3.23 \(\int \frac{1}{d x^3} \, dx\)

Optimal. Leaf size=10 \[ -\frac{1}{2 d x^2} \]

[Out]

-1/(2*d*x^2)

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Rubi [A]  time = 0.0014073, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {12, 30} \[ -\frac{1}{2 d x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(d*x^3),x]

[Out]

-1/(2*d*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{d x^3} \, dx &=\frac{\int \frac{1}{x^3} \, dx}{d}\\ &=-\frac{1}{2 d x^2}\\ \end{align*}

Mathematica [A]  time = 0.0003004, size = 10, normalized size = 1. \[ -\frac{1}{2 d x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(d*x^3),x]

[Out]

-1/(2*d*x^2)

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Maple [A]  time = 0.002, size = 9, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,d{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/d/x^3,x)

[Out]

-1/2/d/x^2

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Maxima [A]  time = 1.00634, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2 \, d x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(d*x^2)

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Fricas [A]  time = 1.49792, size = 19, normalized size = 1.9 \begin{align*} -\frac{1}{2 \, d x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(d*x^2)

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Sympy [A]  time = 0.062868, size = 8, normalized size = 0.8 \begin{align*} - \frac{1}{2 d x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/d/x**3,x)

[Out]

-1/(2*d*x**2)

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Giac [A]  time = 1.2243, size = 11, normalized size = 1.1 \begin{align*} -\frac{1}{2 \, d x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(d*x^2)