3.643 \(\int e^{-\coth ^{-1}(a x)} (c-a^2 c x^2)^{7/2} \, dx\)

Optimal. Leaf size=187 \[ \frac{(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{6 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{8 (1-a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]

[Out]

(-8*(1 - a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + (2*(1 - a*x)^6*(c - a^2*c*x^2)^(7
/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 - a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*
x^7) + ((1 - a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)

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Rubi [A]  time = 0.20121, antiderivative size = 187, 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 = {6192, 6193, 43} \[ \frac{(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{6 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{8 (1-a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(7/2)/E^ArcCoth[a*x],x]

[Out]

(-8*(1 - a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + (2*(1 - a*x)^6*(c - a^2*c*x^2)^(7
/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 - a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*
x^7) + ((1 - a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)

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 )^{7/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int (-1+a x)^4 (1+a x)^3 \, dx}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int \left (8 (-1+a x)^4+12 (-1+a x)^5+6 (-1+a x)^6+(-1+a x)^7\right ) \, dx}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac{8 (1-a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}+\frac{2 (1-a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}-\frac{6 (1-a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}+\frac{(1-a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ \end{align*}

Mathematica [A]  time = 0.0483062, size = 71, normalized size = 0.38 \[ -\frac{c^3 (a x-1)^5 \left (35 a^3 x^3+135 a^2 x^2+185 a x+93\right ) \sqrt{c-a^2 c x^2}}{280 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - a^2*c*x^2)^(7/2)/E^ArcCoth[a*x],x]

[Out]

-(c^3*(-1 + a*x)^5*Sqrt[c - a^2*c*x^2]*(93 + 185*a*x + 135*a^2*x^2 + 35*a^3*x^3))/(280*a^2*Sqrt[1 - 1/(a^2*x^2
)]*x)

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Maple [A]  time = 0.043, size = 100, normalized size = 0.5 \begin{align*}{\frac{x \left ( 35\,{a}^{7}{x}^{7}-40\,{x}^{6}{a}^{6}-140\,{x}^{5}{a}^{5}+168\,{x}^{4}{a}^{4}+210\,{x}^{3}{a}^{3}-280\,{a}^{2}{x}^{2}-140\,ax+280 \right ) }{280\, \left ( ax+1 \right ) ^{3} \left ( ax-1 \right ) ^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x)

[Out]

1/280*x*(35*a^7*x^7-40*a^6*x^6-140*a^5*x^5+168*a^4*x^4+210*a^3*x^3-280*a^2*x^2-140*a*x+280)*(-a^2*c*x^2+c)^(7/
2)*((a*x-1)/(a*x+1))^(1/2)/(a*x+1)^3/(a*x-1)^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{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((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(7/2)*sqrt((a*x - 1)/(a*x + 1)), x)

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Fricas [A]  time = 1.98201, size = 212, normalized size = 1.13 \begin{align*} -\frac{{\left (35 \, a^{7} c^{3} x^{8} - 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} + 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} - 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} + 280 \, c^{3} x\right )} \sqrt{-a^{2} c}}{280 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

-1/280*(35*a^7*c^3*x^8 - 40*a^6*c^3*x^7 - 140*a^5*c^3*x^6 + 168*a^4*c^3*x^5 + 210*a^3*c^3*x^4 - 280*a^2*c^3*x^
3 - 140*a*c^3*x^2 + 280*c^3*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((-a**2*c*x**2+c)**(7/2)*((a*x-1)/(a*x+1))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{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((-a^2*c*x^2+c)^(7/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

integrate((-a^2*c*x^2 + c)^(7/2)*sqrt((a*x - 1)/(a*x + 1)), x)