3.47 \(\int e^{\text {sech}^{-1}(a x^2)} x^5 \, dx\)

Optimal. Leaf size=58 \[ -\frac {\sqrt {1-a x^2}}{6 a^3 \sqrt {\frac {1}{a x^2+1}}}+\frac {x^4}{12 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )} \]

[Out]

1/12*x^4/a+1/6*(1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^6-1/6*(-a*x^2+1)^(1/2)/a^3/(1/(a*x^2+1))^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 71, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6335, 30, 259, 261} \[ -\frac {\sqrt {\frac {1}{a x^2+1}} \sqrt {a x^2+1} \sqrt {1-a^2 x^4}}{6 a^3}+\frac {x^4}{12 a}+\frac {1}{6} x^6 e^{\text {sech}^{-1}\left (a x^2\right )} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x^2]*x^5,x]

[Out]

x^4/(12*a) + (E^ArcSech[a*x^2]*x^6)/6 - (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(6*a^3)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 259

Int[((c_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[(c*x)
^m*(a1*a2 + b1*b2*x^(2*n))^p, x] /; FreeQ[{a1, b1, a2, b2, c, m, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] && (Intege
rQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 6335

Int[E^ArcSech[(a_.)*(x_)^(p_.)]*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*E^ArcSech[a*x^p])/(m + 1), x] + (Dist
[p/(a*(m + 1)), Int[x^(m - p), x], x] + Dist[(p*Sqrt[1 + a*x^p]*Sqrt[1/(1 + a*x^p)])/(a*(m + 1)), Int[x^(m - p
)/(Sqrt[1 + a*x^p]*Sqrt[1 - a*x^p]), x], x]) /; FreeQ[{a, m, p}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int e^{\text {sech}^{-1}\left (a x^2\right )} x^5 \, dx &=\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6+\frac {\int x^3 \, dx}{3 a}+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^3}{\sqrt {1-a x^2} \sqrt {1+a x^2}} \, dx}{3 a}\\ &=\frac {x^4}{12 a}+\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6+\frac {\left (\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2}\right ) \int \frac {x^3}{\sqrt {1-a^2 x^4}} \, dx}{3 a}\\ &=\frac {x^4}{12 a}+\frac {1}{6} e^{\text {sech}^{-1}\left (a x^2\right )} x^6-\frac {\sqrt {\frac {1}{1+a x^2}} \sqrt {1+a x^2} \sqrt {1-a^2 x^4}}{6 a^3}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 56, normalized size = 0.97 \[ \frac {\left (a x^2-1\right ) \sqrt {\frac {1-a x^2}{a x^2+1}} \left (a x^2+1\right )^2}{6 a^3}+\frac {x^4}{4 a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^2]*x^5,x]

[Out]

x^4/(4*a) + ((-1 + a*x^2)*Sqrt[(1 - a*x^2)/(1 + a*x^2)]*(1 + a*x^2)^2)/(6*a^3)

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fricas [A]  time = 0.54, size = 60, normalized size = 1.03 \[ \frac {3 \, a x^{4} + 2 \, {\left (a^{2} x^{6} - x^{2}\right )} \sqrt {\frac {a x^{2} + 1}{a x^{2}}} \sqrt {-\frac {a x^{2} - 1}{a x^{2}}}}{12 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x, algorithm="fricas")

[Out]

1/12*(3*a*x^4 + 2*(a^2*x^6 - x^2)*sqrt((a*x^2 + 1)/(a*x^2))*sqrt(-(a*x^2 - 1)/(a*x^2)))/a^2

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giac [B]  time = 0.17, size = 190, normalized size = 3.28 \[ \frac {{\left (\sqrt {a^{2} x^{2} + a} \sqrt {-a^{2} x^{2} + a} {\left ({\left (a^{2} x^{2} + a\right )} {\left (\frac {2 \, {\left (a^{2} x^{2} + a\right )}}{a^{4}} - \frac {7}{a^{3}}\right )} + \frac {9}{a^{2}}\right )} + \frac {6 \, \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right )}{a}\right )} a - \frac {3 \, {\left (2 \, a^{2} \arcsin \left (\frac {\sqrt {2} \sqrt {a^{2} x^{2} + a}}{2 \, \sqrt {a}}\right ) - \sqrt {a^{2} x^{2} + a} {\left (a^{2} x^{2} - 2 \, a\right )} \sqrt {-a^{2} x^{2} + a}\right )}}{a^{2}} + \frac {3 \, {\left ({\left (a^{2} x^{2} + a\right )}^{2} - 2 \, {\left (a^{2} x^{2} + a\right )} a\right )}}{a^{2}}}{12 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x, algorithm="giac")

[Out]

1/12*((sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2 + a)*((a^2*x^2 + a)*(2*(a^2*x^2 + a)/a^4 - 7/a^3) + 9/a^2) + 6*arcsin(1
/2*sqrt(2)*sqrt(a^2*x^2 + a)/sqrt(a))/a)*a - 3*(2*a^2*arcsin(1/2*sqrt(2)*sqrt(a^2*x^2 + a)/sqrt(a)) - sqrt(a^2
*x^2 + a)*(a^2*x^2 - 2*a)*sqrt(-a^2*x^2 + a))/a^2 + 3*((a^2*x^2 + a)^2 - 2*(a^2*x^2 + a)*a)/a^2)/a^3

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maple [A]  time = 0.05, size = 60, normalized size = 1.03 \[ \frac {\sqrt {-\frac {a \,x^{2}-1}{a \,x^{2}}}\, x^{2} \sqrt {\frac {a \,x^{2}+1}{a \,x^{2}}}\, \left (a^{2} x^{4}-1\right )}{6 a^{2}}+\frac {x^{4}}{4 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x)

[Out]

1/6*(-(a*x^2-1)/a/x^2)^(1/2)*x^2*((a*x^2+1)/a/x^2)^(1/2)*(a^2*x^4-1)/a^2+1/4*x^4/a

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maxima [A]  time = 0.40, size = 42, normalized size = 0.72 \[ \frac {x^{4}}{4 \, a} + \frac {{\left (a^{2} x^{4} - 1\right )} \sqrt {a x^{2} + 1} \sqrt {-a x^{2} + 1}}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x^2+(1/a/x^2-1)^(1/2)*(1/a/x^2+1)^(1/2))*x^5,x, algorithm="maxima")

[Out]

1/4*x^4/a + 1/6*(a^2*x^4 - 1)*sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)/a^3

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mupad [B]  time = 1.64, size = 57, normalized size = 0.98 \[ \sqrt {\frac {1}{a\,x^2}-1}\,\left (\frac {x^6\,\sqrt {\frac {1}{a\,x^2}+1}}{6}-\frac {x^2\,\sqrt {\frac {1}{a\,x^2}+1}}{6\,a^2}\right )+\frac {x^4}{4\,a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*((1/(a*x^2) - 1)^(1/2)*(1/(a*x^2) + 1)^(1/2) + 1/(a*x^2)),x)

[Out]

(1/(a*x^2) - 1)^(1/2)*((x^6*(1/(a*x^2) + 1)^(1/2))/6 - (x^2*(1/(a*x^2) + 1)^(1/2))/(6*a^2)) + x^4/(4*a)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1/a/x**2+(1/a/x**2-1)**(1/2)*(1/a/x**2+1)**(1/2))*x**5,x)

[Out]

Timed out

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