3.120 \(\int (e+f x)^m (a+b \coth ^{-1}(c+d x))^2 \, dx\)

Optimal. Leaf size=22 \[ \text{Unintegrable}\left ((e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2,x\right ) \]

[Out]

Unintegrable[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2, x]

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Rubi [A]  time = 0.0665056, 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 = {} \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2,x]

[Out]

Defer[Subst][Defer[Int][((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCoth[x])^2, x], x, c + d*x]/d

Rubi steps

\begin{align*} \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^m \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ \end{align*}

Mathematica [A]  time = 2.51712, size = 0, normalized size = 0. \[ \int (e+f x)^m \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2,x]

[Out]

Integrate[(e + f*x)^m*(a + b*ArcCoth[c + d*x])^2, x]

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Maple [A]  time = 1.368, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{m} \left ( a+b{\rm arccoth} \left (dx+c\right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x)

[Out]

int((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{2} \operatorname{arcoth}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcoth}\left (d x + c\right ) + a^{2}\right )}{\left (f x + e\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x, algorithm="fricas")

[Out]

integral((b^2*arccoth(d*x + c)^2 + 2*a*b*arccoth(d*x + c) + a^2)*(f*x + e)^m, 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((f*x+e)**m*(a+b*acoth(d*x+c))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcoth}\left (d x + c\right ) + a\right )}^{2}{\left (f x + e\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arccoth(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((b*arccoth(d*x + c) + a)^2*(f*x + e)^m, x)