Integrand size = 12, antiderivative size = 163 \[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=\frac {x^3}{3 a}+\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)}{8 a^4}-\frac {5 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^2}{8 a^4}+\frac {3 \sqrt {\frac {1-a x}{1+a x}} (1+a x)^3}{4 a^4}-\frac {\sqrt {\frac {1-a x}{1+a x}} (1+a x)^4}{4 a^4}+\frac {\arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right )}{4 a^4} \] Output:
1/3*x^3/a+1/8*((-a*x+1)/(a*x+1))^(1/2)*(a*x+1)/a^4-5/8*((-a*x+1)/(a*x+1))^ (1/2)*(a*x+1)^2/a^4+3/4*((-a*x+1)/(a*x+1))^(1/2)*(a*x+1)^3/a^4-1/4*((-a*x+ 1)/(a*x+1))^(1/2)*(a*x+1)^4/a^4+1/4*arctan(((-a*x+1)/(a*x+1))^(1/2))/a^4
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.60 \[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=\frac {8 a^3 x^3+3 a \sqrt {\frac {1-a x}{1+a x}} \left (x+a x^2-2 a^2 x^3-2 a^3 x^4\right )-3 i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right )}{24 a^4} \] Input:
Integrate[x^3/E^ArcSech[a*x],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 = 1.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.26, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6891, 7268, 2335, 27, 2345, 27, 2345, 454, 216}
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.
| \(\displaystyle \int x^3 e^{-\text {sech}^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6891 |
| \(\displaystyle \int \frac {x^3}{\frac {\sqrt {\frac {1-a x}{a x+1}}}{a x}+\sqrt {\frac {1-a x}{a x+1}}+\frac {1}{a x}}dx\) |
| \(\Big \downarrow \) 7268 |
| \(\displaystyle -\frac {4 \int \frac {\sqrt {\frac {1-a x}{a x+1}} \left (1-\sqrt {\frac {1-a x}{a x+1}}\right )^4 \left (\sqrt {\frac {1-a x}{a x+1}}+1\right )^2}{\left (\frac {1-a x}{a x+1}+1\right )^5}d\sqrt {\frac {1-a x}{a x+1}}}{a^4}\) |
| \(\Big \downarrow \) 2335 |
| \(\displaystyle -\frac {4 \left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}-\frac {1}{8} \int \frac {8 \left (-\left (\frac {1-a x}{a x+1}\right )^{5/2}+2 \left (\frac {1-a x}{a x+1}\right )^{3/2}-\sqrt {\frac {1-a x}{a x+1}}-\frac {6 (1-a x)}{a x+1}+\frac {2 (1-a x)^2}{(a x+1)^2}+1\right )}{\left (\frac {1-a x}{a x+1}+1\right )^4}d\sqrt {\frac {1-a x}{a x+1}}\right )}{a^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}-\int \frac {-\left (\frac {1-a x}{a x+1}\right )^{5/2}+2 \left (\frac {1-a x}{a x+1}\right )^{3/2}-\sqrt {\frac {1-a x}{a x+1}}-\frac {6 (1-a x)}{a x+1}+\frac {2 (1-a x)^2}{(a x+1)^2}+1}{\left (\frac {1-a x}{a x+1}+1\right )^4}d\sqrt {\frac {1-a x}{a x+1}}\right )}{a^4}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {4 \left (\frac {1}{6} \int \frac {3 \left (2 \left (\frac {1-a x}{a x+1}\right )^{3/2}-6 \sqrt {\frac {1-a x}{a x+1}}-\frac {4 (1-a x)}{a x+1}+1\right )}{\left (\frac {1-a x}{a x+1}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}-\frac {9 \sqrt {\frac {1-a x}{a x+1}}+4}{6 \left (\frac {1-a x}{a x+1}+1\right )^3}+\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )}{a^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \int \frac {2 \left (\frac {1-a x}{a x+1}\right )^{3/2}-6 \sqrt {\frac {1-a x}{a x+1}}-\frac {4 (1-a x)}{a x+1}+1}{\left (\frac {1-a x}{a x+1}+1\right )^3}d\sqrt {\frac {1-a x}{a x+1}}-\frac {9 \sqrt {\frac {1-a x}{a x+1}}+4}{6 \left (\frac {1-a x}{a x+1}+1\right )^3}+\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )}{a^4}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {5 \sqrt {\frac {1-a x}{a x+1}}+8}{4 \left (\frac {1-a x}{a x+1}+1\right )^2}-\frac {1}{4} \int \frac {1-8 \sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^2}d\sqrt {\frac {1-a x}{a x+1}}\right )-\frac {9 \sqrt {\frac {1-a x}{a x+1}}+4}{6 \left (\frac {1-a x}{a x+1}+1\right )^3}+\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )}{a^4}\) |
| \(\Big \downarrow \) 454 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {1}{2} \int \frac {1}{\frac {1-a x}{a x+1}+1}d\sqrt {\frac {1-a x}{a x+1}}-\frac {\sqrt {\frac {1-a x}{a x+1}}+8}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )+\frac {5 \sqrt {\frac {1-a x}{a x+1}}+8}{4 \left (\frac {1-a x}{a x+1}+1\right )^2}\right )-\frac {9 \sqrt {\frac {1-a x}{a x+1}}+4}{6 \left (\frac {1-a x}{a x+1}+1\right )^3}+\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )}{a^4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {4 \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {1}{2} \arctan \left (\sqrt {\frac {1-a x}{a x+1}}\right )-\frac {\sqrt {\frac {1-a x}{a x+1}}+8}{2 \left (\frac {1-a x}{a x+1}+1\right )}\right )+\frac {5 \sqrt {\frac {1-a x}{a x+1}}+8}{4 \left (\frac {1-a x}{a x+1}+1\right )^2}\right )-\frac {9 \sqrt {\frac {1-a x}{a x+1}}+4}{6 \left (\frac {1-a x}{a x+1}+1\right )^3}+\frac {\sqrt {\frac {1-a x}{a x+1}}}{\left (\frac {1-a x}{a x+1}+1\right )^4}\right )}{a^4}\) |
Input:
Int[x^3/E^ArcSech[a*x],x]
Output:
(-4*(Sqrt[(1 - a*x)/(1 + a*x)]/(1 + (1 - a*x)/(1 + a*x))^4 - (4 + 9*Sqrt[( 1 - a*x)/(1 + a*x)])/(6*(1 + (1 - a*x)/(1 + a*x))^3) + ((8 + 5*Sqrt[(1 - a *x)/(1 + a*x)])/(4*(1 + (1 - a*x)/(1 + a*x))^2) + (-1/2*(8 + Sqrt[(1 - a*x )/(1 + a*x)])/(1 + (1 - a*x)/(1 + a*x)) - ArcTan[Sqrt[(1 - a*x)/(1 + a*x)] ]/2)/4)/2))/a^4
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*d - b*c*x)/(2*a*b*(p + 1)))*(a + b*x^2)^(p + 1), x] + Simp[c*((2*p + 3)/(2*a *(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d}, x] && L tQ[p, -1] && NeQ[p, -3/2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[E^(ArcSech[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[(1 - u)/(1 + u)] + (1/u)*Sqrt[(1 - u)/(1 + u)])^n, x] /; FreeQ[m, x] && Integer Q[n]
Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears [u, x]}, Simp[lst[[2]]*lst[[4]] Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls t[[2]])], x] /; !FalseQ[lst]]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.45 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.74
| method | result | size |
| default | \(a \left (\frac {x^{3}}{3 a^{2}}-\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (2 x^{3} \sqrt {-a^{2} x^{2}+1}\, a^{3} \operatorname {csgn}\left (a \right )-x \sqrt {-a^{2} x^{2}+1}\, \operatorname {csgn}\left (a \right ) a +\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right )\right ) \operatorname {csgn}\left (a \right )}{8 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )\) | \(120\) |
Input:
int(x^3/(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)),x,method=_RETURNVERBOSE)
Output:
a*(1/3*x^3/a^2-1/8/a^4*(-(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))
Time = 0.08 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.58 \[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=\frac {8 \, a^{3} x^{3} - 3 \, {\left (2 \, a^{4} x^{4} - a^{2} x^{2}\right )} \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 3 \, \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right )}{24 \, a^{4}} \] Input:
integrate(x^3/(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)),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
\[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=a \int \frac {x^{4}}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \] Input:
integrate(x**3/(1/a/x+(-1+1/a/x)**(1/2)*(1+1/a/x)**(1/2)),x)
Output:
a*Integral(x**4/(a*x*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)) + 1), x)
\[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=\int { \frac {x^{3}}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \] Input:
integrate(x^3/(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)),x, algorithm="maxim a")
Output:
integrate(x^3/(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)
\[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=\int { \frac {x^{3}}{\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}} \,d x } \] Input:
integrate(x^3/(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)),x, algorithm="giac" )
Output:
integrate(x^3/(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)
Time = 61.70 (sec) , antiderivative size = 795, normalized size of antiderivative = 4.88 \[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx =\text {Too large to display} \] Input:
int(x^3/((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x)),x)
Output:
(log((a*(1/(a*x) - 1)^(1/2)*1i + a*(1/(a*x) + 1)^(1/2) - 1/x)/(2*a - 2*a*( 1/(a*x) + 1)^(1/2) + 1/x))*3i)/(8*a^4) + (1i/(1024*a^4) - (((1/(a*x) - 1)^ (1/2) - 1i)^2*3i)/(128*a^4*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x) - 1)^ (1/2) - 1i)^4*53i)/(512*a^4*((1/(a*x) + 1)^(1/2) - 1)^4) + (((1/(a*x) - 1) ^(1/2) - 1i)^6*87i)/(256*a^4*((1/(a*x) + 1)^(1/2) - 1)^6) + (((1/(a*x) - 1 )^(1/2) - 1i)^8*657i)/(1024*a^4*((1/(a*x) + 1)^(1/2) - 1)^8) + (((1/(a*x) - 1)^(1/2) - 1i)^10*121i)/(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) + (1i/(32*a^4) + (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(16*a^4*((1/(a*x) + 1)^(1/2) - 1)^2) - (((1/(a*x) - 1)^(1/2) - 1i)^4*15i)/(32*a^4*((1/(a*x) + 1)^(1/2) - 1)^4))/(((1/(a*x) - 1)^(1/2) - 1i)^2/((1/(a*x) + 1)^(1/2) - 1)^2 + (2*((1/(a*x) - 1)^(1/2) - 1i)^4)/((1/(a*x) + 1)^(1/2) - 1)^4 + ((1/ (a*x) - 1)^(1/2) - 1i)^6/((1/(a*x) + 1)^(1/2) - 1)^6) - (log((a*(-(a - 1/x )/a)^(1/2)*2i - 2/x + 2*a*((a + 1/x)/a)^(1/2))/(2*a + 1/x - 2*a*((a + 1/x) /a)^(1/2)))*1i)/(2*a^4) + x^3/(3*a) + (((1/(a*x) - 1)^(1/2) - 1i)^2*1i)/(2 56*a^4*((1/(a*x) + 1)^(1/2) - 1)^2) + (((1/(a*x) - 1)^(1/2) - 1i)^4*1i)...
Time = 0.15 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.42 \[ \int e^{-\text {sech}^{-1}(a x)} x^3 \, dx=\frac {6 \mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )-6 \sqrt {a x +1}\, \sqrt {-a x +1}\, a^{3} x^{3}+3 \sqrt {a x +1}\, \sqrt {-a x +1}\, a x +8 a^{3} x^{3}-8}{24 a^{4}} \] Input:
int(x^3/(1/a/x+(-1+1/a/x)^(1/2)*(1+1/a/x)^(1/2)),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 - 8)/(24*a**4)