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

Optimal. Leaf size=18 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {266, 43} \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + x^2),x]

[Out]

x^2/2 - Log[1 + x^2]/2

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + x^2),x]

[Out]

x^2/2 - Log[1 + x^2]/2

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fricas [A]  time = 0.39, size = 14, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*log(x^2 + 1)

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giac [A]  time = 1.03, size = 14, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*log(x^2 + 1)

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {x^{2}}{2}-\frac {\ln \left (x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 14, normalized size = 0.78 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 - 1/2*log(x^2 + 1)

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mupad [B]  time = 0.02, size = 14, normalized size = 0.78 \[ \frac {x^2}{2}-\frac {\ln \left (x^2+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^2 + 1),x)

[Out]

x^2/2 - log(x^2 + 1)/2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.67 \[ \frac {x^{2}}{2} - \frac {\log {\left (x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/2 - log(x**2 + 1)/2

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