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

Optimal. Leaf size=229 \[ \frac{(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{32 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{8 (a x+1)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]

[Out]

(8*(1 + a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(1 + a*x)^7*(c - a^2*c*x^2)^(
9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 + a*x)^8*(c - a^2*c*x^2)^(9/2))/(a^10*(1 - 1/(a^2*x^2))^(9/
2)*x^9) - (8*(1 + a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 + a*x)^10*(c - a^2*
c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)

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Rubi [A]  time = 0.200254, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6192, 6193, 43} \[ \frac{(a x+1)^{10} \left (c-a^2 c x^2\right )^{9/2}}{10 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{8 (a x+1)^9 \left (c-a^2 c x^2\right )^{9/2}}{9 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{3 (a x+1)^8 \left (c-a^2 c x^2\right )^{9/2}}{a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}-\frac{32 (a x+1)^7 \left (c-a^2 c x^2\right )^{9/2}}{7 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}}+\frac{8 (a x+1)^6 \left (c-a^2 c x^2\right )^{9/2}}{3 a^{10} x^9 \left (1-\frac{1}{a^2 x^2}\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(1 + a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(1 + a*x)^7*(c - a^2*c*x^2)^(
9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 + a*x)^8*(c - a^2*c*x^2)^(9/2))/(a^10*(1 - 1/(a^2*x^2))^(9/
2)*x^9) - (8*(1 + a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 + a*x)^10*(c - a^2*
c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)

Rule 6192

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

Rule 6193

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

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

Mathematica [A]  time = 0.0641679, size = 79, normalized size = 0.34 \[ \frac{c^4 (a x+1)^6 \left (63 a^4 x^4-308 a^3 x^3+588 a^2 x^2-528 a x+193\right ) \sqrt{c-a^2 c x^2}}{630 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCoth[a*x]*(c - a^2*c*x^2)^(9/2),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^2*Sqrt[1
- 1/(a^2*x^2)]*x)

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Maple [A]  time = 0.046, size = 116, 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 ) ^{4} \left ( ax+1 \right ) ^{5}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{9}{2}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)*(-a^2*c*x^2+c)^(9/2),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*x-1)^4/(a*x+1)^5/((a*x-1)/(a*x+1))^(1/2)

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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{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.65864, size = 261, normalized size = 1.14 \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}}{630 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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(1/((a*x-1)/(a*x+1))**(1/2)*(-a**2*c*x**2+c)**(9/2),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{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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