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

Optimal. Leaf size=206 \[ \frac{a \sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{5 a \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (a x+1)}{4 c \sqrt{c-a^2 c x^2}} \]

[Out]

-(Sqrt[1 - a^2*x^2]/(c*x*Sqrt[c - a^2*c*x^2])) + (a*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (
a*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (5*a*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c - a^2*c
*x^2]) + (a*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*Sqrt[c - a^2*c*x^2])

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Rubi [A]  time = 0.226395, antiderivative size = 206, 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} \[ \frac{a \sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{c x \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{5 a \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (a x+1)}{4 c \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/(c*x*Sqrt[c - a^2*c*x^2])) + (a*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) + (
a*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (5*a*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c - a^2*c
*x^2]) + (a*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*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^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1-a^2 x^2} \int \frac{e^{\tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \frac{1}{x^2 (1-a x)^2 (1+a x)} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-a^2 x^2} \int \left (\frac{1}{x^2}+\frac{a}{x}+\frac{a^2}{2 (-1+a x)^2}-\frac{5 a^2}{4 (-1+a x)}+\frac{a^2}{4 (1+a x)}\right ) \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{1-a^2 x^2}}{c x \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{5 a \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}+\frac{a \sqrt{1-a^2 x^2} \log (1+a x)}{4 c \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0584262, size = 76, normalized size = 0.37 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{2 a}{1-a x}+4 a \log (x)-5 a \log (1-a x)+a \log (a x+1)-\frac{4}{x}\right )}{4 c \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*(-4/x + (2*a)/(1 - a*x) + 4*a*Log[x] - 5*a*Log[1 - a*x] + a*Log[1 + a*x]))/(4*c*Sqrt[c - a^
2*c*x^2])

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Maple [A]  time = 0.096, size = 122, normalized size = 0.6 \begin{align*} -{\frac{4\,{a}^{2}\ln \left ( x \right ){x}^{2}+\ln \left ( ax+1 \right ){a}^{2}{x}^{2}-5\,\ln \left ( ax-1 \right ){a}^{2}{x}^{2}-4\,a\ln \left ( x \right ) x-ax\ln \left ( ax+1 \right ) +5\,\ln \left ( ax-1 \right ) xa-6\,ax+4}{ \left ( 4\,{a}^{2}{x}^{2}-4 \right ){c}^{2} \left ( ax-1 \right ) x}\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^2/(-a^2*c*x^2+c)^(3/2),x)

[Out]

-1/4*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(4*a^2*ln(x)*x^2+ln(a*x+1)*a^2*x^2-5*ln(a*x-1)*a^2*x^2-4*a*ln(x
)*x-a*x*ln(a*x+1)+5*ln(a*x-1)*x*a-6*a*x+4)/(a^2*x^2-1)/c^2/(a*x-1)/x

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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{3}{2}} \sqrt{-a^{2} x^{2} + 1} x^{2}}\,{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^2/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((a*x + 1)/((-a^2*c*x^2 + c)^(3/2)*sqrt(-a^2*x^2 + 1)*x^2), 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^{5} c^{2} x^{7} - a^{4} c^{2} x^{6} - 2 \, a^{3} c^{2} x^{5} + 2 \, a^{2} c^{2} x^{4} + a c^{2} x^{3} - c^{2} x^{2}}, 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^2/(-a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

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

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

[Out]

Integral((a*x + 1)/(x**2*sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1))**(3/2)), 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{3}{2}} \sqrt{-a^{2} x^{2} + 1} x^{2}}\,{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^2/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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