Integrand size = 12, antiderivative size = 115 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=-\frac {\sqrt {-1+\frac {1}{a x^2}} \sqrt {1+\frac {1}{a x^2}} x^4}{16 a^2}+\frac {x^6}{6 a}+\frac {1}{8} \sqrt {-1+\frac {1}{a x^2}} \sqrt {1+\frac {1}{a x^2}} x^8-\frac {\arctan \left (\sqrt {-1+\frac {1}{a x^2}} \sqrt {1+\frac {1}{a x^2}}\right )}{16 a^4} \] Output:
-1/16*(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2)*x^4/a^2+1/6*x^6/a+1/8*(-1+1/a/x ^2)^(1/2)*(1+1/a/x^2)^(1/2)*x^8-1/16*arctan((-1+1/a/x^2)^(1/2)*(1+1/a/x^2) ^(1/2))/a^4
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.97 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {8 a^3 x^6-3 a \sqrt {\frac {1-a x^2}{1+a x^2}} \left (x^2+a x^4-2 a^2 x^6-2 a^3 x^8\right )+3 i \log \left (-2 i a x^2+2 \sqrt {\frac {1-a x^2}{1+a x^2}} \left (1+a x^2\right )\right )}{48 a^4} \] Input:
Integrate[E^ArcSech[a*x^2]*x^7,x]
Output:
(8*a^3*x^6 - 3*a*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(x^2 + a*x^4 - 2*a^2*x^6 - 2*a^3*x^8) + (3*I)*Log[(-2*I)*a*x^2 + 2*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2)])/(48*a^4)
Time = 0.48 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6889, 15, 335, 807, 262, 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.
| \(\displaystyle \int x^7 e^{\text {sech}^{-1}\left (a x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6889 |
| \(\displaystyle \frac {\int x^5dx}{4 a}+\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^5}{\sqrt {1-a x^2} \sqrt {a x^2+1}}dx}{4 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^5}{\sqrt {1-a x^2} \sqrt {a x^2+1}}dx}{4 a}+\frac {x^6}{24 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
| \(\Big \downarrow \) 335 |
| \(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^5}{\sqrt {1-a^2 x^4}}dx}{4 a}+\frac {x^6}{24 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \int \frac {x^4}{\sqrt {1-a^2 x^4}}dx^2}{8 a}+\frac {x^6}{24 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (\frac {\int \frac {1}{\sqrt {1-a^2 x^4}}dx^2}{2 a^2}-\frac {x^2 \sqrt {1-a^2 x^4}}{2 a^2}\right )}{8 a}+\frac {x^6}{24 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \left (\frac {\arcsin \left (a x^2\right )}{2 a^3}-\frac {x^2 \sqrt {1-a^2 x^4}}{2 a^2}\right )}{8 a}+\frac {x^6}{24 a}+\frac {1}{8} x^8 e^{\text {sech}^{-1}\left (a x^2\right )}\) |
Input:
Int[E^ArcSech[a*x^2]*x^7,x]
Output:
x^6/(24*a) + (E^ArcSech[a*x^2]*x^8)/8 + (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a *x^2]*(-1/2*(x^2*Sqrt[1 - a^2*x^4])/a^2 + ArcSin[a*x^2]/(2*a^3)))/(8*a)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p _.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] ))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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]
Time = 0.20 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.19
| method | result | size |
| default | \(\frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (2 x^{6} \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, a^{4}-x^{2} \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, a^{2}+\arctan \left (\frac {x^{2}}{\sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}}\right )\right )}{16 \sqrt {-\frac {a^{2} x^{4}-1}{a^{2}}}\, a^{4}}+\frac {x^{6}}{6 a}\) | \(137\) |
Input:
int((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))*x^7,x,method=_RETURNVER BOSE)
Output:
1/16*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(2*x^6*(-(a^2*x^ 4-1)/a^2)^(1/2)*a^4-x^2*(-(a^2*x^4-1)/a^2)^(1/2)*a^2+arctan(x^2/(-(a^2*x^4 -1)/a^2)^(1/2)))/(-(a^2*x^4-1)/a^2)^(1/2)/a^4+1/6/a*x^6
Time = 0.09 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {8 \, a^{3} x^{6} + 3 \, {\left (2 \, a^{4} x^{8} - a^{2} x^{4}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 6 \, \arctan \left (\frac {a x^{2} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}} - 1}{a x^{2}}\right )}{48 \, a^{4}} \] Input:
integrate((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))*x^7,x, algorithm= "fricas")
Output:
1/48*(8*a^3*x^6 + 3*(2*a^4*x^8 - a^2*x^4)*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(- (a*x^2 - 1)/(a*x^2)) - 6*arctan((a*x^2*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a* x^2 - 1)/(a*x^2)) - 1)/(a*x^2)))/a^4
\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {\int x^{5}\, dx + \int a x^{7} \sqrt {-1 + \frac {1}{a x^{2}}} \sqrt {1 + \frac {1}{a x^{2}}}\, dx}{a} \] Input:
integrate((1/a/x**2+(-1+1/a/x**2)**(1/2)*(1+1/a/x**2)**(1/2))*x**7,x)
Output:
(Integral(x**5, x) + Integral(a*x**7*sqrt(-1 + 1/(a*x**2))*sqrt(1 + 1/(a*x **2)), x))/a
\[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\int { x^{7} {\left (\sqrt {\frac {1}{a x^{2}} + 1} \sqrt {\frac {1}{a x^{2}} - 1} + \frac {1}{a x^{2}}\right )} \,d x } \] Input:
integrate((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))*x^7,x, algorithm= "maxima")
Output:
1/6*x^6/a + integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)*x^5, x)/a
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (95) = 190\).
Time = 0.22 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.76 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {8 \, a^{2} x^{6} + {\left (\sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (a^{2} x^{2} + a\right )} {\left (2 \, {\left (a^{2} x^{2} + a\right )} {\left (\frac {3 \, {\left (a^{2} x^{2} + a\right )}}{a^{6}} - \frac {13}{a^{5}}\right )} + \frac {43}{a^{4}}\right )} - \frac {39}{a^{3}}\right )} - \frac {18 \, \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right )}{a^{2}}\right )} a + \frac {4 \, {\left (6 \, a^{3} \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) + \sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (2 \, a^{2} x^{2} - 5 \, a\right )} {\left (a^{2} x^{2} + a\right )} + 9 \, a^{2}\right )}\right )}}{a^{4}}}{48 \, a^{3}} \] Input:
integrate((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))*x^7,x, algorithm= "giac")
Output:
1/48*(8*a^2*x^6 + (sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2 + a)*((a^2*x^2 + a)*(2* (a^2*x^2 + a)*(3*(a^2*x^2 + a)/a^6 - 13/a^5) + 43/a^4) - 39/a^3) - 18*arcs in(1/2*sqrt(2)*sqrt(a^2*x^2 + a)/sqrt(a))/a^2)*a + 4*(6*a^3*arcsin(1/2*sqr t(2)*sqrt(a^2*x^2 + a)/sqrt(a)) + sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2 + a)*((2 *a^2*x^2 - 5*a)*(a^2*x^2 + a) + 9*a^2))/a^4)/a^3
Time = 52.25 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.53 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}}{16\,a^4}-\frac {\frac {1{}\mathrm {i}}{2048\,a^4}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{256\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4\,11{}\mathrm {i}}{1024\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^6\,7{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^6}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^8\,239{}\mathrm {i}}{2048\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^8}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^{10}\,1{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^{10}}}{\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^8}+\frac {4\,{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^{10}}+\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^{12}}}-\frac {\ln \left (\frac {\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x^2}+1}-1}\right )\,1{}\mathrm {i}}{16\,a^4}+\frac {x^6}{6\,a}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^2\,1{}\mathrm {i}}{512\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^2}-\frac {{\left (\sqrt {\frac {1}{a\,x^2}-1}-\mathrm {i}\right )}^4\,1{}\mathrm {i}}{2048\,a^4\,{\left (\sqrt {\frac {1}{a\,x^2}+1}-1\right )}^4} \] Input:
int(x^7*((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2)),x)
Output:
(log(((1/(a*x^2) - 1)^(1/2) - 1i)^2/((1/(a*x^2) + 1)^(1/2) - 1)^2 + 1)*1i) /(16*a^4) - (1i/(2048*a^4) + (((1/(a*x^2) - 1)^(1/2) - 1i)^2*1i)/(256*a^4* ((1/(a*x^2) + 1)^(1/2) - 1)^2) + (((1/(a*x^2) - 1)^(1/2) - 1i)^4*11i)/(102 4*a^4*((1/(a*x^2) + 1)^(1/2) - 1)^4) + (((1/(a*x^2) - 1)^(1/2) - 1i)^6*7i) /(512*a^4*((1/(a*x^2) + 1)^(1/2) - 1)^6) - (((1/(a*x^2) - 1)^(1/2) - 1i)^8 *239i)/(2048*a^4*((1/(a*x^2) + 1)^(1/2) - 1)^8) + (((1/(a*x^2) - 1)^(1/2) - 1i)^10*1i)/(512*a^4*((1/(a*x^2) + 1)^(1/2) - 1)^10))/(((1/(a*x^2) - 1)^( 1/2) - 1i)^4/((1/(a*x^2) + 1)^(1/2) - 1)^4 + (4*((1/(a*x^2) - 1)^(1/2) - 1 i)^6)/((1/(a*x^2) + 1)^(1/2) - 1)^6 + (6*((1/(a*x^2) - 1)^(1/2) - 1i)^8)/( (1/(a*x^2) + 1)^(1/2) - 1)^8 + (4*((1/(a*x^2) - 1)^(1/2) - 1i)^10)/((1/(a* x^2) + 1)^(1/2) - 1)^10 + ((1/(a*x^2) - 1)^(1/2) - 1i)^12/((1/(a*x^2) + 1) ^(1/2) - 1)^12) - (log(((1/(a*x^2) - 1)^(1/2) - 1i)/((1/(a*x^2) + 1)^(1/2) - 1))*1i)/(16*a^4) + x^6/(6*a) - (((1/(a*x^2) - 1)^(1/2) - 1i)^2*1i)/(512 *a^4*((1/(a*x^2) + 1)^(1/2) - 1)^2) - (((1/(a*x^2) - 1)^(1/2) - 1i)^4*1i)/ (2048*a^4*((1/(a*x^2) + 1)^(1/2) - 1)^4)
Time = 0.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.80 \[ \int e^{\text {sech}^{-1}\left (a x^2\right )} x^7 \, dx=\frac {-6 \mathit {atan} \left (\frac {\sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}}{a \,x^{2}-1}\right )+6 \sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}\, a^{3} x^{6}-3 \sqrt {a \,x^{2}+1}\, \sqrt {-a \,x^{2}+1}\, a \,x^{2}+8 a^{3} x^{6}}{48 a^{4}} \] Input:
int((1/a/x^2+(-1+1/a/x^2)^(1/2)*(1+1/a/x^2)^(1/2))*x^7,x)
Output:
( - 6*atan((sqrt(a*x**2 + 1)*sqrt( - a*x**2 + 1))/(a*x**2 - 1)) + 6*sqrt(a *x**2 + 1)*sqrt( - a*x**2 + 1)*a**3*x**6 - 3*sqrt(a*x**2 + 1)*sqrt( - a*x* *2 + 1)*a*x**2 + 8*a**3*x**6)/(48*a**4)