3.1.75 \(\int \frac {z^4}{1+z^2} \, dz\) [75]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, 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 = {308, 209} \begin {gather*} \frac {z^3}{3}-z+\tan ^{-1}(z) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[z^4/(1 + z^2),z]

[Out]

-z + z^3/3 + ArcTan[z]

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 308

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*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[z^4/(1 + z^2),z]

[Out]

-z + z^3/3 + ArcTan[z]

________________________________________________________________________________________

Mathics [A]
time = 1.75, size = 11, normalized size = 0.85 \begin {gather*} -z+\frac {z^3}{3}+\text {ArcTan}\left [z\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[z^4/(1 + z^2),z]')

[Out]

-z + z ^ 3 / 3 + ArcTan[z]

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 12, normalized size = 0.92

method	result	size
default	\(-z +\frac {z^{3}}{3}+\arctan \left (z \right )\)	\(12\)
risch	\(-z +\frac {z^{3}}{3}+\arctan \left (z \right )\)	\(12\)
meijerg	\(-\frac {z \left (-5 z^{2}+15\right )}{15}+\arctan \left (z \right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(z^4/(z^2+1),z,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.36, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, z^{3} - z + \arctan \left (z\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.31, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{3} \, z^{3} - z + \arctan \left (z\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 8, normalized size = 0.62 \begin {gather*} \frac {z^{3}}{3} - z + \operatorname {atan}{\left (z \right )} \end {gather*}

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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} \frac {1}{3} z^{3}-z+\arctan z \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*z^3 - z + arctan(z)

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 11, normalized size = 0.85 \begin {gather*} \mathrm {atan}\left (z\right )-z+\frac {z^3}{3} \end {gather*}

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

________________________________________________________________________________________