\(\int \frac {1}{6} (-24+e^{\frac {1}{6} (-14 x^2+7 x^3)} (-28 x+21 x^2) (i \pi +\log (2-\log (2)))) \, dx\) [71]

Optimal result

Integrand size = 46, antiderivative size = 30 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=-4 x+e^{\frac {7}{6} (-2+x) x^2} (i \pi +\log (2-\log (2))) \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {12, 1607, 6838} \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=-4 x+e^{-\frac {7}{6} \left (2 x^2-x^3\right )} (\log (2-\log (2))+i \pi ) \]

[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 1607

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 6838

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{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=-4 x+e^{\frac {7}{6} (-2+x) x^2} (i \pi +\log (2-\log (2))) \]

[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*(-2 + x)*x^2)/6)*(I*Pi + Log[2 - Log[2]])

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67

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

[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*(-2+x)*x^2)-4*x

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=e^{\left (\frac {7}{6} \, x^{3} - \frac {7}{3} \, x^{2}\right )} \log \left (\log \left (2\right ) - 2\right ) - 4 \, x \]

[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

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=- 4 x + \left (e^{- \frac {7 x^{2}}{3}} \log {\left (2 - \log {\left (2 \right )} \right )} + i \pi e^{- \frac {7 x^{2}}{3}}\right ) e^{\frac {7 x^{3}}{6}} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=e^{\left (\frac {7}{6} \, x^{3} - \frac {7}{3} \, x^{2}\right )} \log \left (\log \left (2\right ) - 2\right ) - 4 \, x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=e^{\left (\frac {7}{6} \, x^{3} - \frac {7}{3} \, x^{2}\right )} \log \left (\log \left (2\right ) - 2\right ) - 4 \, x \]

[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

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {1}{6} \left (-24+e^{\frac {1}{6} \left (-14 x^2+7 x^3\right )} \left (-28 x+21 x^2\right ) (i \pi +\log (2-\log (2)))\right ) \, dx=\ln \left (\ln \left (2\right )-2\right )\,{\mathrm {e}}^{\frac {7\,x^3}{6}-\frac {7\,x^2}{3}}-4\,x \]

[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