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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 13, 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 = {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 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 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]

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ \frac {z^3}{3}-z+\tan ^{-1}(z) \]

Antiderivative was successfully verified.

[In]

Integrate[z^4/(1 + z^2),z]

[Out]

-z + z^3/3 + ArcTan[z]

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fricas [A]  time = 0.39, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, z^{3} - z + \arctan \relax (z) \]

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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giac [A]  time = 0.89, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, z^{3} - z + \arctan \relax (z) \]

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)

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maple [A]  time = 0.00, size = 12, normalized size = 0.92 \[ \frac {z^{3}}{3}-z +\arctan \relax (z ) \]

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 = 1.47, size = 11, normalized size = 0.85 \[ \frac {1}{3} \, z^{3} - z + \arctan \relax (z) \]

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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mupad [B]  time = 0.03, size = 11, normalized size = 0.85 \[ \mathrm {atan}\relax (z)-z+\frac {z^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(z^4/(z^2 + 1),z)

[Out]

atan(z) - z + z^3/3

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sympy [A]  time = 0.09, size = 8, normalized size = 0.62 \[ \frac {z^{3}}{3} - z + \operatorname {atan}{\relax (z )} \]

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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