3.856 \(\int e^{-\coth ^{-1}(a x)} (c-\frac{c}{a^2 x^2})^{5/2} \, dx\)

Optimal. Leaf size=238 \[ \frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

[Out]

(c^2*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^2*Sqrt[c - c/(a^2*x^2)])/(3*a^4*Sqrt[1 - 1/
(a^2*x^2)]*x^3) - (c^2*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (2*c^2*Sqrt[c - c/(a^2*x^2)])/
(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^2*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (c^2*Sqrt[c - c/(a^2*x^2
)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])

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Rubi [A]  time = 0.145783, antiderivative size = 238, 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 = {6197, 6193, 88} \[ \frac{c^2 x \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^2 x \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{a^3 x^2 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{3 a^4 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{c^2 \sqrt{c-\frac{c}{a^2 x^2}}}{4 a^5 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{c^2 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(c - c/(a^2*x^2))^(5/2)/E^ArcCoth[a*x],x]

[Out]

(c^2*Sqrt[c - c/(a^2*x^2)])/(4*a^5*Sqrt[1 - 1/(a^2*x^2)]*x^4) - (c^2*Sqrt[c - c/(a^2*x^2)])/(3*a^4*Sqrt[1 - 1/
(a^2*x^2)]*x^3) - (c^2*Sqrt[c - c/(a^2*x^2)])/(a^3*Sqrt[1 - 1/(a^2*x^2)]*x^2) + (2*c^2*Sqrt[c - c/(a^2*x^2)])/
(a^2*Sqrt[1 - 1/(a^2*x^2)]*x) + (c^2*Sqrt[c - c/(a^2*x^2)]*x)/Sqrt[1 - 1/(a^2*x^2)] - (c^2*Sqrt[c - c/(a^2*x^2
)]*Log[x])/(a*Sqrt[1 - 1/(a^2*x^2)])

Rule 6197

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

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

Mathematica [A]  time = 0.0555824, size = 77, normalized size = 0.32 \[ \frac{\left (c-\frac{c}{a^2 x^2}\right )^{5/2} \left (-\frac{a^2}{x^2}+a^5 x+\frac{2 a^3}{x}-a^4 \log (x)-\frac{a}{3 x^3}+\frac{1}{4 x^4}\right )}{a^5 \left (1-\frac{1}{a^2 x^2}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - c/(a^2*x^2))^(5/2)/E^ArcCoth[a*x],x]

[Out]

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

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Maple [A]  time = 0.221, size = 96, normalized size = 0.4 \begin{align*} -{\frac{x \left ( -12\,{x}^{5}{a}^{5}+12\,{a}^{4}\ln \left ( x \right ){x}^{4}-24\,{x}^{3}{a}^{3}+12\,{a}^{2}{x}^{2}+4\,ax-3 \right ) }{ \left ( 12\,ax-12 \right ) \left ({a}^{2}{x}^{2}-1 \right ) ^{2}} \left ({\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}} \right ) ^{{\frac{5}{2}}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*((a*x-1)/(a*x+1))^(1/2)*(c*(a^2*x^2-1)/a^2/x^2)^(5/2)*x*(-12*x^5*a^5+12*a^4*ln(x)*x^4-24*x^3*a^3+12*a^2*
x^2+4*a*x-3)/(a*x-1)/(a^2*x^2-1)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{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((c-c/a^2/x^2)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="maxima")

[Out]

integrate((c - c/(a^2*x^2))^(5/2)*sqrt((a*x - 1)/(a*x + 1)), x)

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Fricas [A]  time = 1.83784, size = 166, normalized size = 0.7 \begin{align*} \frac{{\left (12 \, a^{5} c^{2} x^{5} - 12 \, a^{4} c^{2} x^{4} \log \left (x\right ) + 24 \, a^{3} c^{2} x^{3} - 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt{a^{2} c}}{12 \, a^{6} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a^2/x^2)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="fricas")

[Out]

1/12*(12*a^5*c^2*x^5 - 12*a^4*c^2*x^4*log(x) + 24*a^3*c^2*x^3 - 12*a^2*c^2*x^2 - 4*a*c^2*x + 3*c^2)*sqrt(a^2*c
)/(a^6*x^4)

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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)**(5/2)*((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 (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{5}{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((c-c/a^2/x^2)^(5/2)*((a*x-1)/(a*x+1))^(1/2),x, algorithm="giac")

[Out]

integrate((c - c/(a^2*x^2))^(5/2)*sqrt((a*x - 1)/(a*x + 1)), x)