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

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

[Out]

CannotIntegrate[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x, x]

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Rubi [A]  time = 0.0800919, 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 = {} \[ \int \frac{\coth ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x,x]

[Out]

Defer[Int][ArcCoth[1 - I*d - d*Cot[a + b*x]]/x, x]

Rubi steps

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

Mathematica [A]  time = 0.749697, size = 0, normalized size = 0. \[ \int \frac{\coth ^{-1}(1-i d-d \cot (a+b x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x,x]

[Out]

Integrate[ArcCoth[1 - I*d - d*Cot[a + b*x]]/x, x]

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Maple [A]  time = 0.457, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccoth} \left (1-id-d\cot \left ( bx+a \right ) \right )}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(1-I*d-d*cot(b*x+a))/x,x)

[Out]

int(arccoth(1-I*d-d*cot(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 (-i \, \pi - 4 i \, a - 2 \, \log \left (d\right )\right )} \log \left (x\right ) - \frac{1}{2} i \, \int \frac{\arctan \left (-d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), -d \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right )}{x}\,{d x} + \frac{1}{4} \, \int \frac{\log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} +{\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(1-I*d-d*cot(b*x+a))/x,x, algorithm="maxima")

[Out]

-I*b*x + 1/4*(-I*pi - 4*I*a - 2*log(d))*log(x) - 1/2*I*integrate(arctan2(-d*cos(2*b*x + 2*a) + sin(2*b*x + 2*a
), -d*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1)/x, x) + 1/4*integrate(log((d^2 + 1)*cos(2*b*x + 2*a)^2 + (d^2 +
1)*sin(2*b*x + 2*a)^2 - 2*d*sin(2*b*x + 2*a) - 2*cos(2*b*x + 2*a) + 1)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(1-I*d-d*cot(b*x+a))/x,x, algorithm="fricas")

[Out]

integral(-1/2*log(d*e^(2*I*b*x + 2*I*a)/((d + I)*e^(2*I*b*x + 2*I*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(acoth(1-I*d-d*cot(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{arcoth}\left (-d \cot \left (b x + a\right ) - i \, d + 1\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(1-I*d-d*cot(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(arccoth(-d*cot(b*x + a) - I*d + 1)/x, x)