Integrand size = 19, antiderivative size = 197 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=-\frac {8 b c^5 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {-1+c^2 x^2}}{49 x^6 \sqrt {c^2 x^2}}-\frac {b c \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{1225 x^4 \sqrt {c^2 x^2}}-\frac {4 b c^3 \left (30 c^2 d+49 e\right ) \sqrt {-1+c^2 x^2}}{3675 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5} \]
-1/7*d*(a+b*arccsc(c*x))/x^7-1/5*e*(a+b*arccsc(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)
Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=-\frac {105 a \left (5 d+7 e x^2\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (49 e x^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )+15 d \left (5+6 c^2 x^2+8 c^4 x^4+16 c^6 x^6\right )\right )+105 b \left (5 d+7 e x^2\right ) \csc ^{-1}(c x)}{3675 x^7} \]
-1/3675*(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)*ArcCsc[c*x])/x^7
Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.81, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5762, 27, 359, 245, 245, 242}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 5762 |
| \(\displaystyle \frac {b c x \int -\frac {7 e x^2+5 d}{35 x^8 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b c x \int \frac {7 e x^2+5 d}{x^8 \sqrt {c^2 x^2-1}}dx}{35 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{7} \left (30 c^2 d+49 e\right ) \int \frac {1}{x^6 \sqrt {c^2 x^2-1}}dx+\frac {5 d \sqrt {c^2 x^2-1}}{7 x^7}\right )}{35 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{7} \left (30 c^2 d+49 e\right ) \left (\frac {4}{5} c^2 \int \frac {1}{x^4 \sqrt {c^2 x^2-1}}dx+\frac {\sqrt {c^2 x^2-1}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1}}{7 x^7}\right )}{35 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {b c x \left (\frac {1}{7} \left (30 c^2 d+49 e\right ) \left (\frac {4}{5} c^2 \left (\frac {2}{3} c^2 \int \frac {1}{x^2 \sqrt {c^2 x^2-1}}dx+\frac {\sqrt {c^2 x^2-1}}{3 x^3}\right )+\frac {\sqrt {c^2 x^2-1}}{5 x^5}\right )+\frac {5 d \sqrt {c^2 x^2-1}}{7 x^7}\right )}{35 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{5 x^5}-\frac {b c x \left (\frac {1}{7} \left (\frac {\sqrt {c^2 x^2-1}}{5 x^5}+\frac {4}{5} c^2 \left (\frac {2 c^2 \sqrt {c^2 x^2-1}}{3 x}+\frac {\sqrt {c^2 x^2-1}}{3 x^3}\right )\right ) \left (30 c^2 d+49 e\right )+\frac {5 d \sqrt {c^2 x^2-1}}{7 x^7}\right )}{35 \sqrt {c^2 x^2}}\) |
-1/35*(b*c*x*((5*d*Sqrt[-1 + c^2*x^2])/(7*x^7) + ((30*c^2*d + 49*e)*(Sqrt[ -1 + c^2*x^2]/(5*x^5) + (4*c^2*(Sqrt[-1 + c^2*x^2]/(3*x^3) + (2*c^2*Sqrt[- 1 + c^2*x^2])/(3*x)))/5))/7))/Sqrt[c^2*x^2] - (d*(a + b*ArcCsc[c*x]))/(7*x ^7) - (e*(a + b*ArcCsc[c*x]))/(5*x^5)
3.1.82.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((a_.) + ArcCsc[(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]}, Sim p[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[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])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Time = 0.35 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.74
| method | result | size |
| parts | \(a \left (-\frac {d}{7 x^{7}}-\frac {e}{5 x^{5}}\right )+b \,c^{7} \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{7 x^{7} c^{7}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{5 c^{7} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 c^{10} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{8}}\right )\) | \(145\) |
| derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\) | \(158\) |
| default | \(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 c^{2} e \,x^{2}+75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\) | \(158\) |
a*(-1/7*d/x^7-1/5*e/x^5)+b*c^7*(-1/7*arccsc(c*x)*d/x^7/c^7-1/5/c^7*arccsc( c*x)*e/x^5-1/3675/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)
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=-\frac {735 \, a e x^{2} + 525 \, a d + 105 \, {\left (7 \, b e x^{2} + 5 \, b d\right )} \operatorname {arccsc}\left (c x\right ) + {\left (8 \, {\left (30 \, b c^{6} d + 49 \, b c^{4} e\right )} x^{6} + 4 \, {\left (30 \, b c^{4} d + 49 \, b c^{2} e\right )} x^{4} + 3 \, {\left (30 \, b c^{2} d + 49 \, b e\right )} x^{2} + 75 \, b d\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, x^{7}} \]
-1/3675*(735*a*e*x^2 + 525*a*d + 105*(7*b*e*x^2 + 5*b*d)*arccsc(c*x) + (8* (30*b*c^6*d + 49*b*c^4*e)*x^6 + 4*(30*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
Time = 29.20 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.89 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=- \frac {a d}{7 x^{7}} - \frac {a e}{5 x^{5}} - \frac {b d \operatorname {acsc}{\left (c x \right )}}{7 x^{7}} - \frac {b e \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {b d \left (\begin {cases} \frac {16 c^{7} \sqrt {c^{2} x^{2} - 1}}{35 x} + \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{35 x^{3}} + \frac {6 c^{3} \sqrt {c^{2} x^{2} - 1}}{35 x^{5}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{7 x^{7}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {16 i c^{7} \sqrt {- c^{2} x^{2} + 1}}{35 x} + \frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{35 x^{3}} + \frac {6 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{35 x^{5}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{7 x^{7}} & \text {otherwise} \end {cases}\right )}{7 c} - \frac {b e \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} \]
-a*d/(7*x**7) - a*e/(5*x**5) - b*d*acsc(c*x)/(7*x**7) - b*e*acsc(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*P iecewise((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)
Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {35 \, \operatorname {arccsc}\left (c x\right )}{x^{7}}\right )} - \frac {1}{75} \, b e {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \]
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*arccsc(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*arccsc(c*x )/x^5) - 1/5*a*e/x^5 - 1/7*a*d/x^7
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (169) = 338\).
Time = 0.29 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.86 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=-\frac {1}{3675} \, {\left (75 \, b c^{6} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 315 \, b c^{6} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {525 \, b c^{5} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{x} - 525 \, b c^{6} d {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {1575 \, b c^{5} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 525 \, b c^{6} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 147 \, b c^{4} e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {1575 \, b c^{5} d {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - 490 \, b c^{4} e {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {525 \, b c^{5} d \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {735 \, b c^{3} e {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 735 \, b c^{4} e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {1470 \, b c^{3} e {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {735 \, b c^{3} e \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {735 \, a e}{c x^{5}} + \frac {525 \, a d}{c x^{7}}\right )} c \]
-1/3675*(75*b*c^6*d*(1/(c^2*x^2) - 1)^3*sqrt(-1/(c^2*x^2) + 1) + 315*b*c^6 *d*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1) + 525*b*c^5*d*(1/(c^2*x^2) - 1)^3*arcsin(1/(c*x))/x - 525*b*c^6*d*(-1/(c^2*x^2) + 1)^(3/2) + 1575*b*c^ 5*d*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))/x + 525*b*c^6*d*sqrt(-1/(c^2*x^2) + 1) + 147*b*c^4*e*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1) + 1575*b*c^5 *d*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x - 490*b*c^4*e*(-1/(c^2*x^2) + 1)^(3 /2) + 525*b*c^5*d*arcsin(1/(c*x))/x + 735*b*c^3*e*(1/(c^2*x^2) - 1)^2*arcs in(1/(c*x))/x + 735*b*c^4*e*sqrt(-1/(c^2*x^2) + 1) + 1470*b*c^3*e*(1/(c^2* x^2) - 1)*arcsin(1/(c*x))/x + 735*b*c^3*e*arcsin(1/(c*x))/x + 735*a*e/(c*x ^5) + 525*a*d/(c*x^7))*c
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]