3.2.9 \(\int \frac {x^3+x^4}{1+x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac {x^3}{3}+\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \]

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Rubi [A]  time = 0.03, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1593, 801, 635, 203, 260} \begin {gather*} \frac {x^3}{3}+\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3 + x^4)/(1 + x^2),x]

[Out]

-x + x^2/2 + x^3/3 + ArcTan[x] - Log[1 + x^2]/2

Rule 203

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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]

Rubi steps

\begin {align*} \int \frac {x^3+x^4}{1+x^2} \, dx &=\int \frac {x^3 (1+x)}{1+x^2} \, dx\\ &=\int \left (-1+x+x^2+\frac {1-x}{1+x^2}\right ) \, dx\\ &=-x+\frac {x^2}{2}+\frac {x^3}{3}+\int \frac {1-x}{1+x^2} \, dx\\ &=-x+\frac {x^2}{2}+\frac {x^3}{3}+\int \frac {1}{1+x^2} \, dx-\int \frac {x}{1+x^2} \, dx\\ &=-x+\frac {x^2}{2}+\frac {x^3}{3}+\tan ^{-1}(x)-\frac {1}{2} \log \left (1+x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {x^3}{3}+\frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right )-x+\tan ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3 + x^4)/(1 + x^2),x]

[Out]

-x + x^2/2 + x^3/3 + ArcTan[x] - Log[1 + x^2]/2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3+x^4}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(x^3 + x^4)/(1 + x^2),x]

[Out]

IntegrateAlgebraic[(x^3 + x^4)/(1 + x^2), x]

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fricas [A]  time = 0.83, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} - x + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.39, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} - x + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} \frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\arctan \relax (x )-\frac {\ln \left (x^{2}+1\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+x^3)/(x^2+1),x)

[Out]

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

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maxima [A]  time = 2.90, size = 24, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, x^{3} + \frac {1}{2} \, x^{2} - x + \arctan \relax (x) - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} \mathrm {atan}\relax (x)-\frac {\ln \left (x^2+1\right )}{2}-x+\frac {x^2}{2}+\frac {x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + x^4)/(x^2 + 1),x)

[Out]

atan(x) - log(x^2 + 1)/2 - x + x^2/2 + x^3/3

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sympy [A]  time = 0.11, size = 22, normalized size = 0.73 \begin {gather*} \frac {x^{3}}{3} + \frac {x^{2}}{2} - x - \frac {\log {\left (x^{2} + 1 \right )}}{2} + \operatorname {atan}{\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 + x**2/2 - x - log(x**2 + 1)/2 + atan(x)

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