3.5.73 \(\int \frac {\tanh ^{-1}(a x)^2}{(1-a^2 x^2)^{7/2}} \, dx\) [473]

Optimal. Leaf size=208 \[ \frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac {4144 x}{3375 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}} \]

[Out]

2/125*x/(-a^2*x^2+1)^(5/2)+272/3375*x/(-a^2*x^2+1)^(3/2)-2/25*arctanh(a*x)/a/(-a^2*x^2+1)^(5/2)-8/45*arctanh(a
*x)/a/(-a^2*x^2+1)^(3/2)+1/5*x*arctanh(a*x)^2/(-a^2*x^2+1)^(5/2)+4/15*x*arctanh(a*x)^2/(-a^2*x^2+1)^(3/2)+4144
/3375*x/(-a^2*x^2+1)^(1/2)-16/15*arctanh(a*x)/a/(-a^2*x^2+1)^(1/2)+8/15*x*arctanh(a*x)^2/(-a^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6111, 6109, 197, 198} \begin {gather*} \frac {4144 x}{3375 \sqrt {1-a^2 x^2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x]^2/(1 - a^2*x^2)^(7/2),x]

[Out]

(2*x)/(125*(1 - a^2*x^2)^(5/2)) + (272*x)/(3375*(1 - a^2*x^2)^(3/2)) + (4144*x)/(3375*Sqrt[1 - a^2*x^2]) - (2*
ArcTanh[a*x])/(25*a*(1 - a^2*x^2)^(5/2)) - (8*ArcTanh[a*x])/(45*a*(1 - a^2*x^2)^(3/2)) - (16*ArcTanh[a*x])/(15
*a*Sqrt[1 - a^2*x^2]) + (x*ArcTanh[a*x]^2)/(5*(1 - a^2*x^2)^(5/2)) + (4*x*ArcTanh[a*x]^2)/(15*(1 - a^2*x^2)^(3
/2)) + (8*x*ArcTanh[a*x]^2)/(15*Sqrt[1 - a^2*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 6109

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(-b)*p*((a + b*Arc
Tanh[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] + (Dist[b^2*p*(p - 1), Int[(a + b*ArcTanh[c*x])^(p - 2)/(d + e*x
^2)^(3/2), x], x] + Simp[x*((a + b*ArcTanh[c*x])^p/(d*Sqrt[d + e*x^2])), x]) /; FreeQ[{a, b, c, d, e}, x] && E
qQ[c^2*d + e, 0] && GtQ[p, 1]

Rule 6111

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

Rubi steps

\begin {align*} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx &=-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {2}{25} \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx+\frac {4}{5} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx\\ &=\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8}{125} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {8}{45} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx+\frac {8}{15} \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}}+\frac {16}{375} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {16}{135} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx+\frac {16}{15} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 x}{125 \left (1-a^2 x^2\right )^{5/2}}+\frac {272 x}{3375 \left (1-a^2 x^2\right )^{3/2}}+\frac {4144 x}{3375 \sqrt {1-a^2 x^2}}-\frac {2 \tanh ^{-1}(a x)}{25 a \left (1-a^2 x^2\right )^{5/2}}-\frac {8 \tanh ^{-1}(a x)}{45 a \left (1-a^2 x^2\right )^{3/2}}-\frac {16 \tanh ^{-1}(a x)}{15 a \sqrt {1-a^2 x^2}}+\frac {x \tanh ^{-1}(a x)^2}{5 \left (1-a^2 x^2\right )^{5/2}}+\frac {4 x \tanh ^{-1}(a x)^2}{15 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 x \tanh ^{-1}(a x)^2}{15 \sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 94, normalized size = 0.45 \begin {gather*} \frac {4470 a x-8560 a^3 x^3+4144 a^5 x^5-30 \left (149-260 a^2 x^2+120 a^4 x^4\right ) \tanh ^{-1}(a x)+225 a x \left (15-20 a^2 x^2+8 a^4 x^4\right ) \tanh ^{-1}(a x)^2}{3375 a \left (1-a^2 x^2\right )^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x]^2/(1 - a^2*x^2)^(7/2),x]

[Out]

(4470*a*x - 8560*a^3*x^3 + 4144*a^5*x^5 - 30*(149 - 260*a^2*x^2 + 120*a^4*x^4)*ArcTanh[a*x] + 225*a*x*(15 - 20
*a^2*x^2 + 8*a^4*x^4)*ArcTanh[a*x]^2)/(3375*a*(1 - a^2*x^2)^(5/2))

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Maple [A]
time = 0.71, size = 118, normalized size = 0.57

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (1800 \arctanh \left (a x \right )^{2} a^{5} x^{5}+4144 a^{5} x^{5}-3600 a^{4} x^{4} \arctanh \left (a x \right )-4500 \arctanh \left (a x \right )^{2} a^{3} x^{3}-8560 a^{3} x^{3}+7800 a^{2} x^{2} \arctanh \left (a x \right )+3375 \arctanh \left (a x \right )^{2} a x +4470 a x -4470 \arctanh \left (a x \right )\right )}{3375 a \left (a^{2} x^{2}-1\right )^{3}}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/3375/a*(-a^2*x^2+1)^(1/2)*(1800*arctanh(a*x)^2*a^5*x^5+4144*a^5*x^5-3600*a^4*x^4*arctanh(a*x)-4500*arctanh(
a*x)^2*a^3*x^3-8560*a^3*x^3+7800*a^2*x^2*arctanh(a*x)+3375*arctanh(a*x)^2*a*x+4470*a*x-4470*arctanh(a*x))/(a^2
*x^2-1)^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (172) = 344\).
time = 0.52, size = 514, normalized size = 2.47 \begin {gather*} \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {3 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\right )} \operatorname {artanh}\left (a x\right )^{2} + \frac {1}{3375} \, a {\left (\frac {9 \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} x + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a}\right )}}{a} + \frac {9 \, {\left (\frac {8 \, x}{\sqrt {-a^{2} x^{2} + 1}} + \frac {4 \, x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {3}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2} x - {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a}\right )}}{a} + \frac {100 \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a^{2} x + \sqrt {-a^{2} x^{2} + 1} a}\right )}}{a} + \frac {100 \, {\left (\frac {2 \, x}{\sqrt {-a^{2} x^{2} + 1}} - \frac {1}{\sqrt {-a^{2} x^{2} + 1} a^{2} x - \sqrt {-a^{2} x^{2} + 1} a}\right )}}{a} - \frac {1800 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x + a\right )} a} - \frac {1800 \, \sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x - a\right )} a} - \frac {1800 \, \log \left (a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} + \frac {1800 \, \log \left (-a x + 1\right )}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {300 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} + \frac {300 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} a^{2}} - \frac {135 \, \log \left (a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}} + \frac {135 \, \log \left (-a x + 1\right )}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x, algorithm="maxima")

[Out]

1/15*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) + 3*x/(-a^2*x^2 + 1)^(5/2))*arctanh(a*x)^2 + 1/3375*a*
(9*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) - 3/((-a^2*x^2 + 1)^(3/2)*a^2*x + (-a^2*x^2 + 1)^(3/2)*a
))/a + 9*(8*x/sqrt(-a^2*x^2 + 1) + 4*x/(-a^2*x^2 + 1)^(3/2) - 3/((-a^2*x^2 + 1)^(3/2)*a^2*x - (-a^2*x^2 + 1)^(
3/2)*a))/a + 100*(2*x/sqrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x + sqrt(-a^2*x^2 + 1)*a))/a + 100*(2*x/s
qrt(-a^2*x^2 + 1) - 1/(sqrt(-a^2*x^2 + 1)*a^2*x - sqrt(-a^2*x^2 + 1)*a))/a - 1800*sqrt(-a^2*x^2 + 1)/((a^2*x +
a)*a) - 1800*sqrt(-a^2*x^2 + 1)/((a^2*x - a)*a) - 1800*log(a*x + 1)/(sqrt(-a^2*x^2 + 1)*a^2) + 1800*log(-a*x
+ 1)/(sqrt(-a^2*x^2 + 1)*a^2) - 300*log(a*x + 1)/((-a^2*x^2 + 1)^(3/2)*a^2) + 300*log(-a*x + 1)/((-a^2*x^2 + 1
)^(3/2)*a^2) - 135*log(a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2) + 135*log(-a*x + 1)/((-a^2*x^2 + 1)^(5/2)*a^2))

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Fricas [A]
time = 0.36, size = 139, normalized size = 0.67 \begin {gather*} -\frac {{\left (16576 \, a^{5} x^{5} - 34240 \, a^{3} x^{3} + 225 \, {\left (8 \, a^{5} x^{5} - 20 \, a^{3} x^{3} + 15 \, a x\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )^{2} + 17880 \, a x - 60 \, {\left (120 \, a^{4} x^{4} - 260 \, a^{2} x^{2} + 149\right )} \log \left (-\frac {a x + 1}{a x - 1}\right )\right )} \sqrt {-a^{2} x^{2} + 1}}{13500 \, {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x, algorithm="fricas")

[Out]

-1/13500*(16576*a^5*x^5 - 34240*a^3*x^3 + 225*(8*a^5*x^5 - 20*a^3*x^3 + 15*a*x)*log(-(a*x + 1)/(a*x - 1))^2 +
17880*a*x - 60*(120*a^4*x^4 - 260*a^2*x^2 + 149)*log(-(a*x + 1)/(a*x - 1)))*sqrt(-a^2*x^2 + 1)/(a^7*x^6 - 3*a^
5*x^4 + 3*a^3*x^2 - a)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**2/(-a**2*x**2+1)**(7/2),x)

[Out]

Integral(atanh(a*x)**2/(-(a*x - 1)*(a*x + 1))**(7/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^2/(-a^2*x^2+1)^(7/2),x, algorithm="giac")

[Out]

integrate(arctanh(a*x)^2/(-a^2*x^2 + 1)^(7/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {atanh}\left (a\,x\right )}^2}{{\left (1-a^2\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x)^2/(1 - a^2*x^2)^(7/2),x)

[Out]

int(atanh(a*x)^2/(1 - a^2*x^2)^(7/2), x)

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