3.917 \(\int \frac{e^{\tanh ^{-1}(a x)}}{x^3 (c-a^2 c x^2)^3} \, dx\)

Optimal. Leaf size=164 \[ -\frac{16 a \sqrt{1-a^2 x^2}}{5 c^3 x}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}+\frac{24 a x+35}{15 c^3 x^2 \sqrt{1-a^2 x^2}}+\frac{6 a x+7}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a x+1}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{7 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^3} \]

[Out]

(1 + a*x)/(5*c^3*x^2*(1 - a^2*x^2)^(5/2)) + (7 + 6*a*x)/(15*c^3*x^2*(1 - a^2*x^2)^(3/2)) + (35 + 24*a*x)/(15*c
^3*x^2*Sqrt[1 - a^2*x^2]) - (7*Sqrt[1 - a^2*x^2])/(2*c^3*x^2) - (16*a*Sqrt[1 - a^2*x^2])/(5*c^3*x) - (7*a^2*Ar
cTanh[Sqrt[1 - a^2*x^2]])/(2*c^3)

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Rubi [A]  time = 0.194332, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6148, 823, 835, 807, 266, 63, 208} \[ -\frac{16 a \sqrt{1-a^2 x^2}}{5 c^3 x}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}+\frac{24 a x+35}{15 c^3 x^2 \sqrt{1-a^2 x^2}}+\frac{6 a x+7}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{a x+1}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}-\frac{7 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^3),x]

[Out]

(1 + a*x)/(5*c^3*x^2*(1 - a^2*x^2)^(5/2)) + (7 + 6*a*x)/(15*c^3*x^2*(1 - a^2*x^2)^(3/2)) + (35 + 24*a*x)/(15*c
^3*x^2*Sqrt[1 - a^2*x^2]) - (7*Sqrt[1 - a^2*x^2])/(2*c^3*x^2) - (16*a*Sqrt[1 - a^2*x^2])/(5*c^3*x) - (7*a^2*Ar
cTanh[Sqrt[1 - a^2*x^2]])/(2*c^3)

Rule 6148

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[x^m*(1 -
a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || Gt
Q[c, 0]) && IGtQ[(n + 1)/2, 0] &&  !IntegerQ[p - n/2]

Rule 823

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((e*f - d*g)
*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[
(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; Fr
eeQ[{a, c, d, e, f, g, p}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || Integer
sQ[2*m, 2*p])

Rule 807

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& EqQ[Simplify[m + 2*p + 3], 0]

Rule 266

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^3} \, dx &=\frac{\int \frac{1+a x}{x^3 \left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{\int \frac{7 a^2+6 a^3 x}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{\int \frac{35 a^4+24 a^5 x}{x^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{35+24 a x}{15 c^3 x^2 \sqrt{1-a^2 x^2}}+\frac{\int \frac{105 a^6+48 a^7 x}{x^3 \sqrt{1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{35+24 a x}{15 c^3 x^2 \sqrt{1-a^2 x^2}}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}-\frac{\int \frac{-96 a^7-105 a^8 x}{x^2 \sqrt{1-a^2 x^2}} \, dx}{30 a^6 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{35+24 a x}{15 c^3 x^2 \sqrt{1-a^2 x^2}}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}-\frac{16 a \sqrt{1-a^2 x^2}}{5 c^3 x}+\frac{\left (7 a^2\right ) \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx}{2 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{35+24 a x}{15 c^3 x^2 \sqrt{1-a^2 x^2}}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}-\frac{16 a \sqrt{1-a^2 x^2}}{5 c^3 x}+\frac{\left (7 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )}{4 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{35+24 a x}{15 c^3 x^2 \sqrt{1-a^2 x^2}}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}-\frac{16 a \sqrt{1-a^2 x^2}}{5 c^3 x}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{2 c^3}\\ &=\frac{1+a x}{5 c^3 x^2 \left (1-a^2 x^2\right )^{5/2}}+\frac{7+6 a x}{15 c^3 x^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{35+24 a x}{15 c^3 x^2 \sqrt{1-a^2 x^2}}-\frac{7 \sqrt{1-a^2 x^2}}{2 c^3 x^2}-\frac{16 a \sqrt{1-a^2 x^2}}{5 c^3 x}-\frac{7 a^2 \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0671797, size = 133, normalized size = 0.81 \[ \frac{96 a^6 x^6+9 a^5 x^5-249 a^4 x^4+4 a^3 x^3+176 a^2 x^2-105 a^2 x^2 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2} \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-15 a x-15}{30 c^3 x^2 (a x-1)^2 (a x+1) \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^3),x]

[Out]

(-15 - 15*a*x + 176*a^2*x^2 + 4*a^3*x^3 - 249*a^4*x^4 + 9*a^5*x^5 + 96*a^6*x^6 - 105*a^2*x^2*(-1 + a*x)^2*(1 +
a*x)*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(30*c^3*x^2*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

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Maple [B]  time = 0.052, size = 326, normalized size = 2. \begin{align*} -{\frac{1}{{c}^{3}} \left ({\frac{a}{x}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{1}{20\,a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-3}}-{\frac{11\,a}{10} \left ({\frac{1}{3\,a}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-2}}-{\frac{1}{3}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}} \right ) }+{\frac{a}{8} \left ( -{\frac{1}{3\,a \left ( x+{a}^{-1} \right ) ^{2}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}-{\frac{1}{3\,x+3\,{a}^{-1}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }} \right ) }+{\frac{7\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{9\,a}{16\,x+16\,{a}^{-1}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{39\,a}{16}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) } \left ( x-{a}^{-1} \right ) ^{-1}}+{\frac{1}{2\,{x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^3,x)

[Out]

-1/c^3*(a*(-a^2*x^2+1)^(1/2)/x+1/20/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-11/10*a*(1/3/a/(x-1/a)^2*(-
a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-1/3/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2))+1/8*a*(-1/3/a/(x+1/a)^2*(-a^2
*(x+1/a)^2+2*a*(x+1/a))^(1/2)-1/3/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))+7/2*a^2*arctanh(1/(-a^2*x^2+1)^(
1/2))-9/16*a/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+39/16*a/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/2
*(-a^2*x^2+1)^(1/2)/x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((a*x + 1)/((a^2*c*x^2 - c)^3*sqrt(-a^2*x^2 + 1)*x^3), x)

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Fricas [A]  time = 1.62287, size = 505, normalized size = 3.08 \begin{align*} \frac{116 \, a^{7} x^{7} - 116 \, a^{6} x^{6} - 232 \, a^{5} x^{5} + 232 \, a^{4} x^{4} + 116 \, a^{3} x^{3} - 116 \, a^{2} x^{2} + 105 \,{\left (a^{7} x^{7} - a^{6} x^{6} - 2 \, a^{5} x^{5} + 2 \, a^{4} x^{4} + a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{x}\right ) -{\left (96 \, a^{6} x^{6} + 9 \, a^{5} x^{5} - 249 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 176 \, a^{2} x^{2} - 15 \, a x - 15\right )} \sqrt{-a^{2} x^{2} + 1}}{30 \,{\left (a^{5} c^{3} x^{7} - a^{4} c^{3} x^{6} - 2 \, a^{3} c^{3} x^{5} + 2 \, a^{2} c^{3} x^{4} + a c^{3} x^{3} - c^{3} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^3,x, algorithm="fricas")

[Out]

1/30*(116*a^7*x^7 - 116*a^6*x^6 - 232*a^5*x^5 + 232*a^4*x^4 + 116*a^3*x^3 - 116*a^2*x^2 + 105*(a^7*x^7 - a^6*x
^6 - 2*a^5*x^5 + 2*a^4*x^4 + a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (96*a^6*x^6 + 9*a^5*x^5 - 24
9*a^4*x^4 + 4*a^3*x^3 + 176*a^2*x^2 - 15*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^5*c^3*x^7 - a^4*c^3*x^6 - 2*a^3*c^3*
x^5 + 2*a^2*c^3*x^4 + a*c^3*x^3 - c^3*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a}{- a^{6} x^{8} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt{- a^{2} x^{2} + 1} + x^{2} \sqrt{- a^{2} x^{2} + 1}}\, dx + \int \frac{1}{- a^{6} x^{9} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{7} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{5} \sqrt{- a^{2} x^{2} + 1} + x^{3} \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a/(-a**6*x**8*sqrt(-a**2*x**2 + 1) + 3*a**4*x**6*sqrt(-a**2*x**2 + 1) - 3*a**2*x**4*sqrt(-a**2*x**2
+ 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**6*x**9*sqrt(-a**2*x**2 + 1) + 3*a**4*x**7*sqrt(-a**2*x
**2 + 1) - 3*a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c**3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt{-a^{2} x^{2} + 1} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c)^3,x, algorithm="giac")

[Out]

integrate(-(a*x + 1)/((a^2*c*x^2 - c)^3*sqrt(-a^2*x^2 + 1)*x^3), x)