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

Optimal. Leaf size=194 \[ \frac{16384 c^5 x \sqrt{1-\frac{1}{a^2 x^2}}}{693 \sqrt{c-a c x}}+\frac{4096}{693} c^4 x \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-a c x}+\frac{512}{231} c^3 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{3/2}+\frac{640}{693} c^2 x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{5/2}+\frac{40}{99} c x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{7/2}+\frac{2}{11} x \sqrt{1-\frac{1}{a^2 x^2}} (c-a c x)^{9/2} \]

[Out]

(16384*c^5*Sqrt[1 - 1/(a^2*x^2)]*x)/(693*Sqrt[c - a*c*x]) + (4096*c^4*Sqrt[1 - 1/(a^2*x^2)]*x*Sqrt[c - a*c*x])
/693 + (512*c^3*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(3/2))/231 + (640*c^2*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^
(5/2))/693 + (40*c*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(7/2))/99 + (2*Sqrt[1 - 1/(a^2*x^2)]*x*(c - a*c*x)^(9/2
))/11

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Rubi [A]  time = 0.225662, antiderivative size = 311, normalized size of antiderivative = 1.6, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6176, 6181, 94, 89, 78, 37} \[ -\frac{512 \sqrt{\frac{1}{a x}+1} (c-a c x)^{9/2}}{231 a^3 x^2 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{1024 \sqrt{\frac{1}{a x}+1} (c-a c x)^{9/2}}{99 a^4 x^3 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{22016 \sqrt{\frac{1}{a x}+1} (c-a c x)^{9/2}}{693 a^5 x^4 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 x \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^5 (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{40 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^4 (c-a c x)^{9/2}}{99 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{640 \sqrt{\frac{1}{a x}+1} \left (a-\frac{1}{x}\right )^3 (c-a c x)^{9/2}}{693 a^5 x \left (1-\frac{1}{a x}\right )^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-40*(a - x^(-1))^4*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(99*a^5*(1 - 1/(a*x))^(9/2)) - (22016*Sqrt[1 + 1/(a*x
)]*(c - a*c*x)^(9/2))/(693*a^5*(1 - 1/(a*x))^(9/2)*x^4) + (1024*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(99*a^4*(
1 - 1/(a*x))^(9/2)*x^3) - (512*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(231*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (640*(
a - x^(-1))^3*Sqrt[1 + 1/(a*x)]*(c - a*c*x)^(9/2))/(693*a^5*(1 - 1/(a*x))^(9/2)*x) + (2*(a - x^(-1))^5*Sqrt[1
+ 1/(a*x)]*x*(c - a*c*x)^(9/2))/(11*a^5*(1 - 1/(a*x))^(9/2))

Rule 6176

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

Rule 6181

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

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^{9/2} \, dx &=\frac{(c-a c x)^{9/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a x}\right )^{9/2} x^{9/2} \, dx}{\left (1-\frac{1}{a x}\right )^{9/2} x^{9/2}}\\ &=-\frac{\left (\left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^5}{x^{13/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\left (1-\frac{1}{a x}\right )^{9/2}}\\ &=\frac{2 \left (a-\frac{1}{x}\right )^5 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (20 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^4}{x^{11/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{11 a \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{2 \left (a-\frac{1}{x}\right )^5 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{\left (320 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^3}{x^{9/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{99 a^2 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{640 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (1280 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{\left (1-\frac{x}{a}\right )^2}{x^{7/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{512 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2} x^2}+\frac{640 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (512 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{-\frac{7}{a}+\frac{5 x}{2 a^2}}{x^{5/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{1024 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2} x^3}-\frac{512 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2} x^2}+\frac{640 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}+\frac{\left (11008 \left (\frac{1}{x}\right )^{9/2} (c-a c x)^{9/2}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{693 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}\\ &=-\frac{40 \left (a-\frac{1}{x}\right )^4 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}-\frac{22016 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac{1}{a x}\right )^{9/2} x^4}+\frac{1024 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{99 a^4 \left (1-\frac{1}{a x}\right )^{9/2} x^3}-\frac{512 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{231 a^3 \left (1-\frac{1}{a x}\right )^{9/2} x^2}+\frac{640 \left (a-\frac{1}{x}\right )^3 \sqrt{1+\frac{1}{a x}} (c-a c x)^{9/2}}{693 a^5 \left (1-\frac{1}{a x}\right )^{9/2} x}+\frac{2 \left (a-\frac{1}{x}\right )^5 \sqrt{1+\frac{1}{a x}} x (c-a c x)^{9/2}}{11 a^5 \left (1-\frac{1}{a x}\right )^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0514603, size = 86, normalized size = 0.44 \[ \frac{2 c^4 \sqrt{\frac{1}{a x}+1} \left (63 a^5 x^5-455 a^4 x^4+1510 a^3 x^3-3198 a^2 x^2+5419 a x-11531\right ) \sqrt{c-a c x}}{693 a \sqrt{1-\frac{1}{a x}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*c^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x]*(-11531 + 5419*a*x - 3198*a^2*x^2 + 1510*a^3*x^3 - 455*a^4*x^4 + 63*a
^5*x^5))/(693*a*Sqrt[1 - 1/(a*x)])

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Maple [A]  time = 0.045, size = 80, normalized size = 0.4 \begin{align*}{\frac{ \left ( 2\,ax+2 \right ) \left ( 63\,{x}^{5}{a}^{5}-455\,{x}^{4}{a}^{4}+1510\,{x}^{3}{a}^{3}-3198\,{a}^{2}{x}^{2}+5419\,ax-11531 \right ) }{693\,a \left ( ax-1 \right ) ^{5}} \left ( -acx+c \right ) ^{{\frac{9}{2}}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(a*x+1)*(63*a^5*x^5-455*a^4*x^4+1510*a^3*x^3-3198*a^2*x^2+5419*a*x-11531)*(-a*c*x+c)^(9/2)*((a*x-1)/(a*x
+1))^(1/2)/a/(a*x-1)^5

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Maxima [A]  time = 1.08039, size = 173, normalized size = 0.89 \begin{align*} \frac{2 \,{\left (63 \, a^{6} \sqrt{-c} c^{4} x^{6} - 392 \, a^{5} \sqrt{-c} c^{4} x^{5} + 1055 \, a^{4} \sqrt{-c} c^{4} x^{4} - 1688 \, a^{3} \sqrt{-c} c^{4} x^{3} + 2221 \, a^{2} \sqrt{-c} c^{4} x^{2} - 6112 \, a \sqrt{-c} c^{4} x - 11531 \, \sqrt{-c} c^{4}\right )}{\left (a x - 1\right )}}{693 \,{\left (a^{2} x - a\right )} \sqrt{a x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*a^6*sqrt(-c)*c^4*x^6 - 392*a^5*sqrt(-c)*c^4*x^5 + 1055*a^4*sqrt(-c)*c^4*x^4 - 1688*a^3*sqrt(-c)*c^4*
x^3 + 2221*a^2*sqrt(-c)*c^4*x^2 - 6112*a*sqrt(-c)*c^4*x - 11531*sqrt(-c)*c^4)*(a*x - 1)/((a^2*x - a)*sqrt(a*x
+ 1))

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Fricas [A]  time = 1.87519, size = 244, normalized size = 1.26 \begin{align*} \frac{2 \,{\left (63 \, a^{6} c^{4} x^{6} - 392 \, a^{5} c^{4} x^{5} + 1055 \, a^{4} c^{4} x^{4} - 1688 \, a^{3} c^{4} x^{3} + 2221 \, a^{2} c^{4} x^{2} - 6112 \, a c^{4} x - 11531 \, c^{4}\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{693 \,{\left (a^{2} x - a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*a^6*c^4*x^6 - 392*a^5*c^4*x^5 + 1055*a^4*c^4*x^4 - 1688*a^3*c^4*x^3 + 2221*a^2*c^4*x^2 - 6112*a*c^4*
x - 11531*c^4)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1))/(a^2*x - 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((-a*c*x+c)**(9/2)*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.28056, size = 181, normalized size = 0.93 \begin{align*} -\frac{2 \,{\left (63 \,{\left (a c x + c\right )}^{5} \sqrt{-a c x - c} - 770 \,{\left (a c x + c\right )}^{4} \sqrt{-a c x - c} c + 3960 \,{\left (a c x + c\right )}^{3} \sqrt{-a c x - c} c^{2} - 11088 \,{\left (a c x + c\right )}^{2} \sqrt{-a c x - c} c^{3} - 18480 \,{\left (-a c x - c\right )}^{\frac{3}{2}} c^{4} - 22176 \, \sqrt{-a c x - c} c^{5}\right )}{\left | c \right |}}{693 \, a c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/693*(63*(a*c*x + c)^5*sqrt(-a*c*x - c) - 770*(a*c*x + c)^4*sqrt(-a*c*x - c)*c + 3960*(a*c*x + c)^3*sqrt(-a*
c*x - c)*c^2 - 11088*(a*c*x + c)^2*sqrt(-a*c*x - c)*c^3 - 18480*(-a*c*x - c)^(3/2)*c^4 - 22176*sqrt(-a*c*x - c
)*c^5)*abs(c)/(a*c^2)