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

Optimal result

Integrand size = 7, antiderivative size = 10 \[ \int \frac {1}{c+d x} \, dx=\frac {\log (c+d x)}{d} \]

[Out]

ln(d*x+c)/d

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {31} \[ \int \frac {1}{c+d x} \, dx=\frac {\log (c+d x)}{d} \]

[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*} \text {integral}& = \frac {\log (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c+d x} \, dx=\frac {\log (c+d x)}{d} \]

[In]

Integrate[(c + d*x)^(-1),x]

[Out]

Log[c + d*x]/d

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 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	\(\frac {\ln \left (d x +c \right )}{d}\)	\(11\)
parallelrisch	\(\frac {\ln \left (d x +c \right )}{d}\)	\(11\)

[In]

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

[Out]

ln(d*x+c)/d

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c+d x} \, dx=\frac {\log \left (d x + c\right )}{d} \]

[In]

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

[Out]

log(d*x + c)/d

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int \frac {1}{c+d x} \, dx=\frac {\log {\left (c + d x \right )}}{d} \]

[In]

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

[Out]

log(c + d*x)/d

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c+d x} \, dx=\frac {\log \left (d x + c\right )}{d} \]

[In]

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

[Out]

log(d*x + c)/d

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10 \[ \int \frac {1}{c+d x} \, dx=\frac {\log \left ({\left | d x + c \right |}\right )}{d} \]

[In]

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

[Out]

log(abs(d*x + c))/d

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{c+d x} \, dx=\frac {\ln \left (c+d\,x\right )}{d} \]

[In]

int(1/(c + d*x),x)

[Out]

log(c + d*x)/d