\(\int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx\) [174]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\text {Int}\left (\frac {\cot ^{-1}(c+d \cot (a+b x))}{x},x\right ) \]

[Out]

CannotIntegrate(arccot(c+d*cot(b*x+a))/x,x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx \]

[In]

Int[ArcCot[c + d*Cot[a + b*x]]/x,x]

[Out]

Defer[Int][ArcCot[c + d*Cot[a + b*x]]/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx \]

[In]

Integrate[ArcCot[c + d*Cot[a + b*x]]/x,x]

[Out]

Integrate[ArcCot[c + d*Cot[a + b*x]]/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {arccot}\left (c +d \cot \left (b x +a \right )\right )}{x}d x\]

[In]

int(arccot(c+d*cot(b*x+a))/x,x)

[Out]

int(arccot(c+d*cot(b*x+a))/x,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left (d \cot \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

integrate(arccot(c+d*cot(b*x+a))/x,x, algorithm="fricas")

[Out]

integral(arccot(d*cot(b*x + a) + c)/x, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\text {Timed out} \]

[In]

integrate(acot(c+d*cot(b*x+a))/x,x)

[Out]

Timed out

Maxima [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\text {Timed out} \]

[In]

integrate(arccot(c+d*cot(b*x+a))/x,x, algorithm="maxima")

[Out]

Timed out

Giac [N/A]

Not integrable

Time = 3.50 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left (d \cot \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

integrate(arccot(c+d*cot(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(arccot(d*cot(b*x + a) + c)/x, x)

Mupad [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {\cot ^{-1}(c+d \cot (a+b x))}{x} \, dx=\int \frac {\mathrm {acot}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right )}{x} \,d x \]

[In]

int(acot(c + d*cot(a + b*x))/x,x)

[Out]

int(acot(c + d*cot(a + b*x))/x, x)