3.91.65 \(\int \frac {-1-4 x^2+e^{2+x+(-16-8 x-x^2) \log (7)} (-2 x^2+(16 x^2+4 x^3) \log (7))}{2 x^2} \, dx\) [9065]

Optimal. Leaf size=28 \[ e-e^{2+x-(4+x)^2 \log (7)}+\frac {1}{2 x}-2 x \]

[Out]

exp(1)-2*x+1/2/x-exp(x+2-ln(7)*(4+x)^2)

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Rubi [A]
time = 0.14, antiderivative size = 47, normalized size of antiderivative = 1.68, number of steps used = 7, number of rules used = 4, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 14, 2325, 2268} \begin {gather*} -\frac {7^{-x^2} \log (49) e^{x (1-8 \log (7))+2 (1-8 \log (7))}}{2 \log (7)}-2 x+\frac {1}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 4*x^2 + E^(2 + x + (-16 - 8*x - x^2)*Log[7])*(-2*x^2 + (16*x^2 + 4*x^3)*Log[7]))/(2*x^2),x]

[Out]

1/(2*x) - 2*x - (E^(2*(1 - 8*Log[7]) + x*(1 - 8*Log[7]))*Log[49])/(2*7^x^2*Log[7])

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2268

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[e*(F^(a + b*x + c*x^2)/(2
*c*Log[F])), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 2325

Int[(u_.)*(F_)^(v_)*(G_)^(w_), x_Symbol] :> With[{z = v*Log[F] + w*Log[G]}, Int[u*NormalizeIntegrand[E^z, x],
x] /; BinomialQ[z, x] || (PolynomialQ[z, x] && LeQ[Exponent[z, x], 2])] /; FreeQ[{F, G}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-1-4 x^2+e^{2+x+\left (-16-8 x-x^2\right ) \log (7)} \left (-2 x^2+\left (16 x^2+4 x^3\right ) \log (7)\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {-1-4 x^2}{x^2}+2\ 7^{-(4+x)^2} e^{2+x} (-1+8 \log (7)+x \log (49))\right ) \, dx\\ &=\frac {1}{2} \int \frac {-1-4 x^2}{x^2} \, dx+\int 7^{-(4+x)^2} e^{2+x} (-1+8 \log (7)+x \log (49)) \, dx\\ &=\frac {1}{2} \int \left (-4-\frac {1}{x^2}\right ) \, dx+\int \exp \left (2 (1-8 \log (7))+x (1-8 \log (7))-x^2 \log (7)\right ) (-1+8 \log (7)+x \log (49)) \, dx\\ &=\frac {1}{2 x}-2 x-\frac {7^{-x^2} e^{2 (1-8 \log (7))+x (1-8 \log (7))} \log (49)}{2 \log (7)}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.21, size = 27, normalized size = 0.96 \begin {gather*} -7^{-(4+x)^2} e^{2+x}+\frac {1}{2 x}-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 4*x^2 + E^(2 + x + (-16 - 8*x - x^2)*Log[7])*(-2*x^2 + (16*x^2 + 4*x^3)*Log[7]))/(2*x^2),x]

[Out]

-(E^(2 + x)/7^(4 + x)^2) + 1/(2*x) - 2*x

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 1.03, size = 190, normalized size = 6.79

method	result	size
risch	\(-2 x +\frac {1}{2 x}-\left (\frac {1}{7}\right )^{\left (4+x \right )^{2}} {\mathrm e}^{2+x}\)	\(23\)
norman	\(\frac {\frac {1}{2}-2 x^{2}-x \,{\mathrm e}^{\left (-x^{2}-8 x -16\right ) \ln \left (7\right )+2+x}}{x}\)	\(32\)
default	\(-{\mathrm e}^{-x^{2} \ln \left (7\right )+\left (-8 \ln \left (7\right )+1\right ) x -16 \ln \left (7\right )+2}+\frac {\left (-8 \ln \left (7\right )+1\right ) \sqrt {\pi }\, {\mathrm e}^{-16 \ln \left (7\right )+2+\frac {\left (-8 \ln \left (7\right )+1\right )^{2}}{4 \ln \left (7\right )}} \erf \left (\sqrt {\ln \left (7\right )}\, x -\frac {-8 \ln \left (7\right )+1}{2 \sqrt {\ln \left (7\right )}}\right )}{2 \sqrt {\ln \left (7\right )}}+4 \sqrt {\ln \left (7\right )}\, \sqrt {\pi }\, {\mathrm e}^{-16 \ln \left (7\right )+2+\frac {\left (-8 \ln \left (7\right )+1\right )^{2}}{4 \ln \left (7\right )}} \erf \left (\sqrt {\ln \left (7\right )}\, x -\frac {-8 \ln \left (7\right )+1}{2 \sqrt {\ln \left (7\right )}}\right )-\frac {\sqrt {\pi }\, {\mathrm e}^{-16 \ln \left (7\right )+2+\frac {\left (-8 \ln \left (7\right )+1\right )^{2}}{4 \ln \left (7\right )}} \erf \left (\sqrt {\ln \left (7\right )}\, x -\frac {-8 \ln \left (7\right )+1}{2 \sqrt {\ln \left (7\right )}}\right )}{2 \sqrt {\ln \left (7\right )}}-2 x +\frac {1}{2 x}\)	\(190\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((4*x^3+16*x^2)*ln(7)-2*x^2)*exp((-x^2-8*x-16)*ln(7)+2+x)-4*x^2-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

-exp(-x^2*ln(7)+(-8*ln(7)+1)*x-16*ln(7)+2)+1/2/ln(7)^(1/2)*(-8*ln(7)+1)*Pi^(1/2)*exp(-16*ln(7)+2+1/4*(-8*ln(7)
+1)^2/ln(7))*erf(ln(7)^(1/2)*x-1/2*(-8*ln(7)+1)/ln(7)^(1/2))+4*ln(7)^(1/2)*Pi^(1/2)*exp(-16*ln(7)+2+1/4*(-8*ln
(7)+1)^2/ln(7))*erf(ln(7)^(1/2)*x-1/2*(-8*ln(7)+1)/ln(7)^(1/2))-1/2*Pi^(1/2)*exp(-16*ln(7)+2+1/4*(-8*ln(7)+1)^
2/ln(7))/ln(7)^(1/2)*erf(ln(7)^(1/2)*x-1/2*(-8*ln(7)+1)/ln(7)^(1/2))-2*x+1/2/x

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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.51, size = 231, normalized size = 8.25 \begin {gather*} \frac {4}{33232930569601} \, \sqrt {\pi } \operatorname {erf}\left (x \sqrt {\log \left (7\right )} + \frac {8 \, \log \left (7\right ) - 1}{2 \, \sqrt {\log \left (7\right )}}\right ) e^{\left (\frac {{\left (8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )} + 2\right )} \sqrt {\log \left (7\right )} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )}^{2}}{\log \left (7\right )}}\right ) - 1\right )} {\left (8 \, \log \left (7\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )}^{2}}{\log \left (7\right )}} \left (-\log \left (7\right )\right )^{\frac {3}{2}}} + \frac {2 \, e^{\left (-\frac {{\left (2 \, x \log \left (7\right ) + 8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )}\right )} \log \left (7\right )}{\left (-\log \left (7\right )\right )^{\frac {3}{2}}}\right )} e^{\left (\frac {{\left (8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )} + 2\right )} \log \left (7\right )}{66465861139202 \, \sqrt {-\log \left (7\right )}} - \frac {\sqrt {\pi } \operatorname {erf}\left (x \sqrt {\log \left (7\right )} + \frac {8 \, \log \left (7\right ) - 1}{2 \, \sqrt {\log \left (7\right )}}\right ) e^{\left (\frac {{\left (8 \, \log \left (7\right ) - 1\right )}^{2}}{4 \, \log \left (7\right )} + 2\right )}}{66465861139202 \, \sqrt {\log \left (7\right )}} - 2 \, x + \frac {1}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x)-4*x^2-1)/x^2,x, algorithm="maxima")

[Out]

4/33232930569601*sqrt(pi)*erf(x*sqrt(log(7)) + 1/2*(8*log(7) - 1)/sqrt(log(7)))*e^(1/4*(8*log(7) - 1)^2/log(7)
+ 2)*sqrt(log(7)) - 1/66465861139202*(sqrt(pi)*(2*x*log(7) + 8*log(7) - 1)*(erf(1/2*sqrt((2*x*log(7) + 8*log(
7) - 1)^2/log(7))) - 1)*(8*log(7) - 1)/(sqrt((2*x*log(7) + 8*log(7) - 1)^2/log(7))*(-log(7))^(3/2)) + 2*e^(-1/
4*(2*x*log(7) + 8*log(7) - 1)^2/log(7))*log(7)/(-log(7))^(3/2))*e^(1/4*(8*log(7) - 1)^2/log(7) + 2)*log(7)/sqr
t(-log(7)) - 1/66465861139202*sqrt(pi)*erf(x*sqrt(log(7)) + 1/2*(8*log(7) - 1)/sqrt(log(7)))*e^(1/4*(8*log(7)
- 1)^2/log(7) + 2)/sqrt(log(7)) - 2*x + 1/2/x

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Fricas [A]
time = 0.43, size = 31, normalized size = 1.11 \begin {gather*} -\frac {4 \, x^{2} + 2 \, x e^{\left (-{\left (x^{2} + 8 \, x + 16\right )} \log \left (7\right ) + x + 2\right )} - 1}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x)-4*x^2-1)/x^2,x, algorithm="fricas")

[Out]

-1/2*(4*x^2 + 2*x*e^(-(x^2 + 8*x + 16)*log(7) + x + 2) - 1)/x

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Sympy [A]
time = 0.08, size = 26, normalized size = 0.93 \begin {gather*} - 2 x - e^{x + \left (- x^{2} - 8 x - 16\right ) \log {\left (7 \right )} + 2} + \frac {1}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x**3+16*x**2)*ln(7)-2*x**2)*exp((-x**2-8*x-16)*ln(7)+2+x)-4*x**2-1)/x**2,x)

[Out]

-2*x - exp(x + (-x**2 - 8*x - 16)*log(7) + 2) + 1/(2*x)

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Giac [A]
time = 0.42, size = 31, normalized size = 1.11 \begin {gather*} -\frac {132931722278404 \, x^{2} + 2 \, x e^{\left (-x^{2} \log \left (7\right ) - 8 \, x \log \left (7\right ) + x + 2\right )} - 33232930569601}{66465861139202 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((4*x^3+16*x^2)*log(7)-2*x^2)*exp((-x^2-8*x-16)*log(7)+2+x)-4*x^2-1)/x^2,x, algorithm="giac")

[Out]

-1/66465861139202*(132931722278404*x^2 + 2*x*e^(-x^2*log(7) - 8*x*log(7) + x + 2) - 33232930569601)/x

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Mupad [B]
time = 0.33, size = 29, normalized size = 1.04 \begin {gather*} \frac {1}{2\,x}-2\,x-\frac {{\mathrm {e}}^2\,{\mathrm {e}}^x}{33232930569601\,7^{8\,x}\,7^{x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^2 - (exp(x - log(7)*(8*x + x^2 + 16) + 2)*(log(7)*(16*x^2 + 4*x^3) - 2*x^2))/2 + 1/2)/x^2,x)

[Out]

1/(2*x) - 2*x - (exp(2)*exp(x))/(33232930569601*7^(8*x)*7^(x^2))

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