3.91.29 \(\int \frac {-2 x^2+(e^3 x+2 x^2) (i \pi +\log (4+3 e))+(-2 x^2+(2 e^3 x+2 x^2) (i \pi +\log (4+3 e))) \log (\frac {x^2+(-e^3 x-x^2) (i \pi +\log (4+3 e))}{i \pi +\log (4+3 e)})}{-x+(e^3+x) (i \pi +\log (4+3 e))} \, dx\)

Optimal. Leaf size=34 \[ x^2 \log \left (x \left (-e^3-x+\frac {x}{i \pi +\log (-5+3 (3+e))}\right )\right ) \]

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Rubi [A]  time = 1.31, antiderivative size = 32, normalized size of antiderivative = 0.94, number of steps used = 10, number of rules used = 7, integrand size = 132, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6741, 6688, 6742, 77, 2495, 30, 43} \begin {gather*} x^2 \log \left (-x \left (e^3+x \left (1-\frac {1}{\log (4+3 e)+i \pi }\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2*Log[-(x*(E^3 + x*(1 - (I*Pi + Log[4 + 3*E])^(-1))))]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

Rule 6688

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

Rule 6741

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

Rule 6742

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

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 31, normalized size = 0.91 \begin {gather*} x^2 \log \left (x \left (-e^3+x \left (-1+\frac {1}{i \pi +\log (4+3 e)}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2*Log[x*(-E^3 + x*(-1 + (I*Pi + Log[4 + 3*E])^(-1)))]

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fricas [A]  time = 0.51, size = 36, normalized size = 1.06 \begin {gather*} x^{2} \log \left (\frac {x^{2} - {\left (x^{2} + x e^{3}\right )} \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.74, size = 1346, normalized size = 39.59 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(pi^4*x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e + 4)^2 + 2*x^3*e^3*log(3*e + 4)^2 + pi^2*x^2*e^6 - 2
*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*e + 4)^2 + x^4) + 2*pi^2*x^2*log(pi^2*x^4 + 2*pi^2*
x^3*e^3 + x^4*log(3*e + 4)^2 + 2*x^3*e^3*log(3*e + 4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*
e + 4) + x^2*e^6*log(3*e + 4)^2 + x^4)*log(3*e + 4)^2 + x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e + 4)^2
+ 2*x^3*e^3*log(3*e + 4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*e + 4
)^2 + x^4)*log(3*e + 4)^4 - 2*pi^4*x^2*log(abs(log(-3*e - 4))) - 4*pi^2*x^2*log(3*e + 4)^2*log(abs(log(-3*e -
4))) - 2*x^2*log(3*e + 4)^4*log(abs(log(-3*e - 4))) - 4*pi^2*x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e +
4)^2 + 2*x^3*e^3*log(3*e + 4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*
e + 4)^2 + x^4)*log(3*e + 4) - 4*x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e + 4)^2 + 2*x^3*e^3*log(3*e +
4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*e + 4)^2 + x^4)*log(3*e + 4)
^3 + 8*pi^2*x^2*log(3*e + 4)*log(abs(log(-3*e - 4))) + 8*x^2*log(3*e + 4)^3*log(abs(log(-3*e - 4))) - 4*pi^4*e
^6*sgn(pi*x + pi*e^3) - 4*pi^2*e^6*log(3*e + 4)^2*sgn(pi*x + pi*e^3) + 2*pi^2*x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^
3 + x^4*log(3*e + 4)^2 + 2*x^3*e^3*log(3*e + 4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4)
+ x^2*e^6*log(3*e + 4)^2 + x^4) + 6*x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e + 4)^2 + 2*x^3*e^3*log(3*
e + 4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*e + 4)^2 + x^4)*log(3*e
+ 4)^2 - 4*pi^2*x^2*log(abs(log(-3*e - 4))) - 12*x^2*log(3*e + 4)^2*log(abs(log(-3*e - 4))) + 4*pi^2*e^6*log(3
*e + 4)*sgn(pi*x + pi*e^3) - 4*x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e + 4)^2 + 2*x^3*e^3*log(3*e + 4)
^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*e + 4)^2 + x^4)*log(3*e + 4) +
8*x^2*log(3*e + 4)*log(abs(log(-3*e - 4))) + x^2*log(pi^2*x^4 + 2*pi^2*x^3*e^3 + x^4*log(3*e + 4)^2 + 2*x^3*e
^3*log(3*e + 4)^2 + pi^2*x^2*e^6 - 2*x^4*log(3*e + 4) - 2*x^3*e^3*log(3*e + 4) + x^2*e^6*log(3*e + 4)^2 + x^4)
- 2*x^2*log(abs(log(-3*e - 4))))/(pi^4 + 2*pi^2*log(3*e + 4)^2 + log(3*e + 4)^4 - 4*pi^2*log(3*e + 4) - 4*log
(3*e + 4)^3 + 2*pi^2 + 6*log(3*e + 4)^2 - 4*log(3*e + 4) + 1)

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maple [A]  time = 0.97, size = 39, normalized size = 1.15

method	result	size
norman	\(x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\)	\(39\)
risch	\(x^{2} \ln \left (\frac {\left (-x \,{\mathrm e}^{3}-x^{2}\right ) \ln \left (-3 \,{\mathrm e}-4\right )+x^{2}}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\)	\(39\)
default	\(x^{2} \ln \left (\frac {x \left (-{\mathrm e}^{3} \ln \left (-3 \,{\mathrm e}-4\right )-\ln \left (-3 \,{\mathrm e}-4\right ) x +x \right )}{\ln \left (-3 \,{\mathrm e}-4\right )}\right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x*exp(3)+2*x^2)*ln(-3*exp(1)-4)-2*x^2)*ln(((-x*exp(3)-x^2)*ln(-3*exp(1)-4)+x^2)/ln(-3*exp(1)-4))+(x*e
xp(3)+2*x^2)*ln(-3*exp(1)-4)-2*x^2)/((exp(3)+x)*ln(-3*exp(1)-4)-x),x,method=_RETURNVERBOSE)

[Out]

x^2*ln(((-x*exp(3)-x^2)*ln(-3*exp(1)-4)+x^2)/ln(-3*exp(1)-4))

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maxima [B]  time = 0.41, size = 689, normalized size = 20.26 \begin {gather*} -{\left (\frac {e^{3} \log \left (x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} + e^{3} \log \left (-3 \, e - 4\right )\right ) \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1} - \frac {x}{\log \left (-3 \, e - 4\right ) - 1}\right )} e^{3} \log \left (-3 \, e - 4\right ) - \frac {2 \, e^{6} \log \left (x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} + e^{3} \log \left (-3 \, e - 4\right )\right ) \log \left (-3 \, e - 4\right )^{2}}{\log \left (-3 \, e - 4\right )^{3} - 3 \, \log \left (-3 \, e - 4\right )^{2} + 3 \, \log \left (-3 \, e - 4\right ) - 1} + {\left (\frac {2 \, e^{6} \log \left (x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} + e^{3} \log \left (-3 \, e - 4\right )\right ) \log \left (-3 \, e - 4\right )^{2}}{\log \left (-3 \, e - 4\right )^{3} - 3 \, \log \left (-3 \, e - 4\right )^{2} + 3 \, \log \left (-3 \, e - 4\right ) - 1} + \frac {x^{2} {\left (\log \left (-3 \, e - 4\right ) - 1\right )} - 2 \, x e^{3} \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1}\right )} \log \left (-3 \, e - 4\right ) + \frac {{\left ({\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} + 2 \, \log \left (-3 \, e - 4\right ) - 1\right )} x^{2} \log \relax (x) + {\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} x e^{3} \log \left (-3 \, e - 4\right ) - {\left ({\left (\log \left (-3 \, e - 4\right )^{2} + {\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (\log \left (-3 \, e - 4\right )\right ) - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} - {\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (\log \left (-3 \, e - 4\right )\right ) + 2 \, \log \left (-3 \, e - 4\right ) - 1\right )} x^{2} + {\left ({\left ({\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} + 2 \, \log \left (-3 \, e - 4\right ) - 1\right )} x^{2} - {\left (\log \left (-3 \, e - 4\right )^{3} - \log \left (-3 \, e - 4\right )^{2}\right )} e^{6}\right )} \log \left (-x {\left (\log \left (-3 \, e - 4\right ) - 1\right )} - e^{3} \log \left (-3 \, e - 4\right )\right )}{{\left (\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1\right )} \log \left (-3 \, e - 4\right ) - \log \left (-3 \, e - 4\right )^{2} + 2 \, \log \left (-3 \, e - 4\right ) - 1} - \frac {x^{2} {\left (\log \left (-3 \, e - 4\right ) - 1\right )} - 2 \, x e^{3} \log \left (-3 \, e - 4\right )}{\log \left (-3 \, e - 4\right )^{2} - 2 \, \log \left (-3 \, e - 4\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e^3*log(x*(log(-3*e - 4) - 1) + e^3*log(-3*e - 4))*log(-3*e - 4)/(log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1) - x
/(log(-3*e - 4) - 1))*e^3*log(-3*e - 4) - 2*e^6*log(x*(log(-3*e - 4) - 1) + e^3*log(-3*e - 4))*log(-3*e - 4)^2
/(log(-3*e - 4)^3 - 3*log(-3*e - 4)^2 + 3*log(-3*e - 4) - 1) + (2*e^6*log(x*(log(-3*e - 4) - 1) + e^3*log(-3*e
- 4))*log(-3*e - 4)^2/(log(-3*e - 4)^3 - 3*log(-3*e - 4)^2 + 3*log(-3*e - 4) - 1) + (x^2*(log(-3*e - 4) - 1)
- 2*x*e^3*log(-3*e - 4))/(log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1))*log(-3*e - 4) + (((log(-3*e - 4)^2 - 2*log(-
3*e - 4) + 1)*log(-3*e - 4) - log(-3*e - 4)^2 + 2*log(-3*e - 4) - 1)*x^2*log(x) + (log(-3*e - 4)^2 - 2*log(-3*
e - 4) + 1)*x*e^3*log(-3*e - 4) - ((log(-3*e - 4)^2 + (log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1)*log(log(-3*e - 4
)) - 2*log(-3*e - 4) + 1)*log(-3*e - 4) - log(-3*e - 4)^2 - (log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1)*log(log(-3
*e - 4)) + 2*log(-3*e - 4) - 1)*x^2 + (((log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1)*log(-3*e - 4) - log(-3*e - 4)^
2 + 2*log(-3*e - 4) - 1)*x^2 - (log(-3*e - 4)^3 - log(-3*e - 4)^2)*e^6)*log(-x*(log(-3*e - 4) - 1) - e^3*log(-
3*e - 4)))/((log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1)*log(-3*e - 4) - log(-3*e - 4)^2 + 2*log(-3*e - 4) - 1) - (
x^2*(log(-3*e - 4) - 1) - 2*x*e^3*log(-3*e - 4))/(log(-3*e - 4)^2 - 2*log(-3*e - 4) + 1)

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mupad [B]  time = 8.06, size = 38, normalized size = 1.12 \begin {gather*} x^2\,\ln \left (-\frac {\ln \left (-3\,\mathrm {e}-4\right )\,\left (x^2+{\mathrm {e}}^3\,x\right )-x^2}{\ln \left (-3\,\mathrm {e}-4\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*log(-(log(- 3*exp(1) - 4)*(x*exp(3) + x^2) - x^2)/log(- 3*exp(1) - 4))

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sympy [B]  time = 0.91, size = 107, normalized size = 3.15 \begin {gather*} x^{2} \log {\left (- \frac {i \pi x^{2}}{\log {\left (4 + 3 e \right )} + i \pi } + \frac {x^{2}}{\log {\left (4 + 3 e \right )} + i \pi } - \frac {x^{2} \log {\left (4 + 3 e \right )}}{\log {\left (4 + 3 e \right )} + i \pi } - \frac {i \pi x e^{3}}{\log {\left (4 + 3 e \right )} + i \pi } - \frac {x e^{3} \log {\left (4 + 3 e \right )}}{\log {\left (4 + 3 e \right )} + i \pi } \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*log(-I*pi*x**2/(log(4 + 3*E) + I*pi) + x**2/(log(4 + 3*E) + I*pi) - x**2*log(4 + 3*E)/(log(4 + 3*E) + I*p
i) - I*pi*x*exp(3)/(log(4 + 3*E) + I*pi) - x*exp(3)*log(4 + 3*E)/(log(4 + 3*E) + I*pi))

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