3.9.73 \(\int e^{-3 \coth ^{-1}(a x)} (c-\frac {c}{a^2 x^2})^{5/2} \, dx\) [873]

Optimal. Leaf size=235 \[ -\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 c^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 235, normalized size of antiderivative = 1.00, 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 = {6332, 6328, 76} \begin {gather*} \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 {3 c^2 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \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 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \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}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rule 6328

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symbol] :> Dist[c^p/a^(2*p), Int[(u/x^(
2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p + n/2), 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 6332

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])

Rubi steps

\begin {align*} \int e^{-3 \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^{-3 \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)^4 (1+a x)}{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 {3 a}{x^4}+\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}-\frac {3 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}}}{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 {3 c^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 81, normalized size = 0.34 \begin {gather*} \frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {5 a^4}{4}-\frac {1}{4 x^4}+\frac {a}{x^3}-\frac {a^2}{x^2}-\frac {2 a^3}{x}+a^5 x-3 a^4 \log (x)\right )}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.04, size = 96, normalized size = 0.41

method	result	size
default	\(-\frac {\left (-4 a^{5} x^{5}+12 \ln \left (x \right ) a^{4} x^{4}+8 a^{3} x^{3}+4 a^{2} x^{2}-4 a x +1\right ) x \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{4 \left (a x -1\right )^{3} \left (a^{2} x^{2}-1\right )}\)	\(96\)

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))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/4*(-4*a^5*x^5+12*ln(x)*a^4*x^4+8*a^3*x^3+4*a^2*x^2-4*a*x+1)*x*(c*(a^2*x^2-1)/a^2/x^2)^(5/2)*((a*x-1)/(a*x+1
))^(3/2)/(a*x-1)^3/(a^2*x^2-1)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.34, size = 74, normalized size = 0.31 \begin {gather*} \frac {{\left (4 \, a^{5} c^{2} x^{5} - 12 \, a^{4} c^{2} x^{4} \log \left (x\right ) - 8 \, a^{3} c^{2} x^{3} - 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x - c^{2}\right )} \sqrt {a^{2} c}}{4 \, a^{6} x^{4}} \end {gather*}

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))^(3/2),x, algorithm="fricas")

[Out]

1/4*(4*a^5*c^2*x^5 - 12*a^4*c^2*x^4*log(x) - 8*a^3*c^2*x^3 - 4*a^2*c^2*x^2 + 4*a*c^2*x - c^2)*sqrt(a^2*c)/(a^6
*x^4)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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))**(3/2),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2} \,d x \end {gather*}

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))^(3/2),x)

[Out]

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

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