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

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

[Out]

CannotIntegrate[ArcCot[c - (I - c)*Coth[a + b*x]]/x, x]

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Rubi [A]  time = 0.108297, 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 = {} \[ \int \frac{\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 3.08878, size = 0, normalized size = 0. \[ \int \frac{\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.424, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccot} \left (c- \left ( i-c \right ){\rm coth} \left (bx+a\right )\right )}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} i \, b x - \frac{1}{4} \,{\left (-4 i \, a + 2 \, \arctan \left (1, -c\right ) - i \, \log \left (c^{2} + 1\right )\right )} \log \left (x\right ) + \frac{1}{2} \, \int \frac{\arctan \left (1, -c e^{\left (2 \, b x + 2 \, a\right )}\right )}{x}\,{d x} - \frac{1}{4} i \, \int \frac{\log \left (c^{2} e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*b*x - 1/4*(-4*I*a + 2*arctan2(1, -c) - I*log(c^2 + 1))*log(x) + 1/2*integrate(arctan2(1, -c*e^(2*b*x + 2*a))
/x, x) - 1/4*I*integrate(log(c^2*e^(4*b*x + 4*a) + 1)/x, x)

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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 - i\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c e^{\left (2 \, b x + 2 \, a\right )} - i}\right )}{2 \, x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(c-(I-c)*coth(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)

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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-(I-c)*coth(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 (c - i\right )} \coth \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-(I-c)*coth(b*x+a))/x,x, algorithm="giac")

[Out]

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