3.975 \(\int \frac{e^{\tanh ^{-1}(a x)}}{x^4 (c-a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=297 \[ \frac{a^3 \sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{c x \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3 \sqrt{c-a^2 c x^2}}+\frac{2 a^3 \sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{9 a^3 \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}+\frac{a^3 \sqrt{1-a^2 x^2} \log (a x+1)}{4 c \sqrt{c-a^2 c x^2}} \]

[Out]

-Sqrt[1 - a^2*x^2]/(3*c*x^3*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(2*c*x^2*Sqrt[c - a^2*c*x^2]) - (2*a^
2*Sqrt[1 - a^2*x^2])/(c*x*Sqrt[c - a^2*c*x^2]) + (a^3*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) +
(2*a^3*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (9*a^3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c
- a^2*c*x^2]) + (a^3*Sqrt[1 - a^2*x^2]*Log[1 + a*x])/(4*c*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.241217, antiderivative size = 297, 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} \[ \frac{a^3 \sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{c x \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{\sqrt{1-a^2 x^2}}{3 c x^3 \sqrt{c-a^2 c x^2}}+\frac{2 a^3 \sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{9 a^3 \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}+\frac{a^3 \sqrt{1-a^2 x^2} \log (a x+1)}{4 c \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[a*x]/(x^4*(c - a^2*c*x^2)^(3/2)),x]

[Out]

-Sqrt[1 - a^2*x^2]/(3*c*x^3*Sqrt[c - a^2*c*x^2]) - (a*Sqrt[1 - a^2*x^2])/(2*c*x^2*Sqrt[c - a^2*c*x^2]) - (2*a^
2*Sqrt[1 - a^2*x^2])/(c*x*Sqrt[c - a^2*c*x^2]) + (a^3*Sqrt[1 - a^2*x^2])/(2*c*(1 - a*x)*Sqrt[c - a^2*c*x^2]) +
(2*a^3*Sqrt[1 - a^2*x^2]*Log[x])/(c*Sqrt[c - a^2*c*x^2]) - (9*a^3*Sqrt[1 - a^2*x^2]*Log[1 - a*x])/(4*c*Sqrt[c
- a^2*c*x^2]) + (a^3*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^4 \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^4 \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^4 (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^4}+\frac{a}{x^3}+\frac{2 a^2}{x^2}+\frac{2 a^3}{x}+\frac{a^4}{2 (-1+a x)^2}-\frac{9 a^4}{4 (-1+a x)}+\frac{a^4}{4 (1+a x)}\right ) \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sqrt{1-a^2 x^2}}{3 c x^3 \sqrt{c-a^2 c x^2}}-\frac{a \sqrt{1-a^2 x^2}}{2 c x^2 \sqrt{c-a^2 c x^2}}-\frac{2 a^2 \sqrt{1-a^2 x^2}}{c x \sqrt{c-a^2 c x^2}}+\frac{a^3 \sqrt{1-a^2 x^2}}{2 c (1-a x) \sqrt{c-a^2 c x^2}}+\frac{2 a^3 \sqrt{1-a^2 x^2} \log (x)}{c \sqrt{c-a^2 c x^2}}-\frac{9 a^3 \sqrt{1-a^2 x^2} \log (1-a x)}{4 c \sqrt{c-a^2 c x^2}}+\frac{a^3 \sqrt{1-a^2 x^2} \log (1+a x)}{4 c \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0727007, size = 99, normalized size = 0.33 \[ \frac{\sqrt{1-a^2 x^2} \left (\frac{6 a^3}{1-a x}-\frac{24 a^2}{x}+24 a^3 \log (x)-27 a^3 \log (1-a x)+3 a^3 \log (a x+1)-\frac{6 a}{x^2}-\frac{4}{x^3}\right )}{12 c \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[a*x]/(x^4*(c - a^2*c*x^2)^(3/2)),x]

[Out]

(Sqrt[1 - a^2*x^2]*(-4/x^3 - (6*a)/x^2 - (24*a^2)/x + (6*a^3)/(1 - a*x) + 24*a^3*Log[x] - 27*a^3*Log[1 - a*x]
+ 3*a^3*Log[1 + a*x]))/(12*c*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Maple [A]  time = 0.097, size = 151, normalized size = 0.5 \begin{align*} -{\frac{24\,{a}^{4}\ln \left ( x \right ){x}^{4}+3\,\ln \left ( ax+1 \right ){a}^{4}{x}^{4}-27\,\ln \left ( ax-1 \right ){a}^{4}{x}^{4}-24\,{a}^{3}\ln \left ( x \right ){x}^{3}-3\,{a}^{3}{x}^{3}\ln \left ( ax+1 \right ) +27\,\ln \left ( ax-1 \right ){x}^{3}{a}^{3}-30\,{x}^{3}{a}^{3}+18\,{a}^{2}{x}^{2}+2\,ax+4}{ \left ( 12\,{a}^{2}{x}^{2}-12 \right ){c}^{2}{x}^{3} \left ( ax-1 \right ) }\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^4/(-a^2*c*x^2+c)^(3/2),x)

[Out]

-1/12*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(24*a^4*ln(x)*x^4+3*ln(a*x+1)*a^4*x^4-27*ln(a*x-1)*a^4*x^4-24*
a^3*ln(x)*x^3-3*a^3*x^3*ln(a*x+1)+27*ln(a*x-1)*x^3*a^3-30*x^3*a^3+18*a^2*x^2+2*a*x+4)/(a^2*x^2-1)/c^2/x^3/(a*x
-1)

________________________________________________________________________________________

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^{4}}\,{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^4/(-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^4), x)

________________________________________________________________________________________

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^{9} - a^{4} c^{2} x^{8} - 2 \, a^{3} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{6} + a c^{2} x^{5} - c^{2} x^{4}}, 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^4/(-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^9 - a^4*c^2*x^8 - 2*a^3*c^2*x^7 + 2*a^2*c^2*x^6 +
a*c^2*x^5 - c^2*x^4), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a x + 1}{x^{4} \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**4/(-a**2*c*x**2+c)**(3/2),x)

[Out]

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

________________________________________________________________________________________

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^{4}}\,{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^4/(-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^4), x)