3.903 \(\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx\)

Optimal. Leaf size=264 \[ \frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.304403, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6197, 6193, 88} \[ \frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{x \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{2 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{x^2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 x^3 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 x^4 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 a x^5 \sqrt{1-\frac{1}{a^2 x^2}}}-\frac{4 a^4 \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5,x]

[Out]

Sqrt[c - c/(a^2*x^2)]/(5*a*Sqrt[1 - 1/(a^2*x^2)]*x^5) + (3*Sqrt[c - c/(a^2*x^2)])/(4*Sqrt[1 - 1/(a^2*x^2)]*x^4
) + (4*a*Sqrt[c - c/(a^2*x^2)])/(3*Sqrt[1 - 1/(a^2*x^2)]*x^3) + (2*a^2*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2
*x^2)]*x^2) + (4*a^3*Sqrt[c - c/(a^2*x^2)])/(Sqrt[1 - 1/(a^2*x^2)]*x) - (4*a^4*Sqrt[c - c/(a^2*x^2)]*Log[x])/S
qrt[1 - 1/(a^2*x^2)] + (4*a^4*Sqrt[c - c/(a^2*x^2)]*Log[1 - a*x])/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 \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^5} \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}}}{x^5} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{(1+a x)^2}{x^6 (-1+a x)} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \left (-\frac{1}{x^6}-\frac{3 a}{x^5}-\frac{4 a^2}{x^4}-\frac{4 a^3}{x^3}-\frac{4 a^4}{x^2}-\frac{4 a^5}{x}+\frac{4 a^6}{-1+a x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{5 a \sqrt{1-\frac{1}{a^2 x^2}} x^5}+\frac{3 \sqrt{c-\frac{c}{a^2 x^2}}}{4 \sqrt{1-\frac{1}{a^2 x^2}} x^4}+\frac{4 a \sqrt{c-\frac{c}{a^2 x^2}}}{3 \sqrt{1-\frac{1}{a^2 x^2}} x^3}+\frac{2 a^2 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}} x^2}+\frac{4 a^3 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}} x}-\frac{4 a^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{\sqrt{1-\frac{1}{a^2 x^2}}}+\frac{4 a^4 \sqrt{c-\frac{c}{a^2 x^2}} \log (1-a x)}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0721837, size = 90, normalized size = 0.34 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (\frac{240 a^4 x^4+120 a^3 x^3+80 a^2 x^2+45 a x+12}{60 x^5}-4 a^5 \log (x)+4 a^5 \log (1-a x)\right )}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcCoth[a*x])*Sqrt[c - c/(a^2*x^2)])/x^5,x]

[Out]

(Sqrt[c - c/(a^2*x^2)]*((12 + 45*a*x + 80*a^2*x^2 + 120*a^3*x^3 + 240*a^4*x^4)/(60*x^5) - 4*a^5*Log[x] + 4*a^5
*Log[1 - a*x]))/(a*Sqrt[1 - 1/(a^2*x^2)])

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Maple [A]  time = 0.237, size = 106, normalized size = 0.4 \begin{align*} -{\frac{ \left ( 240\,{a}^{5}\ln \left ( x \right ){x}^{5}-240\,\ln \left ( ax-1 \right ){x}^{5}{a}^{5}-240\,{x}^{4}{a}^{4}-120\,{x}^{3}{a}^{3}-80\,{a}^{2}{x}^{2}-45\,ax-12 \right ) \left ( ax-1 \right ) }{60\, \left ( ax+1 \right ) ^{2}{x}^{4}}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)/x^5,x)

[Out]

-1/60*(240*a^5*ln(x)*x^5-240*ln(a*x-1)*x^5*a^5-240*x^4*a^4-120*x^3*a^3-80*a^2*x^2-45*a*x-12)*(c*(a^2*x^2-1)/a^
2/x^2)^(1/2)*(a*x-1)/(a*x+1)^2/x^4/((a*x-1)/(a*x+1))^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{x^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(c - c/(a^2*x^2))/(x^5*((a*x - 1)/(a*x + 1))^(3/2)), x)

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Fricas [A]  time = 1.62598, size = 257, normalized size = 0.97 \begin{align*} \frac{240 \, a^{6} \sqrt{c} x^{5} \log \left (\frac{2 \, a^{3} c x^{2} - 2 \, a^{2} c x - \sqrt{a^{2} c}{\left (2 \, a x - 1\right )} \sqrt{c} + a c}{a x^{2} - x}\right ) +{\left (240 \, a^{4} x^{4} + 120 \, a^{3} x^{3} + 80 \, a^{2} x^{2} + 45 \, a x + 12\right )} \sqrt{a^{2} c}}{60 \, a^{2} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

1/60*(240*a^6*sqrt(c)*x^5*log((2*a^3*c*x^2 - 2*a^2*c*x - sqrt(a^2*c)*(2*a*x - 1)*sqrt(c) + a*c)/(a*x^2 - x)) +
(240*a^4*x^4 + 120*a^3*x^3 + 80*a^2*x^2 + 45*a*x + 12)*sqrt(a^2*c))/(a^2*x^5)

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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))**(3/2)*(c-c/a**2/x**2)**(1/2)/x**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}}}{x^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((a*x-1)/(a*x+1))^(3/2)*(c-c/a^2/x^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(c - c/(a^2*x^2))/(x^5*((a*x - 1)/(a*x + 1))^(3/2)), x)