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

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[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 [A]
time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccot}\left (c +\left (i c +1\right ) \tan \left (b x +a \right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {acot}\left (c+\mathrm {tan}\left (a+b\,x\right )\,\left (1+c\,1{}\mathrm {i}\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________