3.294 \(\int x^{-1+n} \coth ^{-1}(a+b x^n) \, dx\)

Optimal. Leaf size=47 \[ \frac{\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}+\frac{\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n} \]

[Out]

((a + b*x^n)*ArcCoth[a + b*x^n])/(b*n) + Log[1 - (a + b*x^n)^2]/(2*b*n)

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Rubi [A]  time = 0.0552597, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {6715, 6104, 5911, 260} \[ \frac{\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}+\frac{\left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*ArcCoth[a + b*x^n],x]

[Out]

((a + b*x^n)*ArcCoth[a + b*x^n])/(b*n) + Log[1 - (a + b*x^n)^2]/(2*b*n)

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6104

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcCoth[x])^p, x
], x, c + d*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[p, 0]

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0357095, size = 42, normalized size = 0.89 \[ \frac{\log \left (1-\left (a+b x^n\right )^2\right )+2 \left (a+b x^n\right ) \coth ^{-1}\left (a+b x^n\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*ArcCoth[a + b*x^n],x]

[Out]

(2*(a + b*x^n)*ArcCoth[a + b*x^n] + Log[1 - (a + b*x^n)^2])/(2*b*n)

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Maple [B]  time = 0.119, size = 118, normalized size = 2.5 \begin{align*}{\frac{{x}^{n}\ln \left ( 1+a+b{x}^{n} \right ) }{2\,n}}-{\frac{{x}^{n}\ln \left ( -1+a+b{x}^{n} \right ) }{2\,n}}-{\frac{a}{2\,bn}\ln \left ({x}^{n}+{\frac{a-1}{b}} \right ) }+{\frac{a}{2\,bn}\ln \left ({x}^{n}+{\frac{1+a}{b}} \right ) }+{\frac{1}{2\,bn}\ln \left ({x}^{n}+{\frac{a-1}{b}} \right ) }+{\frac{1}{2\,bn}\ln \left ({x}^{n}+{\frac{1+a}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n-1)*arccoth(a+b*x^n),x)

[Out]

1/2/n*x^n*ln(1+a+b*x^n)-1/2/n*x^n*ln(-1+a+b*x^n)-1/2/b/n*ln(x^n+(a-1)/b)*a+1/2/b/n*ln(x^n+(1+a)/b)*a+1/2/b/n*l
n(x^n+(a-1)/b)+1/2/b/n*ln(x^n+(1+a)/b)

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Maxima [A]  time = 1.06526, size = 54, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )} \operatorname{arcoth}\left (b x^{n} + a\right ) + \log \left (-{\left (b x^{n} + a\right )}^{2} + 1\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arccoth(a+b*x^n),x, algorithm="maxima")

[Out]

1/2*(2*(b*x^n + a)*arccoth(b*x^n + a) + log(-(b*x^n + a)^2 + 1))/(b*n)

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Fricas [B]  time = 1.85904, size = 359, normalized size = 7.64 \begin{align*} \frac{{\left (a + 1\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + 1\right ) -{\left (a - 1\right )} \log \left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a - 1\right ) +{\left (b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right )\right )} \log \left (\frac{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a + 1}{b \cosh \left (n \log \left (x\right )\right ) + b \sinh \left (n \log \left (x\right )\right ) + a - 1}\right )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arccoth(a+b*x^n),x, algorithm="fricas")

[Out]

1/2*((a + 1)*log(b*cosh(n*log(x)) + b*sinh(n*log(x)) + a + 1) - (a - 1)*log(b*cosh(n*log(x)) + b*sinh(n*log(x)
) + a - 1) + (b*cosh(n*log(x)) + b*sinh(n*log(x)))*log((b*cosh(n*log(x)) + b*sinh(n*log(x)) + a + 1)/(b*cosh(n
*log(x)) + b*sinh(n*log(x)) + a - 1)))/(b*n)

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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(x**(-1+n)*acoth(a+b*x**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{n - 1} \operatorname{arcoth}\left (b x^{n} + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arccoth(a+b*x^n),x, algorithm="giac")

[Out]

integrate(x^(n - 1)*arccoth(b*x^n + a), x)