\(\int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx\) [212]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\text {Int}\left (\frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x},x\right ) \]

[Out]

CannotIntegrate(arccot(c+(I+c)*coth(b*x+a))/x,x)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 19, 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 = {} \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx \]

[In]

Int[ArcCot[c + (I + c)*Coth[a + b*x]]/x,x]

[Out]

Defer[Int][ArcCot[c + (I + c)*Coth[a + b*x]]/x, x]

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

Mathematica [N/A]

Not integrable

Time = 3.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx \]

[In]

Integrate[ArcCot[c + (I + c)*Coth[a + b*x]]/x,x]

[Out]

Integrate[ArcCot[c + (I + c)*Coth[a + b*x]]/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.12 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

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

[In]

int(arccot(c+(I+c)*coth(b*x+a))/x,x)

[Out]

int(arccot(c+(I+c)*coth(b*x+a))/x,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (c + i\right )} \coth \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

integral(1/2*I*log((c*e^(2*b*x + 2*a) + I)*e^(-2*b*x - 2*a)/(c + I))/x, x)

Sympy [F(-1)]

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

[In]

integrate(acot(c+(I+c)*coth(b*x+a))/x,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.84 \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (c + i\right )} \coth \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

-I*b*x + 1/4*(-4*I*a + 2*arctan(1/c) - I*log(c^2 + 1))*log(x) - 1/2*integrate(arctan(e^(-2*b*x - 2*a)/c)/x, x)
+ 1/4*I*integrate(log(c^2*e^(4*b*x + 4*a) + 1)/x, x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (c + i\right )} \coth \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(arccot((c + I)*coth(b*x + a) + c)/x, x)

Mupad [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^{-1}(c+(i+c) \coth (a+b x))}{x} \, dx=\int \frac {\mathrm {acot}\left (c+\mathrm {coth}\left (a+b\,x\right )\,\left (c+1{}\mathrm {i}\right )\right )}{x} \,d x \]

[In]

int(acot(c + coth(a + b*x)*(c + 1i))/x,x)

[Out]

int(acot(c + coth(a + b*x)*(c + 1i))/x, x)