3.14.39 \(\int \frac {1}{c+d x} \, dx\) [1339]

Optimal. Leaf size=10 \[ \frac {\log (c+d x)}{d} \]

[Out]

ln(d*x+c)/d

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Rubi [A]
time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {31} \begin {gather*} \frac {\log (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(-1),x]

[Out]

Log[c + d*x]/d

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}{c+d x} \, dx &=\frac {\log (c+d x)}{d}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(-1),x]

[Out]

Log[c + d*x]/d

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Maple [A]
time = 0.12, size = 11, normalized size = 1.10

method	result	size
default	\(\frac {\ln \left (d x +c \right )}{d}\)	\(11\)
norman	\(\frac {\ln \left (d x +c \right )}{d}\)	\(11\)
risch	\(\frac {\ln \left (d x +c \right )}{d}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c),x,method=_RETURNVERBOSE)

[Out]

ln(d*x+c)/d

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Maxima [A]
time = 0.38, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c),x, algorithm="maxima")

[Out]

log(d*x + c)/d

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Fricas [A]
time = 0.53, size = 10, normalized size = 1.00 \begin {gather*} \frac {\log \left (d x + c\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c),x, algorithm="fricas")

[Out]

log(d*x + c)/d

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Sympy [A]
time = 0.01, size = 7, normalized size = 0.70 \begin {gather*} \frac {\log {\left (c + d x \right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c),x)

[Out]

log(c + d*x)/d

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Giac [A]
time = 0.55, size = 11, normalized size = 1.10 \begin {gather*} \frac {\log \left ({\left | d x + c \right |}\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c),x, algorithm="giac")

[Out]

log(abs(d*x + c))/d

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Mupad [B]
time = 0.02, size = 10, normalized size = 1.00 \begin {gather*} \frac {\ln \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c + d*x),x)

[Out]

log(c + d*x)/d

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