3.60 \(\int \frac{1}{1+t} \, dt\)

Optimal. Leaf size=4 \[ \log (t+1) \]

[Out]

Log[1 + t]

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Rubi [A]  time = 0.0004936, antiderivative size = 4, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {31} \[ \log (t+1) \]

Antiderivative was successfully verified.

[In]

Int[(1 + t)^(-1),t]

[Out]

Log[1 + t]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{1+t} \, dt &=\log (1+t)\\ \end{align*}

Mathematica [A]  time = 0.0005086, size = 4, normalized size = 1. \[ \log (t+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + t)^(-1),t]

[Out]

Log[1 + t]

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Maple [A]  time = 0., size = 5, normalized size = 1.3 \begin{align*} \ln \left ( 1+t \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+t),t)

[Out]

ln(1+t)

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Maxima [A]  time = 0.9472, size = 5, normalized size = 1.25 \begin{align*} \log \left (t + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+t),t, algorithm="maxima")

[Out]

log(t + 1)

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Fricas [A]  time = 0.479768, size = 16, normalized size = 4. \begin{align*} \log \left (t + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+t),t, algorithm="fricas")

[Out]

log(t + 1)

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Sympy [A]  time = 0.054785, size = 3, normalized size = 0.75 \begin{align*} \log{\left (t + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+t),t)

[Out]

log(t + 1)

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Giac [A]  time = 1.10235, size = 7, normalized size = 1.75 \begin{align*} \log \left ({\left | t + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+t),t, algorithm="giac")

[Out]

log(abs(t + 1))