3.985 \(\int \frac{e^{\tanh ^{-1}(a x)}}{x^3 (c-a^2 c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=345 \[ \frac{a^2 \sqrt{1-a^2 x^2}}{c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{8 c^2 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}+\frac{3 a^2 \sqrt{1-a^2 x^2} \log (x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{39 a^2 \sqrt{1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt{c-a^2 c x^2}}-\frac{9 a^2 \sqrt{1-a^2 x^2} \log (a x+1)}{16 c^2 \sqrt{c-a^2 c x^2}} \]

[Out]

-Sqrt[1 - a^2*x^2]/(2*c^2*x^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(c^2*x*Sqrt[c - a^2*c*x^2]) + (a^2*
Sqrt[1 - a^2*x^2])/(8*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(c^2*(1 - a*x)*Sqrt[c - a
^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(8*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*a^2*Sqrt[1 - a^2*x^2]*Log[x])/
(c^2*Sqrt[c - a^2*c*x^2]) - (39*a^2*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*c^2*Sqrt[c - a^2*c*x^2]) - (9*a^2*Sqrt
[1 - a^2*x^2]*Log[1 + a*x])/(16*c^2*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.256693, antiderivative size = 345, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6153, 6150, 88} \[ \frac{a^2 \sqrt{1-a^2 x^2}}{c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{8 c^2 (a x+1) \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}+\frac{3 a^2 \sqrt{1-a^2 x^2} \log (x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{39 a^2 \sqrt{1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt{c-a^2 c x^2}}-\frac{9 a^2 \sqrt{1-a^2 x^2} \log (a x+1)}{16 c^2 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^(5/2)),x]

[Out]

-Sqrt[1 - a^2*x^2]/(2*c^2*x^2*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(c^2*x*Sqrt[c - a^2*c*x^2]) + (a^2*
Sqrt[1 - a^2*x^2])/(8*c^2*(1 - a*x)^2*Sqrt[c - a^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(c^2*(1 - a*x)*Sqrt[c - a
^2*c*x^2]) + (a^2*Sqrt[1 - a^2*x^2])/(8*c^2*(1 + a*x)*Sqrt[c - a^2*c*x^2]) + (3*a^2*Sqrt[1 - a^2*x^2]*Log[x])/
(c^2*Sqrt[c - a^2*c*x^2]) - (39*a^2*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(16*c^2*Sqrt[c - a^2*c*x^2]) - (9*a^2*Sqrt
[1 - a^2*x^2]*Log[1 + a*x])/(16*c^2*Sqrt[c - a^2*c*x^2])

Rule 6153

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

Rule 6150

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^3 \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)}}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{x^3 (1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{x^3}+\frac{a}{x^2}+\frac{3 a^2}{x}-\frac{a^3}{4 (-1+a x)^3}+\frac{a^3}{(-1+a x)^2}-\frac{39 a^3}{16 (-1+a x)}-\frac{a^3}{8 (1+a x)^2}-\frac{9 a^3}{16 (1+a x)}\right ) \, dx}{c^2 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{1-a^2 x^2}}{2 c^2 x^2 \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2}}{c^2 x \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{c^2 (1-a x) \sqrt{c-a^2 c x^2}}+\frac{a^2 \sqrt{1-a^2 x^2}}{8 c^2 (1+a x) \sqrt{c-a^2 c x^2}}+\frac{3 a^2 \sqrt{1-a^2 x^2} \log (x)}{c^2 \sqrt{c-a^2 c x^2}}-\frac{39 a^2 \sqrt{1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt{c-a^2 c x^2}}-\frac{9 a^2 \sqrt{1-a^2 x^2} \log (1+a x)}{16 c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.124727, size = 115, normalized size = 0.33 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{16 a^2}{1-a x}+\frac{2 a^2}{a x+1}+\frac{2 a^2}{(a x-1)^2}+48 a^2 \log (x)-39 a^2 \log (1-a x)-9 a^2 \log (a x+1)-\frac{16 a}{x}-\frac{8}{x^2}\right )}{16 c^2 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)^(5/2)),x]

[Out]

(Sqrt[1 - a^2*x^2]*(-8/x^2 - (16*a)/x + (16*a^2)/(1 - a*x) + (2*a^2)/(-1 + a*x)^2 + (2*a^2)/(1 + a*x) + 48*a^2
*Log[x] - 39*a^2*Log[1 - a*x] - 9*a^2*Log[1 + a*x]))/(16*c^2*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.105, size = 242, normalized size = 0.7 \begin{align*} -{\frac{48\,{a}^{5}\ln \left ( x \right ){x}^{5}-9\,\ln \left ( ax+1 \right ){x}^{5}{a}^{5}-39\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-48\,{a}^{4}\ln \left ( x \right ){x}^{4}+9\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}+39\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-30\,{x}^{4}{a}^{4}-48\,{a}^{3}\ln \left ( x \right ){x}^{3}+9\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +39\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}+6\,{x}^{3}{a}^{3}+48\,{a}^{2}\ln \left ( x \right ){x}^{2}-9\,\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-39\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}+44\,{a}^{2}{x}^{2}-8\,ax-8}{ \left ( 16\,{a}^{2}{x}^{2}-16 \right ){c}^{3} \left ( ax-1 \right ) ^{2} \left ( ax+1 \right ){x}^{2}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({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)^(5/2),x)

[Out]

-1/16*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(48*a^5*ln(x)*x^5-9*ln(a*x+1)*x^5*a^5-39*ln(a*x-1)*x^5*a^5-48*
a^4*ln(x)*x^4+9*ln(a*x+1)*a^4*x^4+39*ln(a*x-1)*a^4*x^4-30*x^4*a^4-48*a^3*ln(x)*x^3+9*a^3*x^3*ln(a*x+1)+39*ln(a
*x-1)*x^3*a^3+6*x^3*a^3+48*a^2*ln(x)*x^2-9*ln(a*x+1)*a^2*x^2-39*ln(a*x-1)*a^2*x^2+44*a^2*x^2-8*a*x-8)/(a^2*x^2
-1)/c^3/(a*x-1)^2/(a*x+1)/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 )}^{\frac{5}{2}} \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)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)/((-a^2*c*x^2 + c)^(5/2)*sqrt(-a^2*x^2 + 1)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{a^{7} c^{3} x^{10} - a^{6} c^{3} x^{9} - 3 \, a^{5} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{3} c^{3} x^{6} - 3 \, a^{2} c^{3} x^{5} - a c^{3} x^{4} + c^{3} x^{3}}, x\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)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^7*c^3*x^10 - a^6*c^3*x^9 - 3*a^5*c^3*x^8 + 3*a^4*c^3*x^7 +
3*a^3*c^3*x^6 - 3*a^2*c^3*x^5 - a*c^3*x^4 + c^3*x^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{x^{3} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, dx \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)**(5/2),x)

[Out]

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

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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 )}^{\frac{5}{2}} \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)^(5/2),x, algorithm="giac")

[Out]

integrate((a*x + 1)/((-a^2*c*x^2 + c)^(5/2)*sqrt(-a^2*x^2 + 1)*x^3), x)