3.258 \(\int \frac {\tanh ^{-1}(a x)^p}{1-a^2 x^2} \, dx\)

Optimal. Leaf size=17 \[ \frac {\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \]

[Out]

arctanh(a*x)^(1+p)/a/(1+p)

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5948} \[ \frac {\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^p/(1 - a^2*x^2),x]

[Out]

ArcTanh[a*x]^(1 + p)/(a*(1 + p))

Rule 5948

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^p}{1-a^2 x^2} \, dx &=\frac {\tanh ^{-1}(a x)^{1+p}}{a (1+p)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {\tanh ^{-1}(a x)^{p+1}}{a (p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^p/(1 - a^2*x^2),x]

[Out]

ArcTanh[a*x]^(1 + p)/(a*(1 + p))

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fricas [B]  time = 0.51, size = 84, normalized size = 4.94 \[ \frac {\cosh \left (p \log \left (\frac {1}{2} \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )\right ) \log \left (-\frac {a x + 1}{a x - 1}\right ) + \log \left (-\frac {a x + 1}{a x - 1}\right ) \sinh \left (p \log \left (\frac {1}{2} \, \log \left (-\frac {a x + 1}{a x - 1}\right )\right )\right )}{2 \, {\left (a p + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^p/(-a^2*x^2+1),x, algorithm="fricas")

[Out]

1/2*(cosh(p*log(1/2*log(-(a*x + 1)/(a*x - 1))))*log(-(a*x + 1)/(a*x - 1)) + log(-(a*x + 1)/(a*x - 1))*sinh(p*l
og(1/2*log(-(a*x + 1)/(a*x - 1)))))/(a*p + a)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{p}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^p/(-a^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-arctanh(a*x)^p/(a^2*x^2 - 1), x)

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maple [A]  time = 0.02, size = 18, normalized size = 1.06 \[ \frac {\arctanh \left (a x \right )^{1+p}}{a \left (1+p \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^p/(-a^2*x^2+1),x)

[Out]

arctanh(a*x)^(1+p)/a/(1+p)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\operatorname {artanh}\left (a x\right )^{p}}{a^{2} x^{2} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^p/(-a^2*x^2+1),x, algorithm="maxima")

[Out]

-integrate(arctanh(a*x)^p/(a^2*x^2 - 1), x)

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mupad [B]  time = 1.05, size = 33, normalized size = 1.94 \[ \left \{\begin {array}{cl} \frac {\ln \left (\mathrm {atanh}\left (a\,x\right )\right )}{a} & \text {\ if\ \ }p=-1\\ \frac {{\mathrm {atanh}\left (a\,x\right )}^{p+1}}{a\,\left (p+1\right )} & \text {\ if\ \ }p\neq -1 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-atanh(a*x)^p/(a^2*x^2 - 1),x)

[Out]

piecewise(p == -1, log(atanh(a*x))/a, p ~= -1, atanh(a*x)^(p + 1)/(a*(p + 1)))

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sympy [A]  time = 1.53, size = 26, normalized size = 1.53 \[ \begin {cases} \frac {\begin {cases} \frac {\operatorname {atanh}^{p + 1}{\left (a x \right )}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (\operatorname {atanh}{\left (a x \right )} \right )} & \text {otherwise} \end {cases}}{a} & \text {for}\: a \neq 0 \\0^{p} x & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**p/(-a**2*x**2+1),x)

[Out]

Piecewise((Piecewise((atanh(a*x)**(p + 1)/(p + 1), Ne(p, -1)), (log(atanh(a*x)), True))/a, Ne(a, 0)), (0**p*x,
True))

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