Integrand size = 22, antiderivative size = 236 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}-\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{3 a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \] Output:
-1/4*c^2*(c-c/a^2/x^2)^(1/2)/a^5/(1-1/a^2/x^2)^(1/2)/x^4-1/3*c^2*(c-c/a^2/ x^2)^(1/2)/a^4/(1-1/a^2/x^2)^(1/2)/x^3+c^2*(c-c/a^2/x^2)^(1/2)/a^3/(1-1/a^ 2/x^2)^(1/2)/x^2+2*c^2*(c-c/a^2/x^2)^(1/2)/a^2/(1-1/a^2/x^2)^(1/2)/x+c^2*( c-c/a^2/x^2)^(1/2)*x/(1-1/a^2/x^2)^(1/2)+c^2*(c-c/a^2/x^2)^(1/2)*ln(x)/a/( 1-1/a^2/x^2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.31 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (-\frac {1}{4 a^5 x^4}-\frac {1}{3 a^4 x^3}+\frac {1}{a^3 x^2}+\frac {2}{a^2 x}+x+\frac {\log (x)}{a}\right )}{\left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \] Input:
Integrate[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2),x]
Output:
((c - c/(a^2*x^2))^(5/2)*(-1/4*1/(a^5*x^4) - 1/(3*a^4*x^3) + 1/(a^3*x^2) + 2/(a^2*x) + x + Log[x]/a))/(1 - 1/(a^2*x^2))^(5/2)
Time = 0.67 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.33, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, 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.
\(\displaystyle \int \left (c-\frac {c}{a^2 x^2}\right )^{5/2} e^{\coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6751 |
\(\displaystyle \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2}dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 6747 |
\(\displaystyle \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {(1-a x)^2 (a x+1)^3}{x^5}dx}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \int \left (a^5+\frac {a^4}{x}-\frac {2 a^3}{x^2}-\frac {2 a^2}{x^3}+\frac {a}{x^4}+\frac {1}{x^5}\right )dx}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} \left (a^5 x+a^4 \log (x)+\frac {2 a^3}{x}+\frac {a^2}{x^2}-\frac {a}{3 x^3}-\frac {1}{4 x^4}\right )}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}}\) |
Input:
Int[E^ArcCoth[a*x]*(c - c/(a^2*x^2))^(5/2),x]
Output:
(c^2*Sqrt[c - c/(a^2*x^2)]*(-1/4*1/x^4 - a/(3*x^3) + a^2/x^2 + (2*a^3)/x + a^5*x + a^4*Log[x]))/(a^5*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 = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.41
method | result | size |
default | \(\frac {\left (12 a^{5} x^{5}+12 \ln \left (x \right ) x^{4} a^{4}+24 a^{3} x^{3}+12 a^{2} x^{2}-4 a x -3\right ) {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {5}{2}} x}{12 \left (a x +1\right ) \left (a^{2} x^{2}-1\right )^{2} \sqrt {\frac {a x -1}{a x +1}}}\) | \(96\) |
Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/12*(12*a^5*x^5+12*ln(x)*x^4*a^4+24*a^3*x^3+12*a^2*x^2-4*a*x-3)*(c*(a^2*x ^2-1)/a^2/x^2)^(5/2)*x/(a*x+1)/(a^2*x^2-1)^2/((a*x-1)/(a*x+1))^(1/2)
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.31 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {{\left (12 \, a^{5} c^{2} x^{5} + 12 \, a^{4} c^{2} x^{4} \log \left (x\right ) + 24 \, a^{3} c^{2} x^{3} + 12 \, a^{2} c^{2} x^{2} - 4 \, a c^{2} x - 3 \, c^{2}\right )} \sqrt {a^{2} c}}{12 \, a^{6} x^{4}} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^(5/2),x, algorithm="fric as")
Output:
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)
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate(1/((a*x-1)/(a*x+1))**(1/2)*(c-c/a**2/x**2)**(5/2),x)
Output:
Timed out
\[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int { \frac {{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {5}{2}}}{\sqrt {\frac {a x - 1}{a x + 1}}} \,d x } \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^(5/2),x, algorithm="maxi ma")
Output:
integrate((c - c/(a^2*x^2))^(5/2)/sqrt((a*x - 1)/(a*x + 1)), x)
Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.45 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {1}{6} \, {\left (\frac {12 \, c^{2} x \mathrm {sgn}\left (x\right )}{a \mathrm {sgn}\left (a x + 1\right )} + \frac {12 \, c^{2} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{2} \mathrm {sgn}\left (a x + 1\right )} + \frac {24 \, a^{3} c^{2} x^{3} \mathrm {sgn}\left (x\right ) + 12 \, a^{2} c^{2} x^{2} \mathrm {sgn}\left (x\right ) - 4 \, a c^{2} x \mathrm {sgn}\left (x\right ) - 3 \, c^{2} \mathrm {sgn}\left (x\right )}{a^{6} x^{4} \mathrm {sgn}\left (a x + 1\right )}\right )} \sqrt {c} {\left | a \right |} \] Input:
integrate(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^(5/2),x, algorithm="giac ")
Output:
1/6*(12*c^2*x*sgn(x)/(a*sgn(a*x + 1)) + 12*c^2*log(abs(x))*sgn(x)/(a^2*sgn (a*x + 1)) + (24*a^3*c^2*x^3*sgn(x) + 12*a^2*c^2*x^2*sgn(x) - 4*a*c^2*x*sg n(x) - 3*c^2*sgn(x))/(a^6*x^4*sgn(a*x + 1)))*sqrt(c)*abs(a)
Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \] Input:
int((c - c/(a^2*x^2))^(5/2)/((a*x - 1)/(a*x + 1))^(1/2),x)
Output:
int((c - c/(a^2*x^2))^(5/2)/((a*x - 1)/(a*x + 1))^(1/2), x)
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.22 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx=\frac {\sqrt {c}\, c^{2} \left (12 \,\mathrm {log}\left (x \right ) a^{4} x^{4}+12 a^{5} x^{5}+24 a^{3} x^{3}+12 a^{2} x^{2}-4 a x -3\right )}{12 a^{5} x^{4}} \] Input:
int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a^2/x^2)^(5/2),x)
Output:
(sqrt(c)*c**2*(12*log(x)*a**4*x**4 + 12*a**5*x**5 + 24*a**3*x**3 + 12*a**2 *x**2 - 4*a*x - 3))/(12*a**5*x**4)