\(\int \frac {(d+e x^2) (a+b \sec ^{-1}(c x))}{x^8} \, dx\) [75]

Optimal result

Integrand size = 19, antiderivative size = 197 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\frac {8 b c^5 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 \sqrt {c^2 x^2}}+\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}+\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5} \]

[Out]

-1/7*d*(a+b*arcsec(c*x))/x^7-1/5*e*(a+b*arcsec(c*x))/x^5+8/3675*b*c^5*(30*c^2*d+49*e)*(c^2*x^2-1)^(1/2)/(c^2*x
^2)^(1/2)+1/49*b*c*d*(c^2*x^2-1)^(1/2)/x^6/(c^2*x^2)^(1/2)+1/1225*b*c*(30*c^2*d+49*e)*(c^2*x^2-1)^(1/2)/x^4/(c
^2*x^2)^(1/2)+4/3675*b*c^3*(30*c^2*d+49*e)*(c^2*x^2-1)^(1/2)/x^2/(c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 5346, 12, 464, 277, 270} \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}+\frac {b c \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{1225 x^4 \sqrt {c^2 x^2}}+\frac {b c d \sqrt {c^2 x^2-1}}{49 x^6 \sqrt {c^2 x^2}}+\frac {8 b c^5 \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 \sqrt {c^2 x^2}}+\frac {4 b c^3 \sqrt {c^2 x^2-1} \left (30 c^2 d+49 e\right )}{3675 x^2 \sqrt {c^2 x^2}} \]

[In]

Int[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^8,x]

[Out]

(8*b*c^5*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(3675*Sqrt[c^2*x^2]) + (b*c*d*Sqrt[-1 + c^2*x^2])/(49*x^6*Sqrt[
c^2*x^2]) + (b*c*(30*c^2*d + 49*e)*Sqrt[-1 + c^2*x^2])/(1225*x^4*Sqrt[c^2*x^2]) + (4*b*c^3*(30*c^2*d + 49*e)*S
qrt[-1 + c^2*x^2])/(3675*x^2*Sqrt[c^2*x^2]) - (d*(a + b*ArcSec[c*x]))/(7*x^7) - (e*(a + b*ArcSec[c*x]))/(5*x^5
)

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 270

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

Rule 277

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

Rule 464

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

Rule 5346

Int[((a_.) + ArcSec[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSec[c*x], u, x] - Dist[b*c*(x/Sqrt[c^2*x^2]), Int[SimplifyI
ntegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&  !(ILtQ
[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (I
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rubi steps \begin{align*} \text {integral}& = -\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac {(b c x) \int \frac {-5 d-7 e x^2}{35 x^8 \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}} \\ & = -\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac {(b c x) \int \frac {-5 d-7 e x^2}{x^8 \sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {c^2 x^2}} \\ & = \frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac {\left (b c \left (-30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^6 \sqrt {-1+c^2 x^2}} \, dx}{245 \sqrt {c^2 x^2}} \\ & = \frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}+\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac {\left (4 b c^3 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^4 \sqrt {-1+c^2 x^2}} \, dx}{1225 \sqrt {c^2 x^2}} \\ & = \frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}+\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5}-\frac {\left (8 b c^5 \left (-30 c^2 d-49 e\right ) x\right ) \int \frac {1}{x^2 \sqrt {-1+c^2 x^2}} \, dx}{3675 \sqrt {c^2 x^2}} \\ & = \frac {8 b c^5 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 \sqrt {c^2 x^2}}+\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}+\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \sec ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \sec ^{-1}(c x)\right )}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\frac {-105 a \left (5 d+7 e x^2\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (49 e x^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )+15 d \left (5+6 c^2 x^2+8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (5 d+7 e x^2\right ) \sec ^{-1}(c x)}{3675 x^7} \]

[In]

Integrate[((d + e*x^2)*(a + b*ArcSec[c*x]))/x^8,x]

[Out]

(-105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(5 + 6*c^2*
x^2 + 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(5*d + 7*e*x^2)*ArcSec[c*x])/(3675*x^7)

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74

method	result	size
parts	\(a \left (-\frac {e}{5 x^{5}}-\frac {d}{7 x^{7}}\right )+b \,c^{7} \left (-\frac {\operatorname {arcsec}\left (c x \right ) e}{5 c^{7} x^{5}}-\frac {\operatorname {arcsec}\left (c x \right ) d}{7 x^{7} c^{7}}+\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 c^{10} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{8}}\right )\)	\(145\)
derivativedivides	\(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arcsec}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arcsec}\left (c x \right ) e}{5 c^{5} x^{5}}+\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\)	\(158\)
default	\(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arcsec}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arcsec}\left (c x \right ) e}{5 c^{5} x^{5}}+\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\)	\(158\)

[In]

int((e*x^2+d)*(a+b*arcsec(c*x))/x^8,x,method=_RETURNVERBOSE)

[Out]

a*(-1/5*e/x^5-1/7*d/x^7)+b*c^7*(-1/5/c^7*arcsec(c*x)*e/x^5-1/7*arcsec(c*x)*d/x^7/c^7+1/3675/c^10*(c^2*x^2-1)*(
240*c^8*d*x^6+392*c^6*e*x^6+120*c^6*d*x^4+196*c^4*e*x^4+90*c^4*d*x^2+147*c^2*e*x^2+75*c^2*d)/((c^2*x^2-1)/c^2/
x^2)^(1/2)/x^8)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=-\frac {735 \, a e x^{2} + 525 \, a d + 105 \, {\left (7 \, b e x^{2} + 5 \, b d\right )} \operatorname {arcsec}\left (c x\right ) - {\left (8 \, {\left (30 \, b c^{6} d + 49 \, b c^{4} e\right )} x^{6} + 4 \, {\left (30 \, b c^{4} d + 49 \, b c^{2} e\right )} x^{4} + 3 \, {\left (30 \, b c^{2} d + 49 \, b e\right )} x^{2} + 75 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, x^{7}} \]

[In]

integrate((e*x^2+d)*(a+b*arcsec(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/3675*(735*a*e*x^2 + 525*a*d + 105*(7*b*e*x^2 + 5*b*d)*arcsec(c*x) - (8*(30*b*c^6*d + 49*b*c^4*e)*x^6 + 4*(3
0*b*c^4*d + 49*b*c^2*e)*x^4 + 3*(30*b*c^2*d + 49*b*e)*x^2 + 75*b*d)*sqrt(c^2*x^2 - 1))/x^7

Sympy [A] (verification not implemented)

Time = 29.28 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.88 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=- \frac {a d}{7 x^{7}} - \frac {a e}{5 x^{5}} - \frac {b d \operatorname {asec}{\left (c x \right )}}{7 x^{7}} - \frac {b e \operatorname {asec}{\left (c x \right )}}{5 x^{5}} + \frac {b d \left (\begin {cases} \frac {16 c^{7} \sqrt {c^{2} x^{2} - 1}}{35 x} + \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{35 x^{3}} + \frac {6 c^{3} \sqrt {c^{2} x^{2} - 1}}{35 x^{5}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{7 x^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {16 i c^{7} \sqrt {- c^{2} x^{2} + 1}}{35 x} + \frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{35 x^{3}} + \frac {6 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{35 x^{5}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{7 x^{7}} & \text {otherwise} \end {cases}\right )}{7 c} + \frac {b e \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} \]

[In]

integrate((e*x**2+d)*(a+b*asec(c*x))/x**8,x)

[Out]

-a*d/(7*x**7) - a*e/(5*x**5) - b*d*asec(c*x)/(7*x**7) - b*e*asec(c*x)/(5*x**5) + b*d*Piecewise((16*c**7*sqrt(c
**2*x**2 - 1)/(35*x) + 8*c**5*sqrt(c**2*x**2 - 1)/(35*x**3) + 6*c**3*sqrt(c**2*x**2 - 1)/(35*x**5) + c*sqrt(c*
*2*x**2 - 1)/(7*x**7), Abs(c**2*x**2) > 1), (16*I*c**7*sqrt(-c**2*x**2 + 1)/(35*x) + 8*I*c**5*sqrt(-c**2*x**2
+ 1)/(35*x**3) + 6*I*c**3*sqrt(-c**2*x**2 + 1)/(35*x**5) + I*c*sqrt(-c**2*x**2 + 1)/(7*x**7), True))/(7*c) + b
*e*Piecewise((8*c**5*sqrt(c**2*x**2 - 1)/(15*x) + 4*c**3*sqrt(c**2*x**2 - 1)/(15*x**3) + c*sqrt(c**2*x**2 - 1)
/(5*x**5), Abs(c**2*x**2) > 1), (8*I*c**5*sqrt(-c**2*x**2 + 1)/(15*x) + 4*I*c**3*sqrt(-c**2*x**2 + 1)/(15*x**3
) + I*c*sqrt(-c**2*x**2 + 1)/(5*x**5), True))/(5*c)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=-\frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {35 \, \operatorname {arcsec}\left (c x\right )}{x^{7}}\right )} + \frac {1}{75} \, b e {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsec}\left (c x\right )}{x^{5}}\right )} - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \]

[In]

integrate((e*x^2+d)*(a+b*arcsec(c*x))/x^8,x, algorithm="maxima")

[Out]

-1/245*b*d*((5*c^8*(-1/(c^2*x^2) + 1)^(7/2) - 21*c^8*(-1/(c^2*x^2) + 1)^(5/2) + 35*c^8*(-1/(c^2*x^2) + 1)^(3/2
) - 35*c^8*sqrt(-1/(c^2*x^2) + 1))/c + 35*arcsec(c*x)/x^7) + 1/75*b*e*((3*c^6*(-1/(c^2*x^2) + 1)^(5/2) - 10*c^
6*(-1/(c^2*x^2) + 1)^(3/2) + 15*c^6*sqrt(-1/(c^2*x^2) + 1))/c - 15*arcsec(c*x)/x^5) - 1/5*a*e/x^5 - 1/7*a*d/x^
7

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\frac {1}{3675} \, {\left (240 \, b c^{6} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 392 \, b c^{4} e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {120 \, b c^{4} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} + \frac {196 \, b c^{2} e \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{2}} + \frac {90 \, b c^{2} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{4}} + \frac {147 \, b e \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{4}} - \frac {735 \, b e \arccos \left (\frac {1}{c x}\right )}{c x^{5}} + \frac {75 \, b d \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x^{6}} - \frac {735 \, a e}{c x^{5}} - \frac {525 \, b d \arccos \left (\frac {1}{c x}\right )}{c x^{7}} - \frac {525 \, a d}{c x^{7}}\right )} c \]

[In]

integrate((e*x^2+d)*(a+b*arcsec(c*x))/x^8,x, algorithm="giac")

[Out]

1/3675*(240*b*c^6*d*sqrt(-1/(c^2*x^2) + 1) + 392*b*c^4*e*sqrt(-1/(c^2*x^2) + 1) + 120*b*c^4*d*sqrt(-1/(c^2*x^2
) + 1)/x^2 + 196*b*c^2*e*sqrt(-1/(c^2*x^2) + 1)/x^2 + 90*b*c^2*d*sqrt(-1/(c^2*x^2) + 1)/x^4 + 147*b*e*sqrt(-1/
(c^2*x^2) + 1)/x^4 - 735*b*e*arccos(1/(c*x))/(c*x^5) + 75*b*d*sqrt(-1/(c^2*x^2) + 1)/x^6 - 735*a*e/(c*x^5) - 5
25*b*d*arccos(1/(c*x))/(c*x^7) - 525*a*d/(c*x^7))*c

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \sec ^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]

[In]

int(((d + e*x^2)*(a + b*acos(1/(c*x))))/x^8,x)

[Out]

int(((d + e*x^2)*(a + b*acos(1/(c*x))))/x^8, x)