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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\cot ^{-1}(c-(-c+i) \coth (a+b x))}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx \]

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*}

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Mathematica [A]  time = 3.56, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{-1}(c-(i-c) \coth (a+b x))}{x} \, dx \]

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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fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {i \, \log \left (\frac {{\left (c - i\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c e^{\left (2 \, b x + 2 \, a\right )} - i}\right )}{2 \, x}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left ({\left (c - i\right )} \coth \left (b x + a\right ) + c\right )}{x}\,{d x} \]

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)

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maple [A]  time = 2.09, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccot}\left (c -\left (i-c \right ) \coth \left (b x +a \right )\right )}{x}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ i \, b x + \frac {1}{2} \, \pi \log \relax (x) - \frac {1}{4} \, {\left (2 \, \pi - 4 i \, a - 2 \, \arctan \left (\frac {1}{c}\right ) - i \, \log \left (c^{2} + 1\right )\right )} \log \relax (x) - \frac {1}{2} \, \int \frac {\arctan \left (\frac {e^{\left (-2 \, b x - 2 \, a\right )}}{c}\right )}{x}\,{d x} - \frac {1}{4} i \, \int \frac {\log \left (c^{2} e^{\left (4 \, b x + 4 \, a\right )} + 1\right )}{x}\,{d x} \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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