\(\int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx\) [199]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\text {Int}\left (\frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (c - i\right )} \tanh \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.32 \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (c - i\right )} \tanh \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

integrate(arccot(c-(I-c)*tanh(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 = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (c - i\right )} \tanh \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}(c-(i-c) \tanh (a+b x))}{x} \, dx=\int \frac {\mathrm {acot}\left (c+\mathrm {tanh}\left (a+b\,x\right )\,\left (c-\mathrm {i}\right )\right )}{x} \,d x \]

[In]

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

[Out]

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