Integrand size = 10, antiderivative size = 115 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=-\frac {\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^2}{8 a^2}+\frac {x^3}{3 a}+\frac {1}{4} \sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x^4-\frac {\arctan \left (\sqrt {-1+\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{8 a^4} \] Output:
-1/8*(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)*x^2/a^2+1/3*x^3/a+1/4*(-1+1/a/x)^(1/ 2)*(1+1/a/x)^(1/2)*x^4-1/8*arctan((-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a^4
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.84 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {8 a^3 x^3-3 a \sqrt {\frac {1-a x}{1+a x}} \left (x+a x^2-2 a^2 x^3-2 a^3 x^4\right )+3 i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{24 a^4} \] Input:
Integrate[E^ArcSech[a*x]*x^3,x]
Output:
(8*a^3*x^3 - 3*a*Sqrt[(1 - a*x)/(1 + a*x)]*(x + a*x^2 - 2*a^2*x^3 - 2*a^3* x^4) + (3*I)*Log[(-2*I)*a*x + 2*Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x)])/(24* a^4)
Time = 0.41 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6889, 15, 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.
\(\displaystyle \int x^3 e^{\text {sech}^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6889 |
\(\displaystyle \frac {\int x^2dx}{4 a}+\frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {x^2}{\sqrt {1-a x} \sqrt {a x+1}}dx}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \int \frac {x^2}{\sqrt {1-a x} \sqrt {a x+1}}dx}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}+\frac {x^3}{12 a}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (-\frac {\int -\frac {1}{\sqrt {1-a x} \sqrt {a x+1}}dx}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}+\frac {x^3}{12 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {\int \frac {1}{\sqrt {1-a x} \sqrt {a x+1}}dx}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}+\frac {x^3}{12 a}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}+\frac {x^3}{12 a}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {\sqrt {\frac {1}{a x+1}} \sqrt {a x+1} \left (\frac {\arcsin (a x)}{2 a^3}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{2 a^2}\right )}{4 a}+\frac {1}{4} x^4 e^{\text {sech}^{-1}(a x)}+\frac {x^3}{12 a}\) |
Input:
Int[E^ArcSech[a*x]*x^3,x]
Output:
x^3/(12*a) + (E^ArcSech[a*x]*x^4)/4 + (Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]* (-1/2*(x*Sqrt[1 - a*x]*Sqrt[1 + a*x])/a^2 + ArcSin[a*x]/(2*a^3)))/(4*a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[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 = 118, normalized size of antiderivative = 1.03
method | result | size |
default | \(\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (2 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{3} \operatorname {csgn}\left (a \right )-x \sqrt {-a^{2} x^{2}+1}\, \operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{8 \sqrt {-a^{2} x^{2}+1}\, a^{3}}+\frac {x^{3}}{3 a}\) | \(118\) |
Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^3,x,method=_RETURNVERBOSE)
Output:
1/8*(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(2*x^3*(-a^2*x^2+1)^(1/2)*a ^3*csgn(a)-x*(-a^2*x^2+1)^(1/2)*csgn(a)*a+arctan(csgn(a)*a*x/(-a^2*x^2+1)^ (1/2)))*csgn(a)/(-a^2*x^2+1)^(1/2)/a^3+1/3*x^3/a
Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {8 \, a^{3} x^{3} + 3 \, {\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 3 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{24 \, a^{4}} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^3,x, algorithm="frica s")
Output:
1/24*(8*a^3*x^3 + 3*(2*a^4*x^4 - a^2*x^2)*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 3*arctan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))))/a^4
\[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {\int x^{2}\, dx + \int a x^{3} \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \] Input:
integrate((1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2))*x**3,x)
Output:
(Integral(x**2, x) + Integral(a*x**3*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a
\[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\int { x^{3} {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )} \,d x } \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^3,x, algorithm="maxim a")
Output:
1/3*x^3/a + integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x^2, x)/a
Exception generated. \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^3,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,[0,2,2,2,0,0]%%%}+%%%{1,[0,1,0,1,1,1]%%%} / %%%{1,[0,0,2 ,3,0,0]%%
Time = 50.46 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.53 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{8\,a^4}-\frac {\frac {1{}\mathrm {i}}{1024\,a^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{128\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,11{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6\,7{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8\,239{}\mathrm {i}}{1024\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}\,1{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^{12}}}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}}{8\,a^4}+\frac {x^3}{3\,a}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{1024\,a^4\,{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4} \] Input:
int(x^3*((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)/(8* a^4) - (1i/(1024*a^4) + (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(128*a^4*((1/(a* x) + 1)^(1/2) - 1)^2) + (((1/(a*x) - 1)^(1/2) - 1i)^4*11i)/(512*a^4*((1/(a *x) + 1)^(1/2) - 1)^4) + (((1/(a*x) - 1)^(1/2) - 1i)^6*7i)/(256*a^4*((1/(a *x) + 1)^(1/2) - 1)^6) - (((1/(a*x) - 1)^(1/2) - 1i)^8*239i)/(1024*a^4*((1 /(a*x) + 1)^(1/2) - 1)^8) + (((1/(a*x) - 1)^(1/2) - 1i)^10*1i)/(256*a^4*(( 1/(a*x) + 1)^(1/2) - 1)^10))/(((1/(a*x) - 1)^(1/2) - 1i)^4/((1/(a*x) + 1)^ (1/2) - 1)^4 + (4*((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 + (4*((1/ (a*x) - 1)^(1/2) - 1i)^10)/((1/(a*x) + 1)^(1/2) - 1)^10 + ((1/(a*x) - 1)^( 1/2) - 1i)^12/((1/(a*x) + 1)^(1/2) - 1)^12) - (log(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1))*1i)/(8*a^4) + x^3/(3*a) - (((1/(a*x) - 1)^( 1/2) - 1i)^2*1i)/(256*a^4*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x) - 1)^( 1/2) - 1i)^4*1i)/(1024*a^4*((1/(a*x) + 1)^(1/2) - 1)^4)
Time = 0.15 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.58 \[ \int e^{\text {sech}^{-1}(a x)} x^3 \, dx=\frac {-6 \mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )+6 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{3} x^{3}-3 \sqrt {a x +1}\, \sqrt {-a x +1}\, a x +8 a^{3} x^{3}}{24 a^{4}} \] Input:
int((1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2))*x^3,x)
Output:
( - 6*asin(sqrt( - a*x + 1)/sqrt(2)) + 6*sqrt(a*x + 1)*sqrt( - a*x + 1)*a* *3*x**3 - 3*sqrt(a*x + 1)*sqrt( - a*x + 1)*a*x + 8*a**3*x**3)/(24*a**4)