3.1.71 \(\int \frac {1}{6} (-24+e^{\frac {1}{6} (-14 x^2+7 x^3)} (-28 x+21 x^2) (i \pi +\log (2-\log (2)))) \, dx\)

Optimal. Leaf size=30 \[ -4 x+e^{\frac {7}{6} (-2+x) x^2} (i \pi +\log (2-\log (2))) \]

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Rubi [A]  time = 0.14, antiderivative size = 35, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 1593, 6706} \begin {gather*} -4 x+e^{-\frac {7}{6} \left (2 x^2-x^3\right )} (\log (2-\log (2))+i \pi ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-24 + E^((-14*x^2 + 7*x^3)/6)*(-28*x + 21*x^2)*(I*Pi + Log[2 - Log[2]]))/6,x]

[Out]

-4*x + (I*Pi + Log[2 - Log[2]])/E^((7*(2*x^2 - x^3))/6)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{6} \int \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx\\ &=-4 x+\frac {1}{6} (i \pi +\log (2-\log (2))) \int e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) \, dx\\ &=-4 x+\frac {1}{6} (i \pi +\log (2-\log (2))) \int e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} x (-28+21 x) \, dx\\ &=-4 x+e^{-\frac {7}{6} \left (2 x^2-x^3\right )} (i \pi +\log (2-\log (2)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 35, normalized size = 1.17 \begin {gather*} -4 x+e^{-\frac {7 x^2}{3}+\frac {7 x^3}{6}} (i \pi +\log (2-\log (2))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24 + E^((-14*x^2 + 7*x^3)/6)*(-28*x + 21*x^2)*(I*Pi + Log[2 - Log[2]]))/6,x]

[Out]

-4*x + E^((-7*x^2)/3 + (7*x^3)/6)*(I*Pi + Log[2 - Log[2]])

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fricas [A]  time = 0.59, size = 22, normalized size = 0.73 \begin {gather*} e^{\left (\frac {7}{6} \, x^{3} - \frac {7}{3} \, x^{2}\right )} \log \left (\log \relax (2) - 2\right ) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(21*x^2-28*x)*exp(7/6*x^3-7/3*x^2)*log(log(2)-2)-4,x, algorithm="fricas")

[Out]

e^(7/6*x^3 - 7/3*x^2)*log(log(2) - 2) - 4*x

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giac [A]  time = 0.29, size = 22, normalized size = 0.73 \begin {gather*} e^{\left (\frac {7}{6} \, x^{3} - \frac {7}{3} \, x^{2}\right )} \log \left (\log \relax (2) - 2\right ) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(21*x^2-28*x)*exp(7/6*x^3-7/3*x^2)*log(log(2)-2)-4,x, algorithm="giac")

[Out]

e^(7/6*x^3 - 7/3*x^2)*log(log(2) - 2) - 4*x

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maple [A]  time = 0.04, size = 20, normalized size = 0.67

method	result	size
risch	\(\ln \left (\ln \relax (2)-2\right ) {\mathrm e}^{\frac {7 \left (x -2\right ) x^{2}}{6}}-4 x\)	\(20\)
default	\(-4 x +{\mathrm e}^{\frac {7}{6} x^{3}-\frac {7}{3} x^{2}} \ln \left (\ln \relax (2)-2\right )\)	\(23\)
norman	\(-4 x +{\mathrm e}^{\frac {7}{6} x^{3}-\frac {7}{3} x^{2}} \ln \left (\ln \relax (2)-2\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/6*(21*x^2-28*x)*exp(7/6*x^3-7/3*x^2)*ln(ln(2)-2)-4,x,method=_RETURNVERBOSE)

[Out]

ln(ln(2)-2)*exp(7/6*(x-2)*x^2)-4*x

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maxima [A]  time = 0.42, size = 22, normalized size = 0.73 \begin {gather*} e^{\left (\frac {7}{6} \, x^{3} - \frac {7}{3} \, x^{2}\right )} \log \left (\log \relax (2) - 2\right ) - 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(21*x^2-28*x)*exp(7/6*x^3-7/3*x^2)*log(log(2)-2)-4,x, algorithm="maxima")

[Out]

e^(7/6*x^3 - 7/3*x^2)*log(log(2) - 2) - 4*x

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mupad [B]  time = 0.10, size = 22, normalized size = 0.73 \begin {gather*} \ln \left (\ln \relax (2)-2\right )\,{\mathrm {e}}^{\frac {7\,x^3}{6}-\frac {7\,x^2}{3}}-4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- (log(log(2) - 2)*exp((7*x^3)/6 - (7*x^2)/3)*(28*x - 21*x^2))/6 - 4,x)

[Out]

log(log(2) - 2)*exp((7*x^3)/6 - (7*x^2)/3) - 4*x

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sympy [A]  time = 3.54, size = 41, normalized size = 1.37 \begin {gather*} - 4 x - \left (- e^{- \frac {7 x^{2}}{3}} \log {\left (2 - \log {\relax (2 )} \right )} - i \pi e^{- \frac {7 x^{2}}{3}}\right ) e^{\frac {7 x^{3}}{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/6*(21*x**2-28*x)*exp(7/6*x**3-7/3*x**2)*ln(ln(2)-2)-4,x)

[Out]

-4*x - (-exp(-7*x**2/3)*log(2 - log(2)) - I*pi*exp(-7*x**2/3))*exp(7*x**3/6)

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