\(\int \log (x+x^3) \, dx\) [239]

Optimal result

Integrand size = 6, antiderivative size = 16 \[ \int \log \left (x+x^3\right ) \, dx=-3 x+2 \arctan (x)+x \log \left (x+x^3\right ) \]

[Out]

-3*x+2*arctan(x)+x*ln(x^3+x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2603, 396, 209} \[ \int \log \left (x+x^3\right ) \, dx=2 \arctan (x)+x \log \left (x^3+x\right )-3 x \]

[In]

Int[Log[x + x^3],x]

[Out]

-3*x + 2*ArcTan[x] + x*Log[x + x^3]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = x \log \left (x+x^3\right )-\int \frac {1+3 x^2}{1+x^2} \, dx \\ & = -3 x+x \log \left (x+x^3\right )+2 \int \frac {1}{1+x^2} \, dx \\ & = -3 x+2 \tan ^{-1}(x)+x \log \left (x+x^3\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \log \left (x+x^3\right ) \, dx=-3 x+2 \arctan (x)+x \log \left (x+x^3\right ) \]

[In]

Integrate[Log[x + x^3],x]

[Out]

-3*x + 2*ArcTan[x] + x*Log[x + x^3]

Maple [A] (verified)

Time = 1.17 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
default	\(-3 x +2 \arctan \left (x \right )+x \ln \left (x^{3}+x \right )\)	\(17\)
risch	\(-3 x +2 \arctan \left (x \right )+x \ln \left (x^{3}+x \right )\)	\(17\)
parts	\(-3 x +2 \arctan \left (x \right )+x \ln \left (x^{3}+x \right )\)	\(17\)
parallelrisch	\(-i \ln \left (x \right )-2 i \ln \left (x -i\right )+i \ln \left (x^{3}+x \right )+x \ln \left (x^{3}+x \right )-3 x\)	\(35\)

[In]

int(ln(x^3+x),x,method=_RETURNVERBOSE)

[Out]

-3*x+2*arctan(x)+x*ln(x^3+x)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \log \left (x+x^3\right ) \, dx=x \log \left (x^{3} + x\right ) - 3 \, x + 2 \, \arctan \left (x\right ) \]

[In]

integrate(log(x^3+x),x, algorithm="fricas")

[Out]

x*log(x^3 + x) - 3*x + 2*arctan(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \log \left (x+x^3\right ) \, dx=x \log {\left (x^{3} + x \right )} - 3 x + 2 \operatorname {atan}{\left (x \right )} \]

[In]

integrate(ln(x**3+x),x)

[Out]

x*log(x**3 + x) - 3*x + 2*atan(x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \log \left (x+x^3\right ) \, dx=x \log \left (x^{3} + x\right ) - 3 \, x + 2 \, \arctan \left (x\right ) \]

[In]

integrate(log(x^3+x),x, algorithm="maxima")

[Out]

x*log(x^3 + x) - 3*x + 2*arctan(x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \log \left (x+x^3\right ) \, dx=x \log \left (x^{3} + x\right ) - 3 \, x + 2 \, \arctan \left (x\right ) \]

[In]

integrate(log(x^3+x),x, algorithm="giac")

[Out]

x*log(x^3 + x) - 3*x + 2*arctan(x)

Mupad [B] (verification not implemented)

Time = 1.52 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \log \left (x+x^3\right ) \, dx=2\,\mathrm {atan}\left (x\right )-3\,x+x\,\ln \left (x^3+x\right ) \]

[In]

int(log(x + x^3),x)

[Out]

2*atan(x) - 3*x + x*log(x + x^3)