3.348 \(\int \frac{\sqrt{\tanh ^{-1}(a x)}}{(1-a^2 x^2)^4} \, dx\)

Optimal. Leaf size=252 \[ \frac{3 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\tanh ^{-1}(a x)}\right )}{512 a}+\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{256 a}+\frac{\sqrt{\frac{\pi }{6}} \text{Erf}\left (\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right )}{768 a}-\frac{3 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\tanh ^{-1}(a x)}\right )}{512 a}-\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{256 a}-\frac{\sqrt{\frac{\pi }{6}} \text{Erfi}\left (\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right )}{768 a}+\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a} \]

[Out]

(5*ArcTanh[a*x]^(3/2))/(24*a) + (3*Sqrt[Pi]*Erf[2*Sqrt[ArcTanh[a*x]]])/(512*a) + (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sq
rt[ArcTanh[a*x]]])/(256*a) + (Sqrt[Pi/6]*Erf[Sqrt[6]*Sqrt[ArcTanh[a*x]]])/(768*a) - (3*Sqrt[Pi]*Erfi[2*Sqrt[Ar
cTanh[a*x]]])/(512*a) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(256*a) - (Sqrt[Pi/6]*Erfi[Sqrt[6]*Sq
rt[ArcTanh[a*x]]])/(768*a) + (15*Sqrt[ArcTanh[a*x]]*Sinh[2*ArcTanh[a*x]])/(64*a) + (3*Sqrt[ArcTanh[a*x]]*Sinh[
4*ArcTanh[a*x]])/(64*a) + (Sqrt[ArcTanh[a*x]]*Sinh[6*ArcTanh[a*x]])/(192*a)

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Rubi [A]  time = 0.30064, antiderivative size = 252, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5968, 3312, 3296, 3308, 2180, 2204, 2205} \[ \frac{3 \sqrt{\pi } \text{Erf}\left (2 \sqrt{\tanh ^{-1}(a x)}\right )}{512 a}+\frac{15 \sqrt{\frac{\pi }{2}} \text{Erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{256 a}+\frac{\sqrt{\frac{\pi }{6}} \text{Erf}\left (\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right )}{768 a}-\frac{3 \sqrt{\pi } \text{Erfi}\left (2 \sqrt{\tanh ^{-1}(a x)}\right )}{512 a}-\frac{15 \sqrt{\frac{\pi }{2}} \text{Erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{256 a}-\frac{\sqrt{\frac{\pi }{6}} \text{Erfi}\left (\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right )}{768 a}+\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^4,x]

[Out]

(5*ArcTanh[a*x]^(3/2))/(24*a) + (3*Sqrt[Pi]*Erf[2*Sqrt[ArcTanh[a*x]]])/(512*a) + (15*Sqrt[Pi/2]*Erf[Sqrt[2]*Sq
rt[ArcTanh[a*x]]])/(256*a) + (Sqrt[Pi/6]*Erf[Sqrt[6]*Sqrt[ArcTanh[a*x]]])/(768*a) - (3*Sqrt[Pi]*Erfi[2*Sqrt[Ar
cTanh[a*x]]])/(512*a) - (15*Sqrt[Pi/2]*Erfi[Sqrt[2]*Sqrt[ArcTanh[a*x]]])/(256*a) - (Sqrt[Pi/6]*Erfi[Sqrt[6]*Sq
rt[ArcTanh[a*x]]])/(768*a) + (15*Sqrt[ArcTanh[a*x]]*Sinh[2*ArcTanh[a*x]])/(64*a) + (3*Sqrt[ArcTanh[a*x]]*Sinh[
4*ArcTanh[a*x]])/(64*a) + (Sqrt[ArcTanh[a*x]]*Sinh[6*ArcTanh[a*x]])/(192*a)

Rule 5968

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol] :> Dist[d^q/c, Subst[Int[(
a + b*x)^p/Cosh[x]^(2*(q + 1)), x], x, ArcTanh[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0]
&& ILtQ[2*(q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{\tanh ^{-1}(a x)}}{\left (1-a^2 x^2\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{x} \cosh ^6(x) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{5 \sqrt{x}}{16}+\frac{15}{32} \sqrt{x} \cosh (2 x)+\frac{3}{16} \sqrt{x} \cosh (4 x)+\frac{1}{32} \sqrt{x} \cosh (6 x)\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{\operatorname{Subst}\left (\int \sqrt{x} \cosh (6 x) \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}+\frac{3 \operatorname{Subst}\left (\int \sqrt{x} \cosh (4 x) \, dx,x,\tanh ^{-1}(a x)\right )}{16 a}+\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \cosh (2 x) \, dx,x,\tanh ^{-1}(a x)\right )}{32 a}\\ &=\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}-\frac{\operatorname{Subst}\left (\int \frac{\sinh (6 x)}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{384 a}-\frac{3 \operatorname{Subst}\left (\int \frac{\sinh (4 x)}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}-\frac{15 \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}\\ &=\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}+\frac{\operatorname{Subst}\left (\int \frac{e^{-6 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{768 a}-\frac{\operatorname{Subst}\left (\int \frac{e^{6 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{768 a}+\frac{3 \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}-\frac{3 \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}+\frac{15 \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}-\frac{15 \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\tanh ^{-1}(a x)\right )}{256 a}\\ &=\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}+\frac{\operatorname{Subst}\left (\int e^{-6 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{384 a}-\frac{\operatorname{Subst}\left (\int e^{6 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{384 a}+\frac{3 \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{128 a}-\frac{3 \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{128 a}+\frac{15 \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{128 a}-\frac{15 \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\tanh ^{-1}(a x)}\right )}{128 a}\\ &=\frac{5 \tanh ^{-1}(a x)^{3/2}}{24 a}+\frac{3 \sqrt{\pi } \text{erf}\left (2 \sqrt{\tanh ^{-1}(a x)}\right )}{512 a}+\frac{15 \sqrt{\frac{\pi }{2}} \text{erf}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{256 a}+\frac{\sqrt{\frac{\pi }{6}} \text{erf}\left (\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right )}{768 a}-\frac{3 \sqrt{\pi } \text{erfi}\left (2 \sqrt{\tanh ^{-1}(a x)}\right )}{512 a}-\frac{15 \sqrt{\frac{\pi }{2}} \text{erfi}\left (\sqrt{2} \sqrt{\tanh ^{-1}(a x)}\right )}{256 a}-\frac{\sqrt{\frac{\pi }{6}} \text{erfi}\left (\sqrt{6} \sqrt{\tanh ^{-1}(a x)}\right )}{768 a}+\frac{15 \sqrt{\tanh ^{-1}(a x)} \sinh \left (2 \tanh ^{-1}(a x)\right )}{64 a}+\frac{3 \sqrt{\tanh ^{-1}(a x)} \sinh \left (4 \tanh ^{-1}(a x)\right )}{64 a}+\frac{\sqrt{\tanh ^{-1}(a x)} \sinh \left (6 \tanh ^{-1}(a x)\right )}{192 a}\\ \end{align*}

Mathematica [A]  time = 0.764107, size = 257, normalized size = 1.02 \[ \frac{\frac{\sqrt{6} \sqrt{\tanh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-6 \tanh ^{-1}(a x)\right )}{a \sqrt{-\tanh ^{-1}(a x)}}+\frac{27 \sqrt{\tanh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-4 \tanh ^{-1}(a x)\right )}{a \sqrt{-\tanh ^{-1}(a x)}}+\frac{135 \sqrt{2} \sqrt{\tanh ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-2 \tanh ^{-1}(a x)\right )}{a \sqrt{-\tanh ^{-1}(a x)}}-\frac{135 \sqrt{2} \text{Gamma}\left (\frac{1}{2},2 \tanh ^{-1}(a x)\right )}{a}-\frac{27 \text{Gamma}\left (\frac{1}{2},4 \tanh ^{-1}(a x)\right )}{a}-\frac{\sqrt{6} \text{Gamma}\left (\frac{1}{2},6 \tanh ^{-1}(a x)\right )}{a}-\frac{1440 a^4 x^5 \sqrt{\tanh ^{-1}(a x)}}{\left (a^2 x^2-1\right )^3}+\frac{3840 a^2 x^3 \sqrt{\tanh ^{-1}(a x)}}{\left (a^2 x^2-1\right )^3}-\frac{3168 x \sqrt{\tanh ^{-1}(a x)}}{\left (a^2 x^2-1\right )^3}+\frac{960 \tanh ^{-1}(a x)^{3/2}}{a}}{4608} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[ArcTanh[a*x]]/(1 - a^2*x^2)^4,x]

[Out]

((-3168*x*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 + (3840*a^2*x^3*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 - (1440*a^
4*x^5*Sqrt[ArcTanh[a*x]])/(-1 + a^2*x^2)^3 + (960*ArcTanh[a*x]^(3/2))/a + (Sqrt[6]*Sqrt[ArcTanh[a*x]]*Gamma[1/
2, -6*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) + (27*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -4*ArcTanh[a*x]])/(a*Sqrt[-Ar
cTanh[a*x]]) + (135*Sqrt[2]*Sqrt[ArcTanh[a*x]]*Gamma[1/2, -2*ArcTanh[a*x]])/(a*Sqrt[-ArcTanh[a*x]]) - (135*Sqr
t[2]*Gamma[1/2, 2*ArcTanh[a*x]])/a - (27*Gamma[1/2, 4*ArcTanh[a*x]])/a - (Sqrt[6]*Gamma[1/2, 6*ArcTanh[a*x]])/
a)/4608

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Maple [F]  time = 0.497, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( -{a}^{2}{x}^{2}+1 \right ) ^{4}}\sqrt{{\it Artanh} \left ( ax \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x)

[Out]

int(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{artanh}\left (a x\right )}}{{\left (a^{2} x^{2} - 1\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x, algorithm="maxima")

[Out]

integrate(sqrt(arctanh(a*x))/(a^2*x^2 - 1)^4, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{atanh}{\left (a x \right )}}}{\left (a x - 1\right )^{4} \left (a x + 1\right )^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x)**(1/2)/(-a**2*x**2+1)**4,x)

[Out]

Integral(sqrt(atanh(a*x))/((a*x - 1)**4*(a*x + 1)**4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{artanh}\left (a x\right )}}{{\left (a^{2} x^{2} - 1\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x)^(1/2)/(-a^2*x^2+1)^4,x, algorithm="giac")

[Out]

integrate(sqrt(arctanh(a*x))/(a^2*x^2 - 1)^4, x)