3.4.37 \(\int \frac {\tanh ^{-1}(c+d \cot (a+b x))}{x} \, dx\) [337]

Optimal. Leaf size=18 \[ \text {Int}\left (\frac {\tanh ^{-1}(c+d \cot (a+b x))}{x},x\right ) \]

[Out]

CannotIntegrate(arctanh(c+d*cot(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 = {} \begin {gather*} \int \frac {\tanh ^{-1}(c+d \cot (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[ArcTanh[c + d*Cot[a + b*x]]/x,x]

[Out]

Defer[Int][ArcTanh[c + d*Cot[a + b*x]]/x, x]

Rubi steps

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

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Mathematica [A]
time = 3.88, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{-1}(c+d \cot (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[ArcTanh[c + d*Cot[a + b*x]]/x,x]

[Out]

Integrate[ArcTanh[c + d*Cot[a + b*x]]/x, x]

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Maple [A]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\arctanh \left (c +d \cot \left (b x +a \right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(c+d*cot(b*x+a))/x,x)

[Out]

int(arctanh(c+d*cot(b*x+a))/x,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(c+d*cot(b*x+a))/x,x, algorithm="maxima")

[Out]

integrate(arctanh(d*cot(b*x + a) + c)/x, x)

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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(arctanh(c+d*cot(b*x+a))/x,x, algorithm="fricas")

[Out]

integral(arctanh(d*cot(b*x + a) + c)/x, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (c + d \cot {\left (a + b x \right )} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(c+d*cot(b*x+a))/x,x)

[Out]

Integral(atanh(c + d*cot(a + b*x))/x, x)

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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(arctanh(c+d*cot(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(arctanh(d*cot(b*x + a) + c)/x, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {\mathrm {atanh}\left (c+d\,\mathrm {cot}\left (a+b\,x\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(c + d*cot(a + b*x))/x,x)

[Out]

int(atanh(c + d*cot(a + b*x))/x, x)

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