3.70.86 \(\int \frac {81-9 x^2}{2 x^2} \, dx\)

Optimal. Leaf size=19 \[ -\frac {(9-3 x)^2}{2 x}+\log ^2(9) \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14} \begin {gather*} -\frac {9 x}{2}-\frac {81}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(81 - 9*x^2)/(2*x^2),x]

[Out]

-81/(2*x) - (9*x)/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {81-9 x^2}{x^2} \, dx\\ &=\frac {1}{2} \int \left (-9+\frac {81}{x^2}\right ) \, dx\\ &=-\frac {81}{2 x}-\frac {9 x}{2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 13, normalized size = 0.68 \begin {gather*} \frac {9}{2} \left (-\frac {9}{x}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(81 - 9*x^2)/(2*x^2),x]

[Out]

(9*(-9/x - x))/2

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 10, normalized size = 0.53 \begin {gather*} -\frac {9 \, {\left (x^{2} + 9\right )}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-9*x^2+81)/x^2,x, algorithm="fricas")

[Out]

-9/2*(x^2 + 9)/x

________________________________________________________________________________________

giac [A]  time = 0.14, size = 9, normalized size = 0.47 \begin {gather*} -\frac {9}{2} \, x - \frac {81}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-9*x^2+81)/x^2,x, algorithm="giac")

[Out]

-9/2*x - 81/2/x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 10, normalized size = 0.53

method	result	size
default	\(-\frac {9 x}{2}-\frac {81}{2 x}\)	\(10\)
risch	\(-\frac {9 x}{2}-\frac {81}{2 x}\)	\(10\)
gosper	\(-\frac {9 \left (x^{2}+9\right )}{2 x}\)	\(11\)
norman	\(\frac {-\frac {81}{2}-\frac {9 x^{2}}{2}}{x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-9*x^2+81)/x^2,x,method=_RETURNVERBOSE)

[Out]

-9/2*x-81/2/x

________________________________________________________________________________________

maxima [A]  time = 0.37, size = 9, normalized size = 0.47 \begin {gather*} -\frac {9}{2} \, x - \frac {81}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-9*x^2+81)/x^2,x, algorithm="maxima")

[Out]

-9/2*x - 81/2/x

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 10, normalized size = 0.53 \begin {gather*} -\frac {9\,\left (x^2+9\right )}{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((9*x^2)/2 - 81/2)/x^2,x)

[Out]

-(9*(x^2 + 9))/(2*x)

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 10, normalized size = 0.53 \begin {gather*} - \frac {9 x}{2} - \frac {81}{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-9*x**2+81)/x**2,x)

[Out]

-9*x/2 - 81/(2*x)

________________________________________________________________________________________