\(\int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx\) [927]

Optimal result

Integrand size = 27, antiderivative size = 263 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6332, 6328, 90} \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a x^5 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^4 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (a x+1)}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{x \sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

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

[Out]

-1/5*Sqrt[c - c/(a^2*x^2)]/(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]
)/Sqrt[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 90

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]))

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*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.34 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5 a x^5}+\frac {3}{4 x^4}-\frac {4 a}{3 x^3}+\frac {2 a^2}{x^2}-\frac {4 a^3}{x}-4 a^4 \log (x)+4 a^4 \log (1+a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.40

method	result	size
default	\(\frac {\left (240 \ln \left (a x +1\right ) x^{5} a^{5}-240 a^{5} \ln \left (x \right ) x^{5}-240 a^{4} x^{4}+120 a^{3} x^{3}-80 a^{2} x^{2}+45 a x -12\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{60 \left (a x -1\right )^{2} x^{4}}\)	\(106\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {240 \, a^{6} \sqrt {c} x^{5} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt {a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {c} + a c}{a x^{2} + x}\right ) - {\left (240 \, a^{4} x^{4} - 120 \, a^{3} x^{3} + 80 \, a^{2} x^{2} - 45 \, a x + 12\right )} \sqrt {a^{2} c}}{60 \, a^{2} x^{5}} \]

[In]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^5} \,d x \]

[In]

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

[Out]

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