3.464 \(\int \frac{\tanh ^{-1}(a x)}{(1-a^2 x^2)^{9/2}} \, dx\)

Optimal. Leaf size=177 \[ -\frac{16}{35 a \sqrt{1-a^2 x^2}}-\frac{8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{1}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}} \]

[Out]

-1/(49*a*(1 - a^2*x^2)^(7/2)) - 6/(175*a*(1 - a^2*x^2)^(5/2)) - 8/(105*a*(1 - a^2*x^2)^(3/2)) - 16/(35*a*Sqrt[
1 - a^2*x^2]) + (x*ArcTanh[a*x])/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x])/(35*(1 - a^2*x^2)^(5/2)) + (8*x*
ArcTanh[a*x])/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x])/(35*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.113113, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5960, 5958} \[ -\frac{16}{35 a \sqrt{1-a^2 x^2}}-\frac{8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{1}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 \sqrt{1-a^2 x^2}}+\frac{8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]/(1 - a^2*x^2)^(9/2),x]

[Out]

-1/(49*a*(1 - a^2*x^2)^(7/2)) - 6/(175*a*(1 - a^2*x^2)^(5/2)) - 8/(105*a*(1 - a^2*x^2)^(3/2)) - 16/(35*a*Sqrt[
1 - a^2*x^2]) + (x*ArcTanh[a*x])/(7*(1 - a^2*x^2)^(7/2)) + (6*x*ArcTanh[a*x])/(35*(1 - a^2*x^2)^(5/2)) + (8*x*
ArcTanh[a*x])/(35*(1 - a^2*x^2)^(3/2)) + (16*x*ArcTanh[a*x])/(35*Sqrt[1 - a^2*x^2])

Rule 5960

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> -Simp[(b*(d + e*x^2)^(q + 1))
/(4*c*d*(q + 1)^2), x] + (Dist[(2*q + 3)/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]), x], x] -
Simp[(x*(d + e*x^2)^(q + 1)*(a + b*ArcTanh[c*x]))/(2*d*(q + 1)), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*
d + e, 0] && LtQ[q, -1] && NeQ[q, -3/2]

Rule 5958

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> -Simp[b/(c*d*Sqrt[d + e*x^2]
), x] + Simp[(x*(a + b*ArcTanh[c*x]))/(d*Sqrt[d + e*x^2]), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx &=-\frac{1}{49 a \left (1-a^2 x^2\right )^{7/2}}+\frac{x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6}{7} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx\\ &=-\frac{1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{6}{175 a \left (1-a^2 x^2\right )^{5/2}}+\frac{x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{24}{35} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=-\frac{1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8}{105 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16}{35} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{1}{49 a \left (1-a^2 x^2\right )^{7/2}}-\frac{6}{175 a \left (1-a^2 x^2\right )^{5/2}}-\frac{8}{105 a \left (1-a^2 x^2\right )^{3/2}}-\frac{16}{35 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)}{7 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x \tanh ^{-1}(a x)}{35 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x \tanh ^{-1}(a x)}{35 \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0762956, size = 81, normalized size = 0.46 \[ \frac{1680 a^6 x^6-5320 a^4 x^4+5726 a^2 x^2-105 a x \left (16 a^6 x^6-56 a^4 x^4+70 a^2 x^2-35\right ) \tanh ^{-1}(a x)-2161}{3675 a \left (1-a^2 x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]/(1 - a^2*x^2)^(9/2),x]

[Out]

(-2161 + 5726*a^2*x^2 - 5320*a^4*x^4 + 1680*a^6*x^6 - 105*a*x*(-35 + 70*a^2*x^2 - 56*a^4*x^4 + 16*a^6*x^6)*Arc
Tanh[a*x])/(3675*a*(1 - a^2*x^2)^(7/2))

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Maple [A]  time = 0.174, size = 99, normalized size = 0.6 \begin{align*} -{\frac{1680\,{\it Artanh} \left ( ax \right ){x}^{7}{a}^{7}-1680\,{x}^{6}{a}^{6}-5880\,{\it Artanh} \left ( ax \right ){x}^{5}{a}^{5}+5320\,{x}^{4}{a}^{4}+7350\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -5726\,{a}^{2}{x}^{2}-3675\,ax{\it Artanh} \left ( ax \right ) +2161}{3675\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{4}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)/(-a^2*x^2+1)^(9/2),x)

[Out]

-1/3675/a*(-a^2*x^2+1)^(1/2)*(1680*arctanh(a*x)*x^7*a^7-1680*x^6*a^6-5880*arctanh(a*x)*x^5*a^5+5320*x^4*a^4+73
50*a^3*x^3*arctanh(a*x)-5726*a^2*x^2-3675*a*x*arctanh(a*x)+2161)/(a^2*x^2-1)^4

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Maxima [A]  time = 0.987439, size = 189, normalized size = 1.07 \begin{align*} -\frac{1}{3675} \, a{\left (\frac{1680}{\sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{280}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}} + \frac{126}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}} a^{2}} + \frac{75}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{7}{2}} a^{2}}\right )} + \frac{1}{35} \,{\left (\frac{16 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{8 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}} + \frac{6 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{5}{2}}} + \frac{5 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{7}{2}}}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/(-a^2*x^2+1)^(9/2),x, algorithm="maxima")

[Out]

-1/3675*a*(1680/(sqrt(-a^2*x^2 + 1)*a^2) + 280/((-a^2*x^2 + 1)^(3/2)*a^2) + 126/((-a^2*x^2 + 1)^(5/2)*a^2) + 7
5/((-a^2*x^2 + 1)^(7/2)*a^2)) + 1/35*(16*x/sqrt(-a^2*x^2 + 1) + 8*x/(-a^2*x^2 + 1)^(3/2) + 6*x/(-a^2*x^2 + 1)^
(5/2) + 5*x/(-a^2*x^2 + 1)^(7/2))*arctanh(a*x)

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Fricas [A]  time = 1.75634, size = 285, normalized size = 1.61 \begin{align*} \frac{{\left (3360 \, a^{6} x^{6} - 10640 \, a^{4} x^{4} + 11452 \, a^{2} x^{2} - 105 \,{\left (16 \, a^{7} x^{7} - 56 \, a^{5} x^{5} + 70 \, a^{3} x^{3} - 35 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 4322\right )} \sqrt{-a^{2} x^{2} + 1}}{7350 \,{\left (a^{9} x^{8} - 4 \, a^{7} x^{6} + 6 \, a^{5} x^{4} - 4 \, a^{3} x^{2} + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/(-a^2*x^2+1)^(9/2),x, algorithm="fricas")

[Out]

1/7350*(3360*a^6*x^6 - 10640*a^4*x^4 + 11452*a^2*x^2 - 105*(16*a^7*x^7 - 56*a^5*x^5 + 70*a^3*x^3 - 35*a*x)*log
(-(a*x + 1)/(a*x - 1)) - 4322)*sqrt(-a^2*x^2 + 1)/(a^9*x^8 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + a)

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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(atanh(a*x)/(-a**2*x**2+1)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.29118, size = 186, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (2 \,{\left (4 \,{\left (2 \, a^{6} x^{2} - 7 \, a^{4}\right )} x^{2} + 35 \, a^{2}\right )} x^{2} - 35\right )} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{70 \,{\left (a^{2} x^{2} - 1\right )}^{4}} - \frac{126 \, a^{2} x^{2} + 1680 \,{\left (a^{2} x^{2} - 1\right )}^{3} - 280 \,{\left (a^{2} x^{2} - 1\right )}^{2} - 201}{3675 \,{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt{-a^{2} x^{2} + 1} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)/(-a^2*x^2+1)^(9/2),x, algorithm="giac")

[Out]

-1/70*sqrt(-a^2*x^2 + 1)*(2*(4*(2*a^6*x^2 - 7*a^4)*x^2 + 35*a^2)*x^2 - 35)*x*log(-(a*x + 1)/(a*x - 1))/(a^2*x^
2 - 1)^4 - 1/3675*(126*a^2*x^2 + 1680*(a^2*x^2 - 1)^3 - 280*(a^2*x^2 - 1)^2 - 201)/((a^2*x^2 - 1)^3*sqrt(-a^2*
x^2 + 1)*a)