Integrand size = 10, antiderivative size = 151 \[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=-\frac {\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^2}{16 a^4}-\frac {\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4}{24 a^2}+\frac {x^5}{5 a}+\frac {1}{6} \sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^6-\frac {\arctan \left (\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a^6} \] Output:
-1/16*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)*x^2/a^4-1/24*(-1+1/a/x)^(1/2)*(1+1/ a/x)^(1/2)*x^4/a^2+1/5*x^5/a+1/6*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)*x^6-1/16 *arctan((-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a^6
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.75 \[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=\frac {48 a^5 x^5-5 a x \sqrt {\frac {1-a x}{1+a x}} \left (3+3 a x+2 a^2 x^2+2 a^3 x^3-8 a^4 x^4-8 a^5 x^5\right )+15 i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{240 a^6} \] Input:
Integrate[E^ArcSech[a*x]*x^5,x]
Output:
(48*a^5*x^5 - 5*a*x*Sqrt[(1 - a*x)/(1 + a*x)]*(3 + 3*a*x + 2*a^2*x^2 + 2*a ^3*x^3 - 8*a^4*x^4 - 8*a^5*x^5) + (15*I)*Log[(-2*I)*a*x + 2*Sqrt[(1 - a*x) /(1 + a*x)]*(1 + a*x)])/(240*a^6)
Time = 0.45 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6889, 15, 111, 27, 101, 25, 39, 223}
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.
\(\displaystyle \int x^5 e^{\text {sech}^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle \frac {\int x^4dx}{6 a}+\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {x^4}{\sqrt {1-a x} \sqrt {a x+1}}dx}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {x^4}{\sqrt {1-a x} \sqrt {a x+1}}dx}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {\int -\frac {3 x^2}{\sqrt {1-a x} \sqrt {a x+1}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}\right )}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {3 \int \frac {x^2}{\sqrt {1-a x} \sqrt {a x+1}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}\right )}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {3 \left (-\frac {\int -\frac {1}{\sqrt {1-a x} \sqrt {a x+1}}dx}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}\right )}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-a x} \sqrt {a x+1}}dx}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}\right )}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {3 \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}\right )}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {3 \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}\right )}{6 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}(a x)}+\frac {x^5}{30 a}\) |
Input:
Int[E^ArcSech[a*x]*x^5,x]
Output:
x^5/(30*a) + (E^ArcSech[a*x]*x^6)/6 + (Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]* (-1/4*(x^3*Sqrt[1 - a*x]*Sqrt[1 + a*x])/a^2 + (3*(-1/2*(x*Sqrt[1 - a*x]*Sq rt[1 + a*x])/a^2 + ArcSin[a*x]/(2*a^3)))/(4*a^2)))/(6*a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(E^ ArcSech[a*x^p]/(m + 1)), x] + (Simp[p/(a*(m + 1)) Int[x^(m - p), x], x] + Simp[p*(Sqrt[1 + a*x^p]/(a*(m + 1)))*Sqrt[1/(1 + a*x^p)] Int[x^(m - p)/( Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (8 \,\operatorname {csgn}\left (a \right ) a^{5} x^{5} \sqrt {-a^{2} x^{2}+1}-2 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{3} \operatorname {csgn}\left (a \right )-3 x \sqrt {-a^{2} x^{2}+1}\, \operatorname {csgn}\left (a \right ) a +3 \arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{48 \sqrt {-a^{2} x^{2}+1}\, a^{5}}+\frac {x^{5}}{5 a}\) | \(142\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^5,x,method=_RETURNVERBOSE)
Output:
1/48*(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(8*csgn(a)*a^5*x^5*(-a^2*x ^2+1)^(1/2)-2*x^3*(-a^2*x^2+1)^(1/2)*a^3*csgn(a)-3*x*(-a^2*x^2+1)^(1/2)*cs gn(a)*a+3*arctan(csgn(a)*a*x/(-a^2*x^2+1)^(1/2)))*csgn(a)/(-a^2*x^2+1)^(1/ 2)/a^5+1/5/a*x^5
Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.68 \[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=\frac {48 \, a^{5} x^{5} + 5 \, {\left (8 \, a^{6} x^{6} - 2 \, a^{4} x^{4} - 3 \, a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 15 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{240 \, a^{6}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^5,x, algorithm="frica s")
Output:
1/240*(48*a^5*x^5 + 5*(8*a^6*x^6 - 2*a^4*x^4 - 3*a^2*x^2)*sqrt((a*x + 1)/( a*x))*sqrt(-(a*x - 1)/(a*x)) - 15*arctan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))))/a^6
\[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=\frac {\int x^{4}\, dx + \int a x^{5} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))*x**5,x)
Output:
(Integral(x**4, x) + Integral(a*x**5*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a
\[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=\int { x^{5} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^5,x, algorithm="maxim a")
Output:
1/5*x^5/a + integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^4, x)/a
Exception generated. \[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^5,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{1,[0,4,2,2,0,0]%%%}+%%%{1,[0,3,0,1,1,1]%%%} / %%%{1,[0,0,2 ,3,0,0]%%
Time = 49.77 (sec) , antiderivative size = 697, normalized size of antiderivative = 4.62 \[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx =\text {Too large to display} \] Input:
int(x^5*((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x)),x)
Output:
(log(((1/(a*x) - 1)^(1/2) - 1i)^2/((1/(a*x) + 1)^(1/2) - 1)^2 + 1)*1i)/(16 *a^6) - (log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))*1i)/(16 *a^6) + (1i/(24576*a^6) - (((1/(a*x) - 1)^(1/2) - 1i)^4*17i)/(4096*a^6*((1 /(a*x) + 1)^(1/2) - 1)^4) - (((1/(a*x) - 1)^(1/2) - 1i)^6*139i)/(6144*a^6* ((1/(a*x) + 1)^(1/2) - 1)^6) - (((1/(a*x) - 1)^(1/2) - 1i)^8*23i)/(1024*a^ 6*((1/(a*x) + 1)^(1/2) - 1)^8) + (((1/(a*x) - 1)^(1/2) - 1i)^10*185i)/(102 4*a^6*((1/(a*x) + 1)^(1/2) - 1)^10) + (((1/(a*x) - 1)^(1/2) - 1i)^12*901i) /(4096*a^6*((1/(a*x) + 1)^(1/2) - 1)^12) + (((1/(a*x) - 1)^(1/2) - 1i)^14* 471i)/(2048*a^6*((1/(a*x) + 1)^(1/2) - 1)^14) + (((1/(a*x) - 1)^(1/2) - 1i )^16*229i)/(8192*a^6*((1/(a*x) + 1)^(1/2) - 1)^16))/(((1/(a*x) - 1)^(1/2) - 1i)^6/((1/(a*x) + 1)^(1/2) - 1)^6 + (6*((1/(a*x) - 1)^(1/2) - 1i)^8)/((1 /(a*x) + 1)^(1/2) - 1)^8 + (15*((1/(a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + (20*((1/(a*x) - 1)^(1/2) - 1i)^12)/((1/(a*x) + 1)^(1/2) - 1)^12 + (15*((1/(a*x) - 1)^(1/2) - 1i)^14)/((1/(a*x) + 1)^(1/2) - 1)^14 + (6*((1/(a*x) - 1)^(1/2) - 1i)^16)/((1/(a*x) + 1)^(1/2) - 1)^16 + ((1/(a *x) - 1)^(1/2) - 1i)^18/((1/(a*x) + 1)^(1/2) - 1)^18) + x^5/(5*a) - (((1/( a*x) - 1)^(1/2) - 1i)^2*27i)/(8192*a^6*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1 /(a*x) - 1)^(1/2) - 1i)^4*1i)/(4096*a^6*((1/(a*x) + 1)^(1/2) - 1)^4) + ((( 1/(a*x) - 1)^(1/2) - 1i)^6*1i)/(24576*a^6*((1/(a*x) + 1)^(1/2) - 1)^6)
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.58 \[ \int e^{\text {sech}^{-1}(a x)} x^5 \, dx=\frac {-30 \mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )+40 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{5} x^{5}-10 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{3} x^{3}-15 \sqrt {a x +1}\, \sqrt {-a x +1}\, a x +48 a^{5} x^{5}}{240 a^{6}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^5,x)
Output:
( - 30*asin(sqrt( - a*x + 1)/sqrt(2)) + 40*sqrt(a*x + 1)*sqrt( - a*x + 1)* a**5*x**5 - 10*sqrt(a*x + 1)*sqrt( - a*x + 1)*a**3*x**3 - 15*sqrt(a*x + 1) *sqrt( - a*x + 1)*a*x + 48*a**5*x**5)/(240*a**6)