3.1210 \(\int e^{-\tanh ^{-1}(a x)} (c-a^2 c x^2)^{9/2} \, dx\)

Optimal. Leaf size=234 \[ -\frac{c^4 (1-a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}+\frac{8 c^4 (1-a x)^9 \sqrt{c-a^2 c x^2}}{9 a \sqrt{1-a^2 x^2}}-\frac{3 c^4 (1-a x)^8 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}+\frac{32 c^4 (1-a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{8 c^4 (1-a x)^6 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}} \]

[Out]

(-8*c^4*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) + (32*c^4*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7
*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) + (8*c^4*(1 - a*x)^9*Sqr
t[c - a^2*c*x^2])/(9*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 - a*x)^10*Sqrt[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])

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Rubi [A]  time = 0.119275, antiderivative size = 234, 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 = {6143, 6140, 43} \[ -\frac{c^4 (1-a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}+\frac{8 c^4 (1-a x)^9 \sqrt{c-a^2 c x^2}}{9 a \sqrt{1-a^2 x^2}}-\frac{3 c^4 (1-a x)^8 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}+\frac{32 c^4 (1-a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{8 c^4 (1-a x)^6 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*c^4*(1 - a*x)^6*Sqrt[c - a^2*c*x^2])/(3*a*Sqrt[1 - a^2*x^2]) + (32*c^4*(1 - a*x)^7*Sqrt[c - a^2*c*x^2])/(7
*a*Sqrt[1 - a^2*x^2]) - (3*c^4*(1 - a*x)^8*Sqrt[c - a^2*c*x^2])/(a*Sqrt[1 - a^2*x^2]) + (8*c^4*(1 - a*x)^9*Sqr
t[c - a^2*c*x^2])/(9*a*Sqrt[1 - a^2*x^2]) - (c^4*(1 - a*x)^10*Sqrt[c - a^2*c*x^2])/(10*a*Sqrt[1 - a^2*x^2])

Rule 6143

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

Rule 6140

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^{9/2} \, dx &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int e^{-\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^{9/2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int (1-a x)^5 (1+a x)^4 \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (c^4 \sqrt{c-a^2 c x^2}\right ) \int \left (16 (1-a x)^5-32 (1-a x)^6+24 (1-a x)^7-8 (1-a x)^8+(1-a x)^9\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{8 c^4 (1-a x)^6 \sqrt{c-a^2 c x^2}}{3 a \sqrt{1-a^2 x^2}}+\frac{32 c^4 (1-a x)^7 \sqrt{c-a^2 c x^2}}{7 a \sqrt{1-a^2 x^2}}-\frac{3 c^4 (1-a x)^8 \sqrt{c-a^2 c x^2}}{a \sqrt{1-a^2 x^2}}+\frac{8 c^4 (1-a x)^9 \sqrt{c-a^2 c x^2}}{9 a \sqrt{1-a^2 x^2}}-\frac{c^4 (1-a x)^{10} \sqrt{c-a^2 c x^2}}{10 a \sqrt{1-a^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0595504, size = 76, normalized size = 0.32 \[ -\frac{c^4 (a x-1)^6 \left (63 a^4 x^4+308 a^3 x^3+588 a^2 x^2+528 a x+193\right ) \sqrt{c-a^2 c x^2}}{630 a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^4*(-1 + a*x)^6*Sqrt[c - a^2*c*x^2]*(193 + 528*a*x + 588*a^2*x^2 + 308*a^3*x^3 + 63*a^4*x^4))/(630*a*Sqrt[1
- a^2*x^2])

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Maple [A]  time = 0.03, size = 113, normalized size = 0.5 \begin{align*}{\frac{x \left ( 63\,{a}^{9}{x}^{9}-70\,{x}^{8}{a}^{8}-315\,{a}^{7}{x}^{7}+360\,{x}^{6}{a}^{6}+630\,{x}^{5}{a}^{5}-756\,{x}^{4}{a}^{4}-630\,{x}^{3}{a}^{3}+840\,{a}^{2}{x}^{2}+315\,ax-630 \right ) }{630\, \left ( ax+1 \right ) ^{5} \left ( ax-1 \right ) ^{5}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*x*(63*a^9*x^9-70*a^8*x^8-315*a^7*x^7+360*a^6*x^6+630*a^5*x^5-756*a^4*x^4-630*a^3*x^3+840*a^2*x^2+315*a*x
-630)*(-a^2*c*x^2+c)^(9/2)*(-a^2*x^2+1)^(1/2)/(a*x+1)^5/(a*x-1)^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.44713, size = 313, normalized size = 1.34 \begin{align*} \frac{{\left (63 \, a^{9} c^{4} x^{10} - 70 \, a^{8} c^{4} x^{9} - 315 \, a^{7} c^{4} x^{8} + 360 \, a^{6} c^{4} x^{7} + 630 \, a^{5} c^{4} x^{6} - 756 \, a^{4} c^{4} x^{5} - 630 \, a^{3} c^{4} x^{4} + 840 \, a^{2} c^{4} x^{3} + 315 \, a c^{4} x^{2} - 630 \, c^{4} x\right )} \sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}}{630 \,{\left (a^{2} x^{2} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/630*(63*a^9*c^4*x^10 - 70*a^8*c^4*x^9 - 315*a^7*c^4*x^8 + 360*a^6*c^4*x^7 + 630*a^5*c^4*x^6 - 756*a^4*c^4*x^
5 - 630*a^3*c^4*x^4 + 840*a^2*c^4*x^3 + 315*a*c^4*x^2 - 630*c^4*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^
2*x^2 - 1)

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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((-a**2*c*x**2+c)**(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{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}} \sqrt{-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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