^{4}} \]

[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)

Maxima [F]

\[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=\int { {\left (300 \, x^{4} + 48 \, x^{3} + {\left (51 \, x + 4\right )} e^{\left (x^{4}\right )} + {\left (12 \, x^{4} + 2 \, x e^{\left (x^{4}\right )} - 3\right )} \log \left (x\right ) - 78\right )} e^{\left (-x^{4}\right )} \,d x } \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (19) = 38\).

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=x^{2} \log \left (x\right ) - 3 \, x e^{\left (-x^{4}\right )} \log \left (x\right ) + 25 \, x^{2} - 75 \, x e^{\left (-x^{4}\right )} + 4 \, x - 12 \, e^{\left (-x^{4}\right )} \]

[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)

Mupad [F(-1)]

Timed out. \[ \int e^{-x^4} \left (-78+48 x^3+300 x^4+e^{x^4} (4+51 x)+\left (-3+2 e^{x^4} x+12 x^4\right ) \log (x)\right ) \, dx=\int {\mathrm {e}}^{-x^4}\,\left ({\mathrm {e}}^{x^4}\,\left (51\,x+4\right )+48\,x^3+300\,x^4+\ln \left (x\right )\,\left (2\,x\,{\mathrm {e}}^{x^4}+12\,x^4-3\right )-78\right ) \,d x \]

[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)