Integrand size = 27, antiderivative size = 263 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{5 a \sqrt {1-\frac {1}{a^2 x^2}} x^5}+\frac {3 \sqrt {c-\frac {c}{a^2 x^2}}}{4 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {4 a \sqrt {c-\frac {c}{a^2 x^2}}}{3 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {2 a^2 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a^3 \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}} x}-\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^4 \sqrt {c-\frac {c}{a^2 x^2}} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}}} \] Output:
-1/5*(c-c/a^2/x^2)^(1/2)/a/(1-1/a^2/x^2)^(1/2)/x^5+3/4*(c-c/a^2/x^2)^(1/2) /(1-1/a^2/x^2)^(1/2)/x^4-4/3*a*(c-c/a^2/x^2)^(1/2)/(1-1/a^2/x^2)^(1/2)/x^3 +2*a^2*(c-c/a^2/x^2)^(1/2)/(1-1/a^2/x^2)^(1/2)/x^2-4*a^3*(c-c/a^2/x^2)^(1/ 2)/(1-1/a^2/x^2)^(1/2)/x-4*a^4*(c-c/a^2/x^2)^(1/2)*ln(x)/(1-1/a^2/x^2)^(1/ 2)+4*a^4*(c-c/a^2/x^2)^(1/2)*ln(a*x+1)/(1-1/a^2/x^2)^(1/2)
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.34 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {1}{5 a x^5}+\frac {3}{4 x^4}-\frac {4 a}{3 x^3}+\frac {2 a^2}{x^2}-\frac {4 a^3}{x}-4 a^4 \log (x)+4 a^4 \log (1+a x)\right )}{\sqrt {1-\frac {1}{a^2 x^2}}} \] Input:
Integrate[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^5),x]
Output:
(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)]
Time = 0.89 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.35, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6751, 6747, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {c-\frac {c}{a^2 x^2}} e^{-3 \coth ^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^5}dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2}{x^6 (a x+1)}dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \int \left (\frac {4 a^6}{a x+1}-\frac {4 a^5}{x}+\frac {4 a^4}{x^2}-\frac {4 a^3}{x^3}+\frac {4 a^2}{x^4}-\frac {3 a}{x^5}+\frac {1}{x^6}\right )dx}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c-\frac {c}{a^2 x^2}} \left (-4 a^5 \log (x)+4 a^5 \log (a x+1)-\frac {4 a^4}{x}+\frac {2 a^3}{x^2}-\frac {4 a^2}{3 x^3}+\frac {3 a}{4 x^4}-\frac {1}{5 x^5}\right )}{a \sqrt {1-\frac {1}{a^2 x^2}}}\) |
Input:
Int[Sqrt[c - c/(a^2*x^2)]/(E^(3*ArcCoth[a*x])*x^5),x]
Output:
(Sqrt[c - c/(a^2*x^2)]*(-1/5*1/x^5 + (3*a)/(4*x^4) - (4*a^2)/(3*x^3) + (2* a^3)/x^2 - (4*a^4)/x - 4*a^5*Log[x] + 4*a^5*Log[1 + a*x]))/(a*Sqrt[1 - 1/( a^2*x^2)])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb ol] :> Simp[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] && !Inte gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[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])
Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.40
method | result | size |
default | \(\frac {\left (240 \ln \left (a x +1\right ) x^{5} a^{5}-240 a^{5} \ln \left (x \right ) x^{5}-240 a^{4} x^{4}+120 a^{3} x^{3}-80 a^{2} x^{2}+45 a x -12\right ) \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{60 \left (a x -1\right )^{2} x^{4}}\) | \(106\) |
Input:
int((c-c/a^2/x^2)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x,method=_RETURNVERBOS E)
Output:
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
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {240 \, a^{6} \sqrt {c} x^{5} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x + \sqrt {a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {c} + a c}{a x^{2} + x}\right ) - {\left (240 \, a^{4} x^{4} - 120 \, a^{3} x^{3} + 80 \, a^{2} x^{2} - 45 \, a x + 12\right )} \sqrt {a^{2} c}}{60 \, a^{2} x^{5}} \] Input:
integrate((c-c/a^2/x^2)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="fr icas")
Output:
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)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\text {Timed out} \] Input:
integrate((c-c/a**2/x**2)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**5,x)
Output:
Timed out
\[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int { \frac {\sqrt {c - \frac {c}{a^{2} x^{2}}} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{x^{5}} \,d x } \] Input:
integrate((c-c/a^2/x^2)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="ma xima")
Output:
integrate(sqrt(c - c/(a^2*x^2))*((a*x - 1)/(a*x + 1))^(3/2)/x^5, x)
Time = 0.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.47 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {1}{60} \, {\left (240 \, a^{3} \log \left ({\left | a x + 1 \right |}\right ) \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - 240 \, a^{3} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - \frac {240 \, a^{4} x^{4} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - 120 \, a^{3} x^{3} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) + 80 \, a^{2} x^{2} \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) - 45 \, a x \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right ) + 12 \, \mathrm {sgn}\left (a x + 1\right ) \mathrm {sgn}\left (x\right )}{a^{2} x^{5}}\right )} \sqrt {c} {\left | a \right |} \] Input:
integrate((c-c/a^2/x^2)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="gi ac")
Output:
1/60*(240*a^3*log(abs(a*x + 1))*sgn(a*x + 1)*sgn(x) - 240*a^3*log(abs(x))* sgn(a*x + 1)*sgn(x) - (240*a^4*x^4*sgn(a*x + 1)*sgn(x) - 120*a^3*x^3*sgn(a *x + 1)*sgn(x) + 80*a^2*x^2*sgn(a*x + 1)*sgn(x) - 45*a*x*sgn(a*x + 1)*sgn( x) + 12*sgn(a*x + 1)*sgn(x))/(a^2*x^5))*sqrt(c)*abs(a)
Timed out. \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\int \frac {\sqrt {c-\frac {c}{a^2\,x^2}}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^5} \,d x \] Input:
int(((c - c/(a^2*x^2))^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^5,x)
Output:
int(((c - c/(a^2*x^2))^(1/2)*((a*x - 1)/(a*x + 1))^(3/2))/x^5, x)
Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.28 \[ \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-\frac {c}{a^2 x^2}}}{x^5} \, dx=\frac {\sqrt {c}\, \left (240 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-240 \,\mathrm {log}\left (a x \right ) a^{5} x^{5}+48 a^{5} x^{5}-240 a^{4} x^{4}+120 a^{3} x^{3}-80 a^{2} x^{2}+45 a x -12\right )}{60 a \,x^{5}} \] Input:
int((c-c/a^2/x^2)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x)
Output:
(sqrt(c)*(240*log(a*x + 1)*a**5*x**5 - 240*log(a*x)*a**5*x**5 + 48*a**5*x* *5 - 240*a**4*x**4 + 120*a**3*x**3 - 80*a**2*x**2 + 45*a*x - 12))/(60*a*x* *5)