\(\int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx\) [165]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx=\text {Int}\left (\frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x},x\right ) \]

[Out]

CannotIntegrate(arccot(c+(1+I*c)*tan(b*x+a))/x,x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx=\int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

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

[In]

int(arccot(c+(I*c+1)*tan(b*x+a))/x,x)

[Out]

int(arccot(c+(I*c+1)*tan(b*x+a))/x,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(acot(c+(1+I*c)*tan(b*x+a))/x,x)

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(arccot(c+(1+I*c)*tan(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 = 1.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx=\int { \frac {\operatorname {arccot}\left ({\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(arccot((I*c + 1)*tan(b*x + a) + c)/x, x)

Mupad [N/A]

Not integrable

Time = 1.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx=\int \frac {\mathrm {acot}\left (c+\mathrm {tan}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right )}{x} \,d x \]

[In]

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

[Out]

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