3.714 \(\int e^{-\tanh ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{9/2} \, dx\)

Optimal. Leaf size=375 \[ -\frac{a^9 x^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac{4 a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac{3 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{2 \left (1-a^2 x^2\right )^{9/2}}-\frac{4 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{5 \left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{3 \left (1-a^2 x^2\right )^{9/2}}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{7 \left (1-a^2 x^2\right )^{9/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{8 \left (1-a^2 x^2\right )^{9/2}}+\frac{a^8 x^9 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}} \]

[Out]

-((c - c/(a^2*x^2))^(9/2)*x)/(8*(1 - a^2*x^2)^(9/2)) + (a*(c - c/(a^2*x^2))^(9/2)*x^2)/(7*(1 - a^2*x^2)^(9/2))
+ (2*a^2*(c - c/(a^2*x^2))^(9/2)*x^3)/(3*(1 - a^2*x^2)^(9/2)) - (4*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a
^2*x^2)^(9/2)) - (3*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(2*(1 - a^2*x^2)^(9/2)) + (2*a^5*(c - c/(a^2*x^2))^(9/2)*
x^6)/(1 - a^2*x^2)^(9/2) + (2*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(1 - a^2*x^2)^(9/2) - (4*a^7*(c - c/(a^2*x^2))^
(9/2)*x^8)/(1 - a^2*x^2)^(9/2) - (a^9*(c - c/(a^2*x^2))^(9/2)*x^10)/(1 - a^2*x^2)^(9/2) + (a^8*(c - c/(a^2*x^2
))^(9/2)*x^9*Log[x])/(1 - a^2*x^2)^(9/2)

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Rubi [A]  time = 0.207394, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6160, 6150, 88} \[ -\frac{a^9 x^{10} \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac{4 a^7 x^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^6 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^5 x^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}}-\frac{3 a^4 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{2 \left (1-a^2 x^2\right )^{9/2}}-\frac{4 a^3 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{5 \left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^2 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{3 \left (1-a^2 x^2\right )^{9/2}}+\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{7 \left (1-a^2 x^2\right )^{9/2}}-\frac{x \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{8 \left (1-a^2 x^2\right )^{9/2}}+\frac{a^8 x^9 \log (x) \left (c-\frac{c}{a^2 x^2}\right )^{9/2}}{\left (1-a^2 x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))^(9/2)/E^ArcTanh[a*x],x]

[Out]

-((c - c/(a^2*x^2))^(9/2)*x)/(8*(1 - a^2*x^2)^(9/2)) + (a*(c - c/(a^2*x^2))^(9/2)*x^2)/(7*(1 - a^2*x^2)^(9/2))
+ (2*a^2*(c - c/(a^2*x^2))^(9/2)*x^3)/(3*(1 - a^2*x^2)^(9/2)) - (4*a^3*(c - c/(a^2*x^2))^(9/2)*x^4)/(5*(1 - a
^2*x^2)^(9/2)) - (3*a^4*(c - c/(a^2*x^2))^(9/2)*x^5)/(2*(1 - a^2*x^2)^(9/2)) + (2*a^5*(c - c/(a^2*x^2))^(9/2)*
x^6)/(1 - a^2*x^2)^(9/2) + (2*a^6*(c - c/(a^2*x^2))^(9/2)*x^7)/(1 - a^2*x^2)^(9/2) - (4*a^7*(c - c/(a^2*x^2))^
(9/2)*x^8)/(1 - a^2*x^2)^(9/2) - (a^9*(c - c/(a^2*x^2))^(9/2)*x^10)/(1 - a^2*x^2)^(9/2) + (a^8*(c - c/(a^2*x^2
))^(9/2)*x^9*Log[x])/(1 - a^2*x^2)^(9/2)

Rule 6160

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbol] :> Dist[(x^(2*p)*(c + d/x^2)^p)/
(1 + (c*x^2)/d)^p, Int[(u*(1 + (c*x^2)/d)^p*E^(n*ArcTanh[a*x]))/x^(2*p), x], x] /; FreeQ[{a, c, d, n, p}, x] &
& EqQ[c + a^2*d, 0] &&  !IntegerQ[p] &&  !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 e^{-\tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \, dx &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2}}{x^9} \, dx}{\left (1-a^2 x^2\right )^{9/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \frac{(1-a x)^5 (1+a x)^4}{x^9} \, dx}{\left (1-a^2 x^2\right )^{9/2}}\\ &=\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9\right ) \int \left (-a^9+\frac{1}{x^9}-\frac{a}{x^8}-\frac{4 a^2}{x^7}+\frac{4 a^3}{x^6}+\frac{6 a^4}{x^5}-\frac{6 a^5}{x^4}-\frac{4 a^6}{x^3}+\frac{4 a^7}{x^2}+\frac{a^8}{x}\right ) \, dx}{\left (1-a^2 x^2\right )^{9/2}}\\ &=-\frac{\left (c-\frac{c}{a^2 x^2}\right )^{9/2} x}{8 \left (1-a^2 x^2\right )^{9/2}}+\frac{a \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^2}{7 \left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^3}{3 \left (1-a^2 x^2\right )^{9/2}}-\frac{4 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^4}{5 \left (1-a^2 x^2\right )^{9/2}}-\frac{3 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^5}{2 \left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^6}{\left (1-a^2 x^2\right )^{9/2}}+\frac{2 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^7}{\left (1-a^2 x^2\right )^{9/2}}-\frac{4 a^7 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^8}{\left (1-a^2 x^2\right )^{9/2}}-\frac{a^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^{10}}{\left (1-a^2 x^2\right )^{9/2}}+\frac{a^8 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} x^9 \log (x)}{\left (1-a^2 x^2\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.105585, size = 115, normalized size = 0.31 \[ \frac{x^9 \left (c-\frac{c}{a^2 x^2}\right )^{9/2} \left (\frac{2 a^6}{x^2}+\frac{2 a^5}{x^3}-\frac{3 a^4}{2 x^4}-\frac{4 a^3}{5 x^5}+\frac{2 a^2}{3 x^6}+a^9 (-x)-\frac{4 a^7}{x}+a^8 \log (x)+\frac{a}{7 x^7}-\frac{1}{8 x^8}\right )}{\left (1-a^2 x^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - c/(a^2*x^2))^(9/2)/E^ArcTanh[a*x],x]

[Out]

((c - c/(a^2*x^2))^(9/2)*x^9*(-1/(8*x^8) + a/(7*x^7) + (2*a^2)/(3*x^6) - (4*a^3)/(5*x^5) - (3*a^4)/(2*x^4) + (
2*a^5)/x^3 + (2*a^6)/x^2 - (4*a^7)/x - a^9*x + a^8*Log[x]))/(1 - a^2*x^2)^(9/2)

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Maple [A]  time = 0.161, size = 118, normalized size = 0.3 \begin{align*} -{\frac{x \left ( -840\,{a}^{9}{x}^{9}+840\,{a}^{8}\ln \left ( x \right ){x}^{8}-3360\,{a}^{7}{x}^{7}+1680\,{x}^{6}{a}^{6}+1680\,{x}^{5}{a}^{5}-1260\,{x}^{4}{a}^{4}-672\,{x}^{3}{a}^{3}+560\,{a}^{2}{x}^{2}+120\,ax-105 \right ) }{840\, \left ({a}^{2}{x}^{2}-1 \right ) ^{5}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{9}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c/a^2/x^2)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)

[Out]

-1/840*(c*(a^2*x^2-1)/a^2/x^2)^(9/2)*x/(a^2*x^2-1)^5*(-a^2*x^2+1)^(1/2)*(-840*a^9*x^9+840*a^8*ln(x)*x^8-3360*a
^7*x^7+1680*x^6*a^6+1680*x^5*a^5-1260*x^4*a^4-672*x^3*a^3+560*a^2*x^2+120*a*x-105)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}}}{a x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*(c - c/(a^2*x^2))^(9/2)/(a*x + 1), x)

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Fricas [A]  time = 2.26478, size = 1355, normalized size = 3.61 \begin{align*} \left [\frac{420 \,{\left (a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} +{\left (a x^{5} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) +{\left (840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} - 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} -{\left (840 \, a^{9} + 3360 \, a^{7} - 1680 \, a^{6} - 1680 \, a^{5} + 1260 \, a^{4} + 672 \, a^{3} - 560 \, a^{2} - 120 \, a + 105\right )} c^{4} x^{8} + 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} - 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x + 105 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \,{\left (a^{10} x^{9} - a^{8} x^{7}\right )}}, -\frac{840 \,{\left (a^{9} c^{4} x^{9} - a^{7} c^{4} x^{7}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + a x\right )} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) -{\left (840 \, a^{9} c^{4} x^{9} + 3360 \, a^{7} c^{4} x^{7} - 1680 \, a^{6} c^{4} x^{6} - 1680 \, a^{5} c^{4} x^{5} -{\left (840 \, a^{9} + 3360 \, a^{7} - 1680 \, a^{6} - 1680 \, a^{5} + 1260 \, a^{4} + 672 \, a^{3} - 560 \, a^{2} - 120 \, a + 105\right )} c^{4} x^{8} + 1260 \, a^{4} c^{4} x^{4} + 672 \, a^{3} c^{4} x^{3} - 560 \, a^{2} c^{4} x^{2} - 120 \, a c^{4} x + 105 \, c^{4}\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{840 \,{\left (a^{10} x^{9} - a^{8} x^{7}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

[1/840*(420*(a^9*c^4*x^9 - a^7*c^4*x^7)*sqrt(-c)*log((a^2*c*x^6 + a^2*c*x^2 - c*x^4 + (a*x^5 - a*x)*sqrt(-a^2*
x^2 + 1)*sqrt(-c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - c)/(a^2*x^4 - x^2)) + (840*a^9*c^4*x^9 + 3360*a^7*c^4*x^7
- 1680*a^6*c^4*x^6 - 1680*a^5*c^4*x^5 - (840*a^9 + 3360*a^7 - 1680*a^6 - 1680*a^5 + 1260*a^4 + 672*a^3 - 560*a
^2 - 120*a + 105)*c^4*x^8 + 1260*a^4*c^4*x^4 + 672*a^3*c^4*x^3 - 560*a^2*c^4*x^2 - 120*a*c^4*x + 105*c^4)*sqrt
(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 - a^8*x^7), -1/840*(840*(a^9*c^4*x^9 - a^7*c^4*x^7)*
sqrt(c)*arctan(sqrt(-a^2*x^2 + 1)*(a*x^3 + a*x)*sqrt(c)*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^4 - (a^2 + 1)
*c*x^2 + c)) - (840*a^9*c^4*x^9 + 3360*a^7*c^4*x^7 - 1680*a^6*c^4*x^6 - 1680*a^5*c^4*x^5 - (840*a^9 + 3360*a^7
- 1680*a^6 - 1680*a^5 + 1260*a^4 + 672*a^3 - 560*a^2 - 120*a + 105)*c^4*x^8 + 1260*a^4*c^4*x^4 + 672*a^3*c^4*
x^3 - 560*a^2*c^4*x^2 - 120*a*c^4*x + 105*c^4)*sqrt(-a^2*x^2 + 1)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)))/(a^10*x^9 -
a^8*x^7)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a**2/x**2)**(9/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{9}{2}}}{a x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(9/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(-a^2*x^2 + 1)*(c - c/(a^2*x^2))^(9/2)/(a*x + 1), x)