3.187 \(\int \frac{x^7 (a+b \text{sech}^{-1}(c x))}{\sqrt{1-c^4 x^4}} \, dx\)

Optimal. Leaf size=316 \[ \frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{30 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{18 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{3 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}} \]

[Out]

-(b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(3*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (b*Sqrt[1 - c^2*x^2]
*(1 + c^2*x^2)^(3/2))/(18*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/
2))/(30*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^8) + ((1 -
c^4*x^4)^(3/2)*(a + b*ArcSech[c*x]))/(6*c^8) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(3*c^9*Sqrt[-
1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)

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Rubi [A]  time = 1.38294, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {266, 43, 6309, 12, 6742, 848, 50, 63, 208, 783} \[ \frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}-\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{5/2}}{30 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \left (c^2 x^2+1\right )^{3/2}}{18 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}-\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{c^2 x^2+1}\right )}{3 c^9 x \sqrt{\frac{1}{c x}-1} \sqrt{\frac{1}{c x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]

[Out]

-(b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(3*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) + (b*Sqrt[1 - c^2*x^2]
*(1 + c^2*x^2)^(3/2))/(18*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (b*Sqrt[1 - c^2*x^2]*(1 + c^2*x^2)^(5/
2))/(30*c^9*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x) - (Sqrt[1 - c^4*x^4]*(a + b*ArcSech[c*x]))/(2*c^8) + ((1 -
c^4*x^4)^(3/2)*(a + b*ArcSech[c*x]))/(6*c^8) + (b*Sqrt[1 - c^2*x^2]*ArcTanh[Sqrt[1 + c^2*x^2]])/(3*c^9*Sqrt[-
1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]*x)

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6309

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcSech[c*x],
v, x] + Dist[(b*Sqrt[1 - c^2*x^2])/(c*x*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]), Int[SimplifyIntegrand[v/(x*Sqrt
[1 - c^2*x^2]), x], x], x] /; InverseFunctionFreeQ[v, x]] /; FreeQ[{a, b, c}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 783

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m +
p)*(f + g*x)*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p
] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rubi steps

\begin{align*} \int \frac{x^7 \left (a+b \text{sech}^{-1}(c x)\right )}{\sqrt{1-c^4 x^4}} \, dx &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\left (-2-c^4 x^4\right ) \sqrt{1-c^4 x^4}}{6 c^8 x \sqrt{1-c^2 x^2}} \, dx}{c \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int \frac{\left (-2-c^4 x^4\right ) \sqrt{1-c^4 x^4}}{x \sqrt{1-c^2 x^2}} \, dx}{6 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2} \left (2+c^4 x^2\right )}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{2 \sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}}+\frac{c^4 x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}}\right ) \, dx,x,x^2\right )}{12 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-c^4 x^2}}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{6 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{1-c^4 x^2}}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )}{12 c^5 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+c^2 x}}{x} \, dx,x,x^2\right )}{6 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int x \sqrt{1+c^2 x} \, dx,x,x^2\right )}{12 c^5 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{3 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+c^2 x}} \, dx,x,x^2\right )}{6 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{1+c^2 x}}{c^2}+\frac{\left (1+c^2 x\right )^{3/2}}{c^2}\right ) \, dx,x,x^2\right )}{12 c^5 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{3 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{18 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{30 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}-\frac{\left (b \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{c^2}+\frac{x^2}{c^2}} \, dx,x,\sqrt{1+c^2 x^2}\right )}{3 c^{11} \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ &=-\frac{b \sqrt{1-c^2 x^2} \sqrt{1+c^2 x^2}}{3 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}+\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{18 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{b \sqrt{1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{30 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}-\frac{\sqrt{1-c^4 x^4} \left (a+b \text{sech}^{-1}(c x)\right )}{2 c^8}+\frac{\left (1-c^4 x^4\right )^{3/2} \left (a+b \text{sech}^{-1}(c x)\right )}{6 c^8}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left (\sqrt{1+c^2 x^2}\right )}{3 c^9 \sqrt{-1+\frac{1}{c x}} \sqrt{1+\frac{1}{c x}} x}\\ \end{align*}

Mathematica [A]  time = 0.359832, size = 178, normalized size = 0.56 \[ \frac{-15 a \sqrt{1-c^4 x^4} \left (c^4 x^4+2\right )+\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4} \left (3 c^4 x^4+c^2 x^2+28\right )}{c x-1}-30 b \log \left (-\sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^4 x^4}-c x+1\right )-15 b \sqrt{1-c^4 x^4} \left (c^4 x^4+2\right ) \text{sech}^{-1}(c x)+30 b \log (x (1-c x))}{90 c^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(a + b*ArcSech[c*x]))/Sqrt[1 - c^4*x^4],x]

[Out]

(-15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]*(28 + c^2*x^2 + 3*c^4*
x^4))/(-1 + c*x) - 15*b*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4)*ArcSech[c*x] + 30*b*Log[x*(1 - c*x)] - 30*b*Log[1 - c*
x - Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[1 - c^4*x^4]])/(90*c^8)

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Maple [F]  time = 2.082, size = 0, normalized size = 0. \begin{align*} \int{{x}^{7} \left ( a+b{\rm arcsech} \left (cx\right ) \right ){\frac{1}{\sqrt{-{c}^{4}{x}^{4}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x)

[Out]

int(x^7*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a{\left (\frac{{\left (-c^{4} x^{4} + 1\right )}^{\frac{3}{2}}}{c^{8}} - \frac{3 \, \sqrt{-c^{4} x^{4} + 1}}{c^{8}}\right )} + \frac{1}{6} \, b{\left (\frac{{\left (c^{8} x^{8} + c^{4} x^{4} - 2\right )} \log \left (\sqrt{c x + 1} \sqrt{-c x + 1} + 1\right )}{\sqrt{c^{2} x^{2} + 1} \sqrt{c x + 1} \sqrt{-c x + 1} c^{8}} - 6 \, \int \frac{6 \, c^{6} x^{13} \log \left (c\right ) + 12 \, c^{6} x^{13} \log \left (\sqrt{x}\right ) +{\left (12 \, c^{6} x^{13} \log \left (\sqrt{x}\right ) +{\left (c^{6} x^{6}{\left (6 \, \log \left (c\right ) + 1\right )} + c^{4} x^{4} + 2 \, c^{2} x^{2} + 2\right )} x^{7}\right )} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}}{6 \,{\left (c^{6} x^{6} e^{\left (\log \left (c x + 1\right ) + \log \left (-c x + 1\right )\right )} + c^{6} x^{6} e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )}\right )} \sqrt{c^{2} x^{2} + 1}}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")

[Out]

1/6*a*((-c^4*x^4 + 1)^(3/2)/c^8 - 3*sqrt(-c^4*x^4 + 1)/c^8) + 1/6*b*((c^8*x^8 + c^4*x^4 - 2)*log(sqrt(c*x + 1)
*sqrt(-c*x + 1) + 1)/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(-c*x + 1)*c^8) - 6*integrate(1/6*(6*c^6*x^13*log(c)
+ 12*c^6*x^13*log(sqrt(x)) + (12*c^6*x^13*log(sqrt(x)) + (c^6*x^6*(6*log(c) + 1) + c^4*x^4 + 2*c^2*x^2 + 2)*x
^7)*e^(1/2*log(c*x + 1) + 1/2*log(-c*x + 1)))/((c^6*x^6*e^(log(c*x + 1) + log(-c*x + 1)) + c^6*x^6*e^(1/2*log(
c*x + 1) + 1/2*log(-c*x + 1)))*sqrt(c^2*x^2 + 1)), x))

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Fricas [A]  time = 1.9024, size = 705, normalized size = 2.23 \begin{align*} -\frac{15 \,{\left (b c^{6} x^{6} - b c^{4} x^{4} + 2 \, b c^{2} x^{2} - 2 \, b\right )} \sqrt{-c^{4} x^{4} + 1} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (3 \, b c^{5} x^{5} + b c^{3} x^{3} + 28 \, b c x\right )} \sqrt{-c^{4} x^{4} + 1} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 15 \,{\left (b c^{2} x^{2} - b\right )} \log \left (\frac{c^{2} x^{2} + \sqrt{-c^{4} x^{4} + 1} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) - 15 \,{\left (b c^{2} x^{2} - b\right )} \log \left (-\frac{c^{2} x^{2} - \sqrt{-c^{4} x^{4} + 1} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c^{2} x^{2} - 1}\right ) + 15 \,{\left (a c^{6} x^{6} - a c^{4} x^{4} + 2 \, a c^{2} x^{2} - 2 \, a\right )} \sqrt{-c^{4} x^{4} + 1}}{90 \,{\left (c^{10} x^{2} - c^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")

[Out]

-1/90*(15*(b*c^6*x^6 - b*c^4*x^4 + 2*b*c^2*x^2 - 2*b)*sqrt(-c^4*x^4 + 1)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2
)) + 1)/(c*x)) - (3*b*c^5*x^5 + b*c^3*x^3 + 28*b*c*x)*sqrt(-c^4*x^4 + 1)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 15*(
b*c^2*x^2 - b)*log((c^2*x^2 + sqrt(-c^4*x^4 + 1)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c^2*x^2 - 1)) - 15*(
b*c^2*x^2 - b)*log(-(c^2*x^2 - sqrt(-c^4*x^4 + 1)*c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c^2*x^2 - 1)) + 15*
(a*c^6*x^6 - a*c^4*x^4 + 2*a*c^2*x^2 - 2*a)*sqrt(-c^4*x^4 + 1))/(c^10*x^2 - c^8)

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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(x**7*(a+b*asech(c*x))/(-c**4*x**4+1)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x^{7}}{\sqrt{-c^{4} x^{4} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(a+b*arcsech(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)*x^7/sqrt(-c^4*x^4 + 1), x)