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

Optimal. Leaf size=23 \[ \text{CannotIntegrate}\left (\frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10437, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx \]

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 = 0.320209, size = 0, normalized size = 0. \[ \int \frac{\cot ^{-1}(c+(1+i c) \tan (a+b x))}{x} \, dx \]

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]

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Maple [A]  time = 0.415, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccot} \left (c+ \left ( 1+ic \right ) \tan \left ( bx+a \right ) \right )}{x}}\, dx \end{align*}

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)

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

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{i \, \log \left (\frac{{\left (c e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{c - i}\right )}{2 \, x}, x\right ) \end{align*}

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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left ({\left (i \, c + 1\right )} \tan \left (b x + a\right ) + c\right )}{x}\,{d x} \end{align*}

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)