Integrand size = 19, antiderivative size = 238 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{49 x^7}+\frac {b \left (30 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{1225 x^5}+\frac {4 b c^2 \left (30 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3675 x^3}+\frac {8 b c^4 \left (30 c^2 d+49 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{3675 x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5} \]
-1/7*d*(a+b*arcsech(c*x))/x^7-1/5*e*(a+b*arcsech(c*x))/x^5+1/49*b*d*(1/(c* x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x^7+1/1225*b*(30*c^2*d+49*e)* (1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x^5+4/3675*b*c^2*(30*c^ 2*d+49*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x^3+8/3675*b* c^4*(30*c^2*d+49*e)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.49 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {-105 a \left (5 d+7 e x^2\right )+b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (49 e x^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )+15 d \left (5+6 c^2 x^2+8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (5 d+7 e x^2\right ) \text {sech}^{-1}(c x)}{3675 x^7} \]
(-105*a*(5*d + 7*e*x^2) + b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*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)*ArcSech[c*x])/(3675*x^7)
Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6855, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {7 e x^2+5 d}{35 x^8 \sqrt {1-c^2 x^2}}dx-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{35} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {7 e x^2+5 d}{x^8 \sqrt {1-c^2 x^2}}dx-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle -\frac {1}{35} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (30 c^2 d+49 e\right ) \int \frac {1}{x^6 \sqrt {1-c^2 x^2}}dx-\frac {5 d \sqrt {1-c^2 x^2}}{7 x^7}\right )-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {1}{35} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (30 c^2 d+49 e\right ) \left (\frac {4}{5} c^2 \int \frac {1}{x^4 \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2}}{7 x^7}\right )-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {1}{35} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (30 c^2 d+49 e\right ) \left (\frac {4}{5} c^2 \left (\frac {2}{3} c^2 \int \frac {1}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-c^2 x^2}}{5 x^5}\right )-\frac {5 d \sqrt {1-c^2 x^2}}{7 x^7}\right )-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {1}{35} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\frac {1}{7} \left (\frac {4}{5} c^2 \left (-\frac {2 c^2 \sqrt {1-c^2 x^2}}{3 x}-\frac {\sqrt {1-c^2 x^2}}{3 x^3}\right )-\frac {\sqrt {1-c^2 x^2}}{5 x^5}\right ) \left (30 c^2 d+49 e\right )-\frac {5 d \sqrt {1-c^2 x^2}}{7 x^7}\right )\) |
-1/35*(b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-5*d*Sqrt[1 - c^2*x^2])/(7*x ^7) + ((30*c^2*d + 49*e)*(-1/5*Sqrt[1 - c^2*x^2]/x^5 + (4*c^2*(-1/3*Sqrt[1 - c^2*x^2]/x^3 - (2*c^2*Sqrt[1 - c^2*x^2])/(3*x)))/5))/7)) - (d*(a + b*Ar cSech[c*x]))/(7*x^7) - (e*(a + b*ArcSech[c*x]))/(5*x^5)
3.1.94.3.1 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_.) + ArcSech[(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]}, Si mp[(a + b*ArcSech[c*x]) u, x] + Simp[b*Sqrt[1 + c*x]*Sqrt[1/(1 + c*x)] Int[SimplifyIntegrand[u/(x*Sqrt[1 - c*x]*Sqrt[1 + c*x]), x], x], x]] /; Fre eQ[{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 = 0.36 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.62
| method | result | size |
| parts | \(a \left (-\frac {e}{5 x^{5}}-\frac {d}{7 x^{7}}\right )+b \,c^{7} \left (-\frac {\operatorname {arcsech}\left (c x \right ) e}{5 c^{7} x^{5}}-\frac {\operatorname {arcsech}\left (c x \right ) d}{7 x^{7} c^{7}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 e \,c^{2} x^{2}+75 c^{2} d \right )}{3675 c^{8} x^{6}}\right )\) | \(147\) |
| derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arcsech}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arcsech}\left (c x \right ) e}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 e \,c^{2} x^{2}+75 c^{2} d \right )}{3675 c^{6} x^{6}}\right )}{c^{2}}\right )\) | \(160\) |
| default | \(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arcsech}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arcsech}\left (c x \right ) e}{5 c^{5} x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (240 c^{8} d \,x^{6}+392 c^{6} e \,x^{6}+120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}+147 e \,c^{2} x^{2}+75 c^{2} d \right )}{3675 c^{6} x^{6}}\right )}{c^{2}}\right )\) | \(160\) |
a*(-1/5*e/x^5-1/7*d/x^7)+b*c^7*(-1/5/c^7*arcsech(c*x)*e/x^5-1/7*arcsech(c* x)*d/x^7/c^7+1/3675/c^8*(-(c*x-1)/c/x)^(1/2)/x^6*((c*x+1)/c/x)^(1/2)*(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))
Time = 0.27 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.63 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=-\frac {735 \, a e x^{2} + 525 \, a d + 105 \, {\left (7 \, b e x^{2} + 5 \, b d\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, {\left (30 \, b c^{7} d + 49 \, b c^{5} e\right )} x^{7} + 4 \, {\left (30 \, b c^{5} d + 49 \, b c^{3} e\right )} x^{5} + 75 \, b c d x + 3 \, {\left (30 \, b c^{3} d + 49 \, b c e\right )} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{3675 \, x^{7}} \]
-1/3675*(735*a*e*x^2 + 525*a*d + 105*(7*b*e*x^2 + 5*b*d)*log((c*x*sqrt(-(c ^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (8*(30*b*c^7*d + 49*b*c^5*e)*x^7 + 4* (30*b*c^5*d + 49*b*c^3*e)*x^5 + 75*b*c*d*x + 3*(30*b*c^3*d + 49*b*c*e)*x^3 )*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/x^7
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{8}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {7}{2}} + 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {35 \, \operatorname {arsech}\left (c x\right )}{x^{7}}\right )} + \frac {1}{75} \, b e {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \]
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*a rcsech(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*arcsech(c*x)/x^5 ) - 1/5*a*e/x^5 - 1/7*a*d/x^7
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \]