3.5.96 \(\int \frac {1}{(1-a^2 x^2)^{9/2} \tanh ^{-1}(a x)^3} \, dx\) [496]

Optimal. Leaf size=107 \[ -\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{128 a}+\frac {189 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{128 a}+\frac {175 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{128 a}+\frac {49 \text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{128 a} \]

[Out]

-1/2/a/(-a^2*x^2+1)^(7/2)/arctanh(a*x)^2-7/2*x/(-a^2*x^2+1)^(7/2)/arctanh(a*x)+35/128*Chi(arctanh(a*x))/a+189/
128*Chi(3*arctanh(a*x))/a+175/128*Chi(5*arctanh(a*x))/a+49/128*Chi(7*arctanh(a*x))/a

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Rubi [A]
time = 0.30, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6113, 6179, 6181, 5556, 3382, 6115, 3393} \begin {gather*} -\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}+\frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{128 a}+\frac {189 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{128 a}+\frac {175 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{128 a}+\frac {49 \text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{128 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]^3),x]

[Out]

-1/2*1/(a*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]^2) - (7*x)/(2*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]) + (35*CoshIntegral[
ArcTanh[a*x]])/(128*a) + (189*CoshIntegral[3*ArcTanh[a*x]])/(128*a) + (175*CoshIntegral[5*ArcTanh[a*x]])/(128*
a) + (49*CoshIntegral[7*ArcTanh[a*x]])/(128*a)

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3393

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 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 6113

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

Rule 6115

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 6179

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

Rule 6181

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

Rubi steps

\begin {align*} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}+\frac {1}{2} (7 a) \int \frac {x}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)^2} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7}{2} \int \frac {1}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)} \, dx+\left (21 a^2\right ) \int \frac {x^2}{\left (1-a^2 x^2\right )^{9/2} \tanh ^{-1}(a x)} \, dx\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7 \text {Subst}\left (\int \frac {\cosh ^7(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {21 \text {Subst}\left (\int \frac {\cosh ^5(x) \sinh ^2(x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7 \text {Subst}\left (\int \left (\frac {35 \cosh (x)}{64 x}+\frac {21 \cosh (3 x)}{64 x}+\frac {7 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{2 a}+\frac {21 \text {Subst}\left (\int \left (-\frac {5 \cosh (x)}{64 x}+\frac {\cosh (3 x)}{64 x}+\frac {3 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\tanh ^{-1}(a x)\right )}{a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {7 \text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}+\frac {21 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {21 \text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {49 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}+\frac {63 \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {147 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}-\frac {105 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{64 a}+\frac {245 \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\tanh ^{-1}(a x)\right )}{128 a}\\ &=-\frac {1}{2 a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {7 x}{2 \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{128 a}+\frac {189 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{128 a}+\frac {175 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{128 a}+\frac {49 \text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{128 a}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 99, normalized size = 0.93 \begin {gather*} \frac {1}{128} \left (-\frac {64}{a \left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)^2}-\frac {448 x}{\left (1-a^2 x^2\right )^{7/2} \tanh ^{-1}(a x)}+\frac {35 \text {Chi}\left (\tanh ^{-1}(a x)\right )}{a}+\frac {189 \text {Chi}\left (3 \tanh ^{-1}(a x)\right )}{a}+\frac {175 \text {Chi}\left (5 \tanh ^{-1}(a x)\right )}{a}+\frac {49 \text {Chi}\left (7 \tanh ^{-1}(a x)\right )}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - a^2*x^2)^(9/2)*ArcTanh[a*x]^3),x]

[Out]

(-64/(a*(1 - a^2*x^2)^(7/2)*ArcTanh[a*x]^2) - (448*x)/((1 - a^2*x^2)^(7/2)*ArcTanh[a*x]) + (35*CoshIntegral[Ar
cTanh[a*x]])/a + (189*CoshIntegral[3*ArcTanh[a*x]])/a + (175*CoshIntegral[5*ArcTanh[a*x]])/a + (49*CoshIntegra
l[7*ArcTanh[a*x]])/a)/128

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(363\) vs. \(2(91)=182\).
time = 4.01, size = 364, normalized size = 3.40

method	result	size
default	\(\frac {189 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2} a^{2} x^{2}+175 \hyperbolicCosineIntegral \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2} a^{2} x^{2}+49 \hyperbolicCosineIntegral \left (7 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2} a^{2} x^{2}+35 \arctanh \left (a x \right )^{2} \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) a^{2} x^{2}-63 \arctanh \left (a x \right ) \sinh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}-7 \arctanh \left (a x \right ) \sinh \left (7 \arctanh \left (a x \right )\right ) a^{2} x^{2}-35 \arctanh \left (a x \right ) \sinh \left (5 \arctanh \left (a x \right )\right ) a^{2} x^{2}-\cosh \left (7 \arctanh \left (a x \right )\right ) a^{2} x^{2}-7 \cosh \left (5 \arctanh \left (a x \right )\right ) a^{2} x^{2}-21 \cosh \left (3 \arctanh \left (a x \right )\right ) a^{2} x^{2}+35 \sqrt {-a^{2} x^{2}+1}\, a x \arctanh \left (a x \right )-189 \hyperbolicCosineIntegral \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-175 \hyperbolicCosineIntegral \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-49 \hyperbolicCosineIntegral \left (7 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}-35 \hyperbolicCosineIntegral \left (\arctanh \left (a x \right )\right ) \arctanh \left (a x \right )^{2}+63 \sinh \left (3 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+7 \sinh \left (7 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+35 \sinh \left (5 \arctanh \left (a x \right )\right ) \arctanh \left (a x \right )+\cosh \left (7 \arctanh \left (a x \right )\right )+7 \cosh \left (5 \arctanh \left (a x \right )\right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cosh \left (3 \arctanh \left (a x \right )\right )}{128 a \arctanh \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}\)	\(364\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/128/a*(189*Chi(3*arctanh(a*x))*arctanh(a*x)^2*a^2*x^2+175*Chi(5*arctanh(a*x))*arctanh(a*x)^2*a^2*x^2+49*Chi(
7*arctanh(a*x))*arctanh(a*x)^2*a^2*x^2+35*arctanh(a*x)^2*Chi(arctanh(a*x))*a^2*x^2-63*arctanh(a*x)*sinh(3*arct
anh(a*x))*a^2*x^2-7*arctanh(a*x)*sinh(7*arctanh(a*x))*a^2*x^2-35*arctanh(a*x)*sinh(5*arctanh(a*x))*a^2*x^2-cos
h(7*arctanh(a*x))*a^2*x^2-7*cosh(5*arctanh(a*x))*a^2*x^2-21*cosh(3*arctanh(a*x))*a^2*x^2+35*(-a^2*x^2+1)^(1/2)
*a*x*arctanh(a*x)-189*Chi(3*arctanh(a*x))*arctanh(a*x)^2-175*Chi(5*arctanh(a*x))*arctanh(a*x)^2-49*Chi(7*arcta
nh(a*x))*arctanh(a*x)^2-35*Chi(arctanh(a*x))*arctanh(a*x)^2+63*sinh(3*arctanh(a*x))*arctanh(a*x)+7*sinh(7*arct
anh(a*x))*arctanh(a*x)+35*sinh(5*arctanh(a*x))*arctanh(a*x)+cosh(7*arctanh(a*x))+7*cosh(5*arctanh(a*x))+35*(-a
^2*x^2+1)^(1/2)+21*cosh(3*arctanh(a*x)))/arctanh(a*x)^2/(a^2*x^2-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^3,x, algorithm="maxima")

[Out]

integrate(1/((-a^2*x^2 + 1)^(9/2)*arctanh(a*x)^3), x)

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Fricas [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(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^3,x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)/((a^10*x^10 - 5*a^8*x^8 + 10*a^6*x^6 - 10*a^4*x^4 + 5*a^2*x^2 - 1)*arctanh(a*x)^3
), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \operatorname {atanh}^{3}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*x**2+1)**(9/2)/atanh(a*x)**3,x)

[Out]

Integral(1/((-(a*x - 1)*(a*x + 1))**(9/2)*atanh(a*x)**3), 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(1/(-a^2*x^2+1)^(9/2)/arctanh(a*x)^3,x, algorithm="giac")

[Out]

integrate(1/((-a^2*x^2 + 1)^(9/2)*arctanh(a*x)^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {atanh}\left (a\,x\right )}^3\,{\left (1-a^2\,x^2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(atanh(a*x)^3*(1 - a^2*x^2)^(9/2)),x)

[Out]

int(1/(atanh(a*x)^3*(1 - a^2*x^2)^(9/2)), x)

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