\(\int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx\) [182]

Optimal result

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

[Out]

CannotIntegrate((Pi-arccot(-c+(1+I*c)*cot(b*x+a)))/x,x)

Rubi [N/A]

Not integrable

Time = 0.10 (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 = {} \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx \]

[In]

Int[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x,x]

[Out]

Defer[Int][ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x, x]

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

Mathematica [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx \]

[In]

Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x,x]

[Out]

Integrate[ArcCot[c - (1 + I*c)*Cot[a + b*x]]/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14

\[\int \frac {\pi -\operatorname {arccot}\left (-c +\left (i c +1\right ) \cot \left (b x +a \right )\right )}{x}d x\]

[In]

int((Pi-arccot(-c+(I*c+1)*cot(b*x+a)))/x,x)

[Out]

int((Pi-arccot(-c+(I*c+1)*cot(b*x+a)))/x,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx=\int { \frac {\pi - \operatorname {arccot}\left ({\left (i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )}{x} \,d x } \]

[In]

integrate((pi-arccot(-c+(1+I*c)*cot(b*x+a)))/x,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate((pi-acot(-c+(1+I*c)*cot(b*x+a)))/x,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((pi-arccot(-c+(1+I*c)*cot(b*x+a)))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for mor
e details)Is

Giac [N/A]

Not integrable

Time = 2.72 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx=\int { \frac {\pi - \operatorname {arccot}\left ({\left (i \, c + 1\right )} \cot \left (b x + a\right ) - c\right )}{x} \,d x } \]

[In]

integrate((pi-arccot(-c+(1+I*c)*cot(b*x+a)))/x,x, algorithm="giac")

[Out]

integrate((pi - arccot((I*c + 1)*cot(b*x + a) - c))/x, x)

Mupad [N/A]

Not integrable

Time = 1.54 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {\cot ^{-1}(c-(1+i c) \cot (a+b x))}{x} \, dx=\int \frac {\Pi +\mathrm {acot}\left (c-\mathrm {cot}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right )}{x} \,d x \]

[In]

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

[Out]

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