Integrand size = 12, antiderivative size = 96 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=-\frac {1}{3 a^2 x^6}-\frac {\sqrt {1+\frac {1}{a^2 x^2}}}{3 a x^5}-\frac {1}{4 x^4}-\frac {a \sqrt {1+\frac {1}{a^2 x^2}}}{12 x^3}+\frac {a^3 \sqrt {1+\frac {1}{a^2 x^2}}}{8 x}-\frac {1}{8} a^4 \text {csch}^{-1}(a x) \] Output:
-1/3/a^2/x^6-1/3*(1+1/a^2/x^2)^(1/2)/a/x^5-1/4/x^4-1/12*a*(1+1/a^2/x^2)^(1 /2)/x^3+1/8*a^3*(1+1/a^2/x^2)^(1/2)/x-1/8*a^4*arccsch(a*x)
Time = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=\frac {\frac {\left (4+3 a^2 x^2\right ) \left (-2-2 a \sqrt {1+\frac {1}{a^2 x^2}} x+a^3 \sqrt {1+\frac {1}{a^2 x^2}} x^3\right )}{x^6}-3 a^6 \text {arcsinh}\left (\frac {1}{a x}\right )}{24 a^2} \] Input:
Integrate[E^(2*ArcCsch[a*x])/x^5,x]
Output:
(((4 + 3*a^2*x^2)*(-2 - 2*a*Sqrt[1 + 1/(a^2*x^2)]*x + a^3*Sqrt[1 + 1/(a^2* x^2)]*x^3))/x^6 - 3*a^6*ArcSinh[1/(a*x)])/(24*a^2)
Time = 0.84 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6892, 7293, 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.
\(\displaystyle \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 6892 |
\(\displaystyle \int \frac {\left (\sqrt {\frac {1}{a^2 x^2}+1}+\frac {1}{a x}\right )^2}{x^5}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2}{a^2 x^7}+\frac {2 \sqrt {\frac {1}{a^2 x^2}+1}}{a x^6}+\frac {1}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{8} a^4 \text {csch}^{-1}(a x)-\frac {1}{3 a^2 x^6}-\frac {\sqrt {\frac {1}{a^2 x^2}+1}}{3 a x^5}-\frac {a \sqrt {\frac {1}{a^2 x^2}+1}}{12 x^3}+\frac {a^3 \sqrt {\frac {1}{a^2 x^2}+1}}{8 x}-\frac {1}{4 x^4}\) |
Input:
Int[E^(2*ArcCsch[a*x])/x^5,x]
Output:
-1/3*1/(a^2*x^6) - Sqrt[1 + 1/(a^2*x^2)]/(3*a*x^5) - 1/(4*x^4) - (a*Sqrt[1 + 1/(a^2*x^2)])/(12*x^3) + (a^3*Sqrt[1 + 1/(a^2*x^2)])/(8*x) - (a^4*ArcCs ch[a*x])/8
Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && IntegerQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(78)=156\).
Time = 0.14 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.31
method | result | size |
default | \(\frac {-\frac {1}{6 x^{6}}-\frac {a^{2}}{4 x^{4}}}{a^{2}}-\frac {a \sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (3 \sqrt {\frac {1}{a^{2}}}\, \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} a^{4} x^{4}-3 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{4} x^{6}+3 \ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) a^{2} x^{6}-6 \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}\, a^{2} x^{2}+8 \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}\right )}{24 x^{5} \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, \sqrt {\frac {1}{a^{2}}}}-\frac {1}{6 a^{2} x^{6}}\) | \(222\) |
Input:
int((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x,method=_RETURNVERBOSE)
Output:
1/a^2*(-1/6/x^6-1/4*a^2/x^4)-1/24*a*((a^2*x^2+1)/a^2/x^2)^(1/2)/x^5*(3*(1/ a^2)^(1/2)*((a^2*x^2+1)/a^2)^(3/2)*a^4*x^4-3*(1/a^2)^(1/2)*((a^2*x^2+1)/a^ 2)^(1/2)*a^4*x^6+3*ln(2*((1/a^2)^(1/2)*((a^2*x^2+1)/a^2)^(1/2)*a^2+1)/a^2/ x)*a^2*x^6-6*((a^2*x^2+1)/a^2)^(3/2)*(1/a^2)^(1/2)*a^2*x^2+8*((a^2*x^2+1)/ a^2)^(3/2)*(1/a^2)^(1/2))/((a^2*x^2+1)/a^2)^(1/2)/(1/a^2)^(1/2)-1/6/a^2/x^ 6
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.36 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=-\frac {3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - 3 \, a^{6} x^{6} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + 6 \, a^{2} x^{2} - {\left (3 \, a^{5} x^{5} - 2 \, a^{3} x^{3} - 8 \, a x\right )} \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 8}{24 \, a^{2} x^{6}} \] Input:
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="fricas")
Output:
-1/24*(3*a^6*x^6*log(a*x*sqrt((a^2*x^2 + 1)/(a^2*x^2)) - a*x + 1) - 3*a^6* x^6*log(a*x*sqrt((a^2*x^2 + 1)/(a^2*x^2)) - a*x - 1) + 6*a^2*x^2 - (3*a^5* x^5 - 2*a^3*x^3 - 8*a*x)*sqrt((a^2*x^2 + 1)/(a^2*x^2)) + 8)/(a^2*x^6)
Time = 1.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.22 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=\begin {cases} \frac {- \frac {a^{2}}{4 x^{4}} - 2 a \left (\begin {cases} \frac {a^{4} \log {\left (2 \sqrt {1 + \frac {1}{a^{2} x^{2}}} \sqrt {\frac {1}{a^{2}}} + \frac {2}{a^{2} x} \right )}}{16 \sqrt {\frac {1}{a^{2}}}} + \sqrt {1 + \frac {1}{a^{2} x^{2}}} \left (- \frac {a^{4}}{16 x} + \frac {a^{2}}{24 x^{3}} + \frac {1}{6 x^{5}}\right ) & \text {for}\: \frac {1}{a^{2}} \neq 0 \\\frac {1}{5 x^{5}} & \text {otherwise} \end {cases}\right ) - \frac {1}{3 x^{6}}}{a^{2}} & \text {for}\: a^{2} \neq 0 \\\text {NaN} & \text {otherwise} \end {cases} \] Input:
integrate((1/a/x+(1+1/a**2/x**2)**(1/2))**2/x**5,x)
Output:
Piecewise(((-a**2/(4*x**4) - 2*a*Piecewise((a**4*log(2*sqrt(1 + 1/(a**2*x* *2))*sqrt(a**(-2)) + 2/(a**2*x))/(16*sqrt(a**(-2))) + sqrt(1 + 1/(a**2*x** 2))*(-a**4/(16*x) + a**2/(24*x**3) + 1/(6*x**5)), Ne(a**(-2), 0)), (1/(5*x **5), True)) - 1/(3*x**6))/a**2, Ne(a**2, 0)), (nan, True))
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (78) = 156\).
Time = 0.03 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.88 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=-\frac {3 \, a^{5} \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - 3 \, a^{5} \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) - \frac {2 \, {\left (3 \, a^{10} x^{5} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 8 \, a^{8} x^{3} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, a^{6} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}\right )}}{a^{6} x^{6} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{3} - 3 \, a^{4} x^{4} {\left (\frac {1}{a^{2} x^{2}} + 1\right )}^{2} + 3 \, a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - 1}}{48 \, a} - \frac {1}{4 \, x^{4}} - \frac {1}{3 \, a^{2} x^{6}} \] Input:
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="maxima")
Output:
-1/48*(3*a^5*log(a*x*sqrt(1/(a^2*x^2) + 1) + 1) - 3*a^5*log(a*x*sqrt(1/(a^ 2*x^2) + 1) - 1) - 2*(3*a^10*x^5*(1/(a^2*x^2) + 1)^(5/2) - 8*a^8*x^3*(1/(a ^2*x^2) + 1)^(3/2) - 3*a^6*x*sqrt(1/(a^2*x^2) + 1))/(a^6*x^6*(1/(a^2*x^2) + 1)^3 - 3*a^4*x^4*(1/(a^2*x^2) + 1)^2 + 3*a^2*x^2*(1/(a^2*x^2) + 1) - 1)) /a - 1/4/x^4 - 1/3/(a^2*x^6)
Time = 0.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.26 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=-\frac {1}{48} \, {\left (3 \, {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} + 1\right ) \mathrm {sgn}\left (x\right ) - 3 \, {\left | a \right |} \log \left (\sqrt {a^{2} x^{2} + 1} - 1\right ) \mathrm {sgn}\left (x\right ) - \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 8 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 3 \, \sqrt {a^{2} x^{2} + 1} {\left | a \right |} \mathrm {sgn}\left (x\right ) - 6 \, {\left (a^{2} x^{2} + 1\right )} a - 2 \, a\right )}}{a^{6} x^{6}}\right )} a^{3} \] Input:
integrate((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x, algorithm="giac")
Output:
-1/48*(3*abs(a)*log(sqrt(a^2*x^2 + 1) + 1)*sgn(x) - 3*abs(a)*log(sqrt(a^2* x^2 + 1) - 1)*sgn(x) - 2*(3*(a^2*x^2 + 1)^(5/2)*abs(a)*sgn(x) - 8*(a^2*x^2 + 1)^(3/2)*abs(a)*sgn(x) - 3*sqrt(a^2*x^2 + 1)*abs(a)*sgn(x) - 6*(a^2*x^2 + 1)*a - 2*a)/(a^6*x^6))*a^3
Time = 24.76 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.93 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=\frac {a^3\,\sqrt {\frac {1}{a^2\,x^2}+1}}{8\,x}-\frac {1}{3\,a^2\,x^6}-\frac {a\,\sqrt {\frac {1}{a^2\,x^2}+1}}{12\,x^3}-\frac {1}{4\,x^4}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{3\,a\,x^5}-\frac {a^3\,\mathrm {asinh}\left (\frac {\sqrt {\frac {1}{a^2}}}{x}\right )}{8\,\sqrt {\frac {1}{a^2}}} \] Input:
int(((1/(a^2*x^2) + 1)^(1/2) + 1/(a*x))^2/x^5,x)
Output:
(a^3*(1/(a^2*x^2) + 1)^(1/2))/(8*x) - 1/(3*a^2*x^6) - (a*(1/(a^2*x^2) + 1) ^(1/2))/(12*x^3) - 1/(4*x^4) - (1/(a^2*x^2) + 1)^(1/2)/(3*a*x^5) - (a^3*as inh((1/a^2)^(1/2)/x))/(8*(1/a^2)^(1/2))
Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.19 \[ \int \frac {e^{2 \text {csch}^{-1}(a x)}}{x^5} \, dx=\frac {3 \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}-2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-8 \sqrt {a^{2} x^{2}+1}+3 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x -1\right ) a^{6} x^{6}-3 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}+1}+a x +1\right ) a^{6} x^{6}-6 a^{2} x^{2}-8}{24 a^{2} x^{6}} \] Input:
int((1/a/x+(1+1/a^2/x^2)^(1/2))^2/x^5,x)
Output:
(3*sqrt(a**2*x**2 + 1)*a**4*x**4 - 2*sqrt(a**2*x**2 + 1)*a**2*x**2 - 8*sqr t(a**2*x**2 + 1) + 3*log(sqrt(a**2*x**2 + 1) + a*x - 1)*a**6*x**6 - 3*log( sqrt(a**2*x**2 + 1) + a*x + 1)*a**6*x**6 - 6*a**2*x**2 - 8)/(24*a**2*x**6)