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

Optimal. Leaf size=135 \[ \frac {a x+1}{5 c^3 x \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \sqrt {1-a^2 x^2}}{5 c^3 x}+\frac {5 a x+8}{5 c^3 x \sqrt {1-a^2 x^2}}+\frac {5 a x+6}{15 c^3 x \left (1-a^2 x^2\right )^{3/2}}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]

[Out]

1/5*(a*x+1)/c^3/x/(-a^2*x^2+1)^(5/2)+1/15*(5*a*x+6)/c^3/x/(-a^2*x^2+1)^(3/2)-a*arctanh((-a^2*x^2+1)^(1/2))/c^3
+1/5*(5*a*x+8)/c^3/x/(-a^2*x^2+1)^(1/2)-16/5*(-a^2*x^2+1)^(1/2)/c^3/x

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Rubi [A]  time = 0.17, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6148, 823, 807, 266, 63, 208} \[ \frac {a x+1}{5 c^3 x \left (1-a^2 x^2\right )^{5/2}}-\frac {16 \sqrt {1-a^2 x^2}}{5 c^3 x}+\frac {5 a x+8}{5 c^3 x \sqrt {1-a^2 x^2}}+\frac {5 a x+6}{15 c^3 x \left (1-a^2 x^2\right )^{3/2}}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + a*x)/(5*c^3*x*(1 - a^2*x^2)^(5/2)) + (6 + 5*a*x)/(15*c^3*x*(1 - a^2*x^2)^(3/2)) + (8 + 5*a*x)/(5*c^3*x*Sq
rt[1 - a^2*x^2]) - (16*Sqrt[1 - a^2*x^2])/(5*c^3*x) - (a*ArcTanh[Sqrt[1 - a^2*x^2]])/c^3

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]

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 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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.06, size = 121, normalized size = 0.90 \[ \frac {48 a^5 x^5-33 a^4 x^4-87 a^3 x^3+52 a^2 x^2-15 a x (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+38 a x-15}{15 c^3 x (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15 + 38*a*x + 52*a^2*x^2 - 87*a^3*x^3 - 33*a^4*x^4 + 48*a^5*x^5 - 15*a*x*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2
*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(15*c^3*x*(-1 + a*x)^2*(1 + a*x)*Sqrt[1 - a^2*x^2])

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fricas [A]  time = 0.65, size = 223, normalized size = 1.65 \[ \frac {23 \, a^{6} x^{6} - 23 \, a^{5} x^{5} - 46 \, a^{4} x^{4} + 46 \, a^{3} x^{3} + 23 \, a^{2} x^{2} - 23 \, a x + 15 \, {\left (a^{6} x^{6} - a^{5} x^{5} - 2 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (48 \, a^{5} x^{5} - 33 \, a^{4} x^{4} - 87 \, a^{3} x^{3} + 52 \, a^{2} x^{2} + 38 \, a x - 15\right )} \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{5} c^{3} x^{6} - a^{4} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{4} + 2 \, a^{2} c^{3} x^{3} + a c^{3} x^{2} - c^{3} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(23*a^6*x^6 - 23*a^5*x^5 - 46*a^4*x^4 + 46*a^3*x^3 + 23*a^2*x^2 - 23*a*x + 15*(a^6*x^6 - a^5*x^5 - 2*a^4*
x^4 + 2*a^3*x^3 + a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (48*a^5*x^5 - 33*a^4*x^4 - 87*a^3*x^3 + 52*
a^2*x^2 + 38*a*x - 15)*sqrt(-a^2*x^2 + 1))/(a^5*c^3*x^6 - a^4*c^3*x^5 - 2*a^3*c^3*x^4 + 2*a^2*c^3*x^3 + a*c^3*
x^2 - c^3*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [B]  time = 0.05, size = 314, normalized size = 2.33 \[ -\frac {a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{x}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a \left (x -\frac {1}{a}\right )^{2}}+\frac {27 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 \left (x -\frac {1}{a}\right )}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{4 a}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 a \left (x +\frac {1}{a}\right )^{2}}+\frac {23 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 \left (x +\frac {1}{a}\right )}}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/c^3*(a*arctanh(1/(-a^2*x^2+1)^(1/2))+(-a^2*x^2+1)^(1/2)/x-1/4/a/(x-1/a)^2*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2
)+27/16/(x-1/a)*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)+1/4/a*(1/5/a/(x-1/a)^3*(-a^2*(x-1/a)^2-2*a*(x-1/a))^(1/2)-2
/5*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)))+1/24
/a/(x+1/a)^2*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2)+23/48/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a))^(1/2))

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maxima [A]  time = 0.36, size = 178, normalized size = 1.32 \[ -\frac {\frac {15 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c^{3}} - \frac {15 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c^{3}} - \frac {2 \, {\left (15 \, {\left (a^{2} x^{2} - 1\right )}^{2} a^{2} - 5 \, {\left (a^{2} x^{2} - 1\right )} a^{2} + 3 \, a^{2}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} c^{3}}}{30 \, a} + \frac {16 \, a^{6} x^{6} - 40 \, a^{4} x^{4} + 30 \, a^{2} x^{2} - 5}{5 \, {\left (a^{4} c^{3} x^{5} - 2 \, a^{2} c^{3} x^{3} + c^{3} x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(15*a^2*log(sqrt(-a^2*x^2 + 1) + 1)/c^3 - 15*a^2*log(sqrt(-a^2*x^2 + 1) - 1)/c^3 - 2*(15*(a^2*x^2 - 1)^2
*a^2 - 5*(a^2*x^2 - 1)*a^2 + 3*a^2)/((-a^2*x^2 + 1)^(5/2)*c^3))/a + 1/5*(16*a^6*x^6 - 40*a^4*x^4 + 30*a^2*x^2
- 5)/((a^4*c^3*x^5 - 2*a^2*c^3*x^3 + c^3*x)*sqrt(a*x + 1)*sqrt(-a*x + 1))

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mupad [B]  time = 0.09, size = 335, normalized size = 2.48 \[ \frac {17\,a^3\,\sqrt {1-a^2\,x^2}}{60\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^3\,x}+\frac {23\,a^2\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {413\,a^2\,\sqrt {1-a^2\,x^2}}{240\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(17*a^3*(1 - a^2*x^2)^(1/2))/(60*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) - (a^3*(1 - a^2*x^2)^(1/2))/(24*(a^2*c
^3 + 2*a^3*c^3*x + a^4*c^3*x^2)) - (1 - a^2*x^2)^(1/2)/(c^3*x) + (a*atan((1 - a^2*x^2)^(1/2)*1i)*1i)/c^3 + (23
*a^2*(1 - a^2*x^2)^(1/2))/(48*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) + (c^3*(-a^2)^(1/2))/a)) + (413*a^2*(1 - a^2*x^
2)^(1/2))/(240*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a)) + (a^2*(1 - a^2*x^2)^(1/2))/(20*(-a^2
)^(1/2)*(3*c^3*x*(-a^2)^(1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(-a^2)^(1/2)))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{- a^{6} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{8} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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