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

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

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, 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, 6103, 5910, 260} \[ \frac {\log \left (1-\left (a+b x^n\right )^2\right )}{2 b n}+\frac {\left (a+b x^n\right ) \tanh ^{-1}\left (a+b x^n\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*ArcTanh[a + b*x^n],x]

[Out]

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

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]

Rule 5910

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

Rule 6103

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 42, normalized size = 0.89 \[ \frac {\log \left (1-\left (a+b x^n\right )^2\right )+2 \left (a+b x^n\right ) \tanh ^{-1}\left (a+b x^n\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.57, size = 109, normalized size = 2.32 \[ \frac {{\left (a + 1\right )} \log \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a + 1\right ) - {\left (a - 1\right )} \log \left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a - 1\right ) + {\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right )} \log \left (-\frac {b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a + 1}{b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right ) + a - 1}\right )}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+n)*arctanh(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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giac [B]  time = 0.21, size = 124, normalized size = 2.64 \[ \frac {{\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | -b x^{n} - a - 1 \right |}}{{\left | b x^{n} + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | -\frac {b x^{n} + a + 1}{b x^{n} + a - 1} + 1 \right |}\right )}{b^{2}} + \frac {\log \left (-\frac {b x^{n} + a + 1}{b x^{n} + a - 1}\right )}{b^{2} {\left (\frac {b x^{n} + a + 1}{b x^{n} + a - 1} - 1\right )}}\right )}}{2 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.07, size = 121, normalized size = 2.57 \[ \frac {x^{n} \ln \left (1+a +b \,x^{n}\right )}{2 n}-\frac {x^{n} \ln \left (1-a -b \,x^{n}\right )}{2 n}-\frac {\ln \left (x^{n}+\frac {a -1}{b}\right ) a}{2 b n}+\frac {\ln \left (x^{n}+\frac {1+a}{b}\right ) a}{2 b n}+\frac {\ln \left (x^{n}+\frac {a -1}{b}\right )}{2 b n}+\frac {\ln \left (x^{n}+\frac {1+a}{b}\right )}{2 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.32, size = 40, normalized size = 0.85 \[ \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {artanh}\left (b x^{n} + a\right ) + \log \left (-{\left (b x^{n} + a\right )}^{2} + 1\right )}{2 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.47, size = 56, normalized size = 1.19 \[ \frac {x^n\,\mathrm {atanh}\left (a+b\,x^n\right )}{n}-\frac {\ln \left (a+b\,x^n-1\right )\,\left (a-1\right )}{2\,b\,n}+\frac {\ln \left (a+b\,x^n+1\right )\,\left (a+1\right )}{2\,b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n - 1)*atanh(a + b*x^n),x)

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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