3.45 \(\int e^{\text{sech}^{-1}(a x^2)} x^7 \, dx\)

Optimal. Leaf size=111 \[ -\frac{x^2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right )}{16 a^4}+\frac{x^6}{24 a}+\frac{1}{8} x^8 e^{\text{sech}^{-1}\left (a x^2\right )} \]

[Out]

x^6/(24*a) + (E^ArcSech[a*x^2]*x^8)/8 - (x^2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(16*a^3
) + (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcSin[a*x^2])/(16*a^4)

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Rubi [A]  time = 0.0616554, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 30, 259, 275, 321, 216} \[ -\frac{x^2 \sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\sqrt{\frac{1}{a x^2+1}} \sqrt{a x^2+1} \sin ^{-1}\left (a x^2\right )}{16 a^4}+\frac{x^6}{24 a}+\frac{1}{8} x^8 e^{\text{sech}^{-1}\left (a x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSech[a*x^2]*x^7,x]

[Out]

x^6/(24*a) + (E^ArcSech[a*x^2]*x^8)/8 - (x^2*Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*Sqrt[1 - a^2*x^4])/(16*a^3
) + (Sqrt[(1 + a*x^2)^(-1)]*Sqrt[1 + a*x^2]*ArcSin[a*x^2])/(16*a^4)

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]

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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[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]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

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]

Rubi steps

\begin{align*} \int e^{\text{sech}^{-1}\left (a x^2\right )} x^7 \, dx &=\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8+\frac{\int x^5 \, dx}{4 a}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^5}{\sqrt{1-a x^2} \sqrt{1+a x^2}} \, dx}{4 a}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \int \frac{x^5}{\sqrt{1-a^2 x^4}} \, dx}{4 a}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx,x,x^2\right )}{8 a}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8-\frac{x^2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\left (\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx,x,x^2\right )}{16 a^3}\\ &=\frac{x^6}{24 a}+\frac{1}{8} e^{\text{sech}^{-1}\left (a x^2\right )} x^8-\frac{x^2 \sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sqrt{1-a^2 x^4}}{16 a^3}+\frac{\sqrt{\frac{1}{1+a x^2}} \sqrt{1+a x^2} \sin ^{-1}\left (a x^2\right )}{16 a^4}\\ \end{align*}

Mathematica [C]  time = 0.181418, size = 111, normalized size = 1. \[ \frac{8 a^3 x^6-3 a \sqrt{\frac{1-a x^2}{a x^2+1}} \left (-2 a^3 x^8-2 a^2 x^6+a x^4+x^2\right )+3 i \log \left (2 \sqrt{\frac{1-a x^2}{a x^2+1}} \left (a x^2+1\right )-2 i a x^2\right )}{48 a^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x^2]*x^7,x]

[Out]

(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)

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Maple [A]  time = 0.322, size = 137, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{16\,{a}^{4}}\sqrt{-{\frac{a{x}^{2}-1}{a{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 ({{x}^{2}{\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{{a}^{2}{x}^{4}-1}{{a}^{2}}}}}}}+{\frac{{x}^{6}}{6\,a}} \end{align*}

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^7,x)

[Out]

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*x^6/a

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x^{6}}{6 \, a} + \frac{-\frac{{\left (-a^{2} x^{4} + 1\right )}^{\frac{3}{2}} x^{2}}{8 \, a^{2}} + \frac{\sqrt{-a^{2} x^{4} + 1} x^{2}}{16 \, a^{2}} + \frac{\arcsin \left (\frac{a^{2} x^{2}}{\sqrt{a^{2}}}\right )}{16 \, \sqrt{a^{2}} a^{2}}}{a} \end{align*}

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^7,x, algorithm="maxima")

[Out]

1/6*x^6/a + integrate(sqrt(a*x^2 + 1)*sqrt(-a*x^2 + 1)*x^5, x)/a

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Fricas [A]  time = 2.12503, size = 251, normalized size = 2.26 \begin{align*} \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}} \end{align*}

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^7,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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**7,x)

[Out]

Timed out

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Giac [B]  time = 1.19495, size = 360, normalized size = 3.24 \begin{align*} \frac{8 \, a^{4} x^{6} - \frac{30 \, a^{4} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right ) - 12 \,{\left (\sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a} a^{2} x^{2} - 2 \, a^{2} \arcsin \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + a}}{2 \, \sqrt{a}}\right )\right )} a^{2} - \sqrt{a^{2} x^{2} + a} \sqrt{-a^{2} x^{2} + a}{\left (15 \, a^{3} +{\left (a^{2} x^{2} - a\right )}{\left (2 \,{\left (3 \, a^{2} x^{2} + 14 \, a\right )}{\left (a^{2} x^{2} - a\right )} + 59 \, a^{2}\right )}\right )} - 8 \,{\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 )} + 3 \, a^{2}\right )}\right )} a}{a^{3}}}{48 \, a^{5}} \end{align*}

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^7,x, algorithm="giac")

[Out]

1/48*(8*a^4*x^6 - (30*a^4*arcsin(1/2*sqrt(2)*sqrt(-a^2*x^2 + a)/sqrt(a)) - 12*(sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2
+ a)*a^2*x^2 - 2*a^2*arcsin(1/2*sqrt(2)*sqrt(-a^2*x^2 + a)/sqrt(a)))*a^2 - sqrt(a^2*x^2 + a)*sqrt(-a^2*x^2 +
a)*(15*a^3 + (a^2*x^2 - a)*(2*(3*a^2*x^2 + 14*a)*(a^2*x^2 - a) + 59*a^2)) - 8*(6*a^3*arcsin(1/2*sqrt(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) + 3*a^2))*a)/a^3
)/a^5