3.102.77 \(\int (-2+e^{x^4} (6 x^2+8 x^6+(-4 x-8 x^5) \log (4))) \, dx\)

Optimal. Leaf size=19 \[ \left (2-2 e^{x^4} x^2\right ) (-x+\log (4)) \]

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Rubi [C]  time = 0.12, antiderivative size = 64, normalized size of antiderivative = 3.37, number of steps used = 10, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2226, 2218, 2211, 2204, 2212} \begin {gather*} -\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-2 e^{x^4} x^2 \log (4)-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2 + E^x^4*(6*x^2 + 8*x^6 + (-4*x - 8*x^5)*Log[4]),x]

[Out]

-2*x - (3*x^3*Gamma[3/4, -x^4])/(2*(-x^4)^(3/4)) - (2*x^7*Gamma[7/4, -x^4])/(-x^4)^(7/4) - 2*E^x^4*x^2*Log[4]

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 2211

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/(d*(m + 1))
, Subst[Int[F^(a + b*x^2), x], x, (c + d*x)^(m + 1)], x] /; FreeQ[{F, a, b, c, d, m, n}, x] && EqQ[n, 2*(m + 1
)]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), 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[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 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 2226

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-2 x+\int e^{x^4} \left (6 x^2+8 x^6+\left (-4 x-8 x^5\right ) \log (4)\right ) \, dx\\ &=-2 x+\int \left (6 e^{x^4} x^2+8 e^{x^4} x^6-4 e^{x^4} x \log (4)-8 e^{x^4} x^5 \log (4)\right ) \, dx\\ &=-2 x+6 \int e^{x^4} x^2 \, dx+8 \int e^{x^4} x^6 \, dx-(4 \log (4)) \int e^{x^4} x \, dx-(8 \log (4)) \int e^{x^4} x^5 \, dx\\ &=-2 x-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-2 e^{x^4} x^2 \log (4)-(2 \log (4)) \operatorname {Subst}\left (\int e^{x^2} \, dx,x,x^2\right )+(4 \log (4)) \int e^{x^4} x \, dx\\ &=-2 x-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-2 e^{x^4} x^2 \log (4)-\sqrt {\pi } \text {erfi}\left (x^2\right ) \log (4)+(2 \log (4)) \operatorname {Subst}\left (\int e^{x^2} \, dx,x,x^2\right )\\ &=-2 x-\frac {3 x^3 \Gamma \left (\frac {3}{4},-x^4\right )}{2 \left (-x^4\right )^{3/4}}-\frac {2 x^7 \Gamma \left (\frac {7}{4},-x^4\right )}{\left (-x^4\right )^{7/4}}-2 e^{x^4} x^2 \log (4)\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.07, size = 64, normalized size = 3.37 \begin {gather*} -2 x+\frac {3 \sqrt [4]{-x^4} \Gamma \left (\frac {3}{4},-x^4\right )}{2 x}-\frac {2 \sqrt [4]{-x^4} \Gamma \left (\frac {7}{4},-x^4\right )}{x}-2 e^{x^4} x^2 \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2 + E^x^4*(6*x^2 + 8*x^6 + (-4*x - 8*x^5)*Log[4]),x]

[Out]

-2*x + (3*(-x^4)^(1/4)*Gamma[3/4, -x^4])/(2*x) - (2*(-x^4)^(1/4)*Gamma[7/4, -x^4])/x - 2*E^x^4*x^2*Log[4]

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fricas [A]  time = 1.20, size = 21, normalized size = 1.11 \begin {gather*} 2 \, {\left (x^{3} - 2 \, x^{2} \log \relax (2)\right )} e^{\left (x^{4}\right )} - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x^5-4*x)*log(2)+8*x^6+6*x^2)*exp(x^4)-2,x, algorithm="fricas")

[Out]

2*(x^3 - 2*x^2*log(2))*e^(x^4) - 2*x

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giac [A]  time = 0.13, size = 24, normalized size = 1.26 \begin {gather*} 2 \, x^{3} e^{\left (x^{4}\right )} - 4 \, x^{2} e^{\left (x^{4}\right )} \log \relax (2) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x^5-4*x)*log(2)+8*x^6+6*x^2)*exp(x^4)-2,x, algorithm="giac")

[Out]

2*x^3*e^(x^4) - 4*x^2*e^(x^4)*log(2) - 2*x

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maple [A]  time = 0.10, size = 23, normalized size = 1.21

method	result	size
risch	\(\left (-4 x^{2} \ln \relax (2)+2 x^{3}\right ) {\mathrm e}^{x^{4}}-2 x\)	\(23\)
default	\(-2 x +2 x^{3} {\mathrm e}^{x^{4}}-4 \ln \relax (2) x^{2} {\mathrm e}^{x^{4}}\)	\(25\)
norman	\(-2 x +2 x^{3} {\mathrm e}^{x^{4}}-4 \ln \relax (2) x^{2} {\mathrm e}^{x^{4}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(-8*x^5-4*x)*ln(2)+8*x^6+6*x^2)*exp(x^4)-2,x,method=_RETURNVERBOSE)

[Out]

(-4*x^2*ln(2)+2*x^3)*exp(x^4)-2*x

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maxima [C]  time = 0.39, size = 80, normalized size = 4.21 \begin {gather*} -\frac {2 \, x^{7} \Gamma \left (\frac {7}{4}, -x^{4}\right )}{\left (-x^{4}\right )^{\frac {7}{4}}} - \frac {3 \, x^{3} \Gamma \left (\frac {3}{4}, -x^{4}\right )}{2 \, \left (-x^{4}\right )^{\frac {3}{4}}} + 2 i \, \sqrt {\pi } \operatorname {erf}\left (i \, x^{2}\right ) \log \relax (2) - 2 \, {\left (2 \, x^{2} e^{\left (x^{4}\right )} + i \, \sqrt {\pi } \operatorname {erf}\left (i \, x^{2}\right )\right )} \log \relax (2) - 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x^5-4*x)*log(2)+8*x^6+6*x^2)*exp(x^4)-2,x, algorithm="maxima")

[Out]

-2*x^7*gamma(7/4, -x^4)/(-x^4)^(7/4) - 3/2*x^3*gamma(3/4, -x^4)/(-x^4)^(3/4) + 2*I*sqrt(pi)*erf(I*x^2)*log(2)
- 2*(2*x^2*e^(x^4) + I*sqrt(pi)*erf(I*x^2))*log(2) - 2*x

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mupad [B]  time = 0.08, size = 24, normalized size = 1.26 \begin {gather*} 2\,x^3\,{\mathrm {e}}^{x^4}-2\,x-4\,x^2\,{\mathrm {e}}^{x^4}\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^4)*(6*x^2 - 2*log(2)*(4*x + 8*x^5) + 8*x^6) - 2,x)

[Out]

2*x^3*exp(x^4) - 2*x - 4*x^2*exp(x^4)*log(2)

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sympy [A]  time = 0.12, size = 20, normalized size = 1.05 \begin {gather*} - 2 x + \left (2 x^{3} - 4 x^{2} \log {\relax (2 )}\right ) e^{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x**5-4*x)*ln(2)+8*x**6+6*x**2)*exp(x**4)-2,x)

[Out]

-2*x + (2*x**3 - 4*x**2*log(2))*exp(x**4)

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