3.75 \(\int \frac{z^4}{1+z^2} \, dz\)

Optimal. Leaf size=13 \[ \frac{z^3}{3}-z+\tan ^{-1}(z) \]

[Out]

-z + z^3/3 + ArcTan[z]

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Rubi [A]  time = 0.0053858, antiderivative size = 13, normalized size of antiderivative = 1., 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 = {302, 203} \[ \frac{z^3}{3}-z+\tan ^{-1}(z) \]

Antiderivative was successfully verified.

[In]

Int[z^4/(1 + z^2),z]

[Out]

-z + z^3/3 + ArcTan[z]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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])

Rubi steps

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

Mathematica [A]  time = 0.0039374, size = 13, normalized size = 1. \[ \frac{z^3}{3}-z+\tan ^{-1}(z) \]

Antiderivative was successfully verified.

[In]

Integrate[z^4/(1 + z^2),z]

[Out]

-z + z^3/3 + ArcTan[z]

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Maple [A]  time = 0.001, size = 12, normalized size = 0.9 \begin{align*} -z+{\frac{{z}^{3}}{3}}+\arctan \left ( z \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(z^4/(z^2+1),z)

[Out]

-z+1/3*z^3+arctan(z)

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Maxima [A]  time = 1.40311, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{3} \, z^{3} - z + \arctan \left (z\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*z^3 - z + arctan(z)

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Fricas [A]  time = 2.09771, size = 34, normalized size = 2.62 \begin{align*} \frac{1}{3} \, z^{3} - z + \arctan \left (z\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*z^3 - z + arctan(z)

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Sympy [A]  time = 0.080637, size = 8, normalized size = 0.62 \begin{align*} \frac{z^{3}}{3} - z + \operatorname{atan}{\left (z \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(z**4/(z**2+1),z)

[Out]

z**3/3 - z + atan(z)

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Giac [A]  time = 1.05979, size = 15, normalized size = 1.15 \begin{align*} \frac{1}{3} \, z^{3} - z + \arctan \left (z\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*z^3 - z + arctan(z)