3.1.60 \(\int \frac {1}{1+t} \, dt\) [60]

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

[Out]

ln(1+t)

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Rubi [A]
time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {31} \begin {gather*} \log (t+1) \end {gather*}

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*}

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Mathematica [A]
time = 0.00, size = 4, normalized size = 1.00 \begin {gather*} \log (1+t) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + t)^(-1),t]

[Out]

Log[1 + t]

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Maple [A]
time = 0.06, size = 5, normalized size = 1.25

method	result	size
default	\(\ln \left (1+t \right )\)	\(5\)
norman	\(\ln \left (1+t \right )\)	\(5\)
meijerg	\(\ln \left (1+t \right )\)	\(5\)
risch	\(\ln \left (1+t \right )\)	\(5\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+t),t,method=_RETURNVERBOSE)

[Out]

ln(1+t)

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Maxima [A]
time = 3.00, size = 4, normalized size = 1.00 \begin {gather*} \log \left (t + 1\right ) \end {gather*}

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 = 1.06, size = 4, normalized size = 1.00 \begin {gather*} \log \left (t + 1\right ) \end {gather*}

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 = 0.01, size = 3, normalized size = 0.75 \begin {gather*} \log {\left (t + 1 \right )} \end {gather*}

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 = 0.45, size = 5, normalized size = 1.25 \begin {gather*} \log \left ({\left | t + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(t + 1))

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Mupad [B]
time = 0.02, size = 4, normalized size = 1.00 \begin {gather*} \ln \left (t+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(t + 1),t)

[Out]

log(t + 1)

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