3.2.87 \(\int \frac {x^7 (a+b \text {sech}^{-1}(c x))}{\sqrt {1-c^4 x^4}} \, dx\) [187]

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

[Out]

1/6*(-c^4*x^4+1)^(3/2)*(a+b*arcsech(c*x))/c^8+1/18*b*(c^2*x^2+1)^(3/2)*(-c^2*x^2+1)^(1/2)/c^9/x/(-1+1/c/x)^(1/
2)/(1+1/c/x)^(1/2)-1/30*b*(c^2*x^2+1)^(5/2)*(-c^2*x^2+1)^(1/2)/c^9/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+1/3*b*ar
ctanh((c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/c^9/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-1/3*b*(-c^2*x^2+1)^(1/2)*(c
^2*x^2+1)^(1/2)/c^9/x/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-1/2*(a+b*arcsech(c*x))*(-c^4*x^4+1)^(1/2)/c^8

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Rubi [A]
time = 0.93, antiderivative size = 316, normalized size of antiderivative = 1.00, 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 = {272, 45, 6444, 12, 6874, 862, 52, 65, 214, 797} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(b*Sqrt[1 - c^2*x^2]*Sqrt[1 + c^2*x^2])/(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 12

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

Rule 45

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 52

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 65

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 214

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

Rule 272

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 797

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/e)*x)^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]))

Rule 862

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/e)*x)^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 6444

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 6874

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

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.23, size = 178, normalized size = 0.56 \begin {gather*} \frac {-15 a \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right )+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4} \left (28+c^2 x^2+3 c^4 x^4\right )}{-1+c x}-15 b \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right ) \text {sech}^{-1}(c x)+30 b \log (x (1-c x))-30 b \log \left (1-c x-\sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^4 x^4}\right )}{90 c^8} \end {gather*}

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 = 1.98, size = 0, normalized size = 0.00 \[\int \frac {x^{7} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}{\sqrt {-c^{4} x^{4}+1}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.40, size = 336, normalized size = 1.06 \begin {gather*} -\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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{7} \left (a + b \operatorname {asech}{\left (c x \right )}\right )}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \end {gather*}

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]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^7\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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