3.4.33 \(\int \frac {\tanh ^{-1}(\cot (a+b x))}{e+f x} \, dx\) [333]

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

[Out]

CannotIntegrate(arctanh(cot(b*x+a))/(f*x+e),x)

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Rubi [A]
time = 0.03, 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}(\cot (a+b x))}{e+f x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[ArcTanh[Cot[a + b*x]]/(e + f*x),x]

[Out]

Defer[Int][ArcTanh[Cot[a + b*x]]/(e + f*x), x]

Rubi steps

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

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Mathematica [A]
time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{-1}(\cot (a+b x))}{e+f x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[ArcTanh[Cot[a + b*x]]/(e + f*x),x]

[Out]

Integrate[ArcTanh[Cot[a + b*x]]/(e + f*x), x]

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Maple [A]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\arctanh \left (\cot \left (b x +a \right )\right )}{f x +e}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(cot(b*x+a))/(f*x+e),x)

[Out]

int(arctanh(cot(b*x+a))/(f*x+e),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(cot(b*x+a))/(f*x+e),x, algorithm="maxima")

[Out]

integrate(arctanh(cot(b*x + a))/(f*x + e), 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(cot(b*x+a))/(f*x+e),x, algorithm="fricas")

[Out]

integral(arctanh(cot(b*x + a))/(f*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(cot(b*x+a))/(f*x+e),x)

[Out]

Integral(atanh(cot(a + b*x))/(e + f*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(cot(b*x+a))/(f*x+e),x, algorithm="giac")

[Out]

integrate(arctanh(cot(b*x + a))/(f*x + e), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(cot(a + b*x))/(e + f*x),x)

[Out]

int(atanh(cot(a + b*x))/(e + f*x), x)

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