3.252 \(\int \frac{\coth ^{-1}(\cot (a+b x))}{e+f x} \, dx\)

Optimal. Leaf size=17 \[ \text{CannotIntegrate}\left (\frac{\coth ^{-1}(\cot (a+b x))}{e+f x},x\right ) \]

[Out]

CannotIntegrate[ArcCoth[Cot[a + b*x]]/(e + f*x), x]

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Rubi [A]  time = 0.0417984, 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 = {} \[ \int \frac{\coth ^{-1}(\cot (a+b x))}{e+f x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.107466, size = 0, normalized size = 0. \[ \int \frac{\coth ^{-1}(\cot (a+b x))}{e+f x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.173, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccoth} \left (\cot \left ( bx+a \right ) \right )}{fx+e}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\cot \left (b x + a\right )\right )}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (\cot \left (b x + a\right )\right )}{f x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acoth(cot(a + b*x))/(e + f*x), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (\cot \left (b x + a\right )\right )}{f x + e}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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