Integrand size = 19, antiderivative size = 205 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=-\frac {8 b c^5 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 \sqrt {-c^2 x^2}}+\frac {b c d \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {b c \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {4 b c^3 \left (30 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{3675 x^2 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5} \] Output:
-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)-1/7*d*(a+b*arccsch(c*x))/x^7-1/5*e*(a+b*arc csch(c*x))/x^5
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.53 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\frac {-105 a \left (5 d+7 e x^2\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (49 e x^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )-15 d \left (-5+6 c^2 x^2-8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (5 d+7 e x^2\right ) \text {csch}^{-1}(c x)}{3675 x^7} \] Input:
Integrate[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^8,x]
Output:
(-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)*ArcCsch[c*x])/(3675*x^7)
Time = 0.35 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6856, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 6856 |
| \(\displaystyle -\frac {b c x \int -\frac {7 e x^2+5 d}{35 x^8 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {7 e x^2+5 d}{x^8 \sqrt {-c^2 x^2-1}}dx}{35 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {b c x \left (\frac {5 d \sqrt {-c^2 x^2-1}}{7 x^7}-\frac {1}{7} \left (30 c^2 d-49 e\right ) \int \frac {1}{x^6 \sqrt {-c^2 x^2-1}}dx\right )}{35 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {b c x \left (\frac {5 d \sqrt {-c^2 x^2-1}}{7 x^7}-\frac {1}{7} \left (30 c^2 d-49 e\right ) \left (\frac {\sqrt {-c^2 x^2-1}}{5 x^5}-\frac {4}{5} c^2 \int \frac {1}{x^4 \sqrt {-c^2 x^2-1}}dx\right )\right )}{35 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {b c x \left (\frac {5 d \sqrt {-c^2 x^2-1}}{7 x^7}-\frac {1}{7} \left (30 c^2 d-49 e\right ) \left (\frac {\sqrt {-c^2 x^2-1}}{5 x^5}-\frac {4}{5} c^2 \left (\frac {\sqrt {-c^2 x^2-1}}{3 x^3}-\frac {2}{3} c^2 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1}}dx\right )\right )\right )}{35 \sqrt {-c^2 x^2}}-\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {d \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}+\frac {b c x \left (\frac {5 d \sqrt {-c^2 x^2-1}}{7 x^7}-\frac {1}{7} \left (\frac {\sqrt {-c^2 x^2-1}}{5 x^5}-\frac {4}{5} c^2 \left (\frac {\sqrt {-c^2 x^2-1}}{3 x^3}-\frac {2 c^2 \sqrt {-c^2 x^2-1}}{3 x}\right )\right ) \left (30 c^2 d-49 e\right )\right )}{35 \sqrt {-c^2 x^2}}\) |
Input:
Int[((d + e*x^2)*(a + b*ArcCsch[c*x]))/x^8,x]
Output:
(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))/(35*Sqrt[-(c^2*x^2)]) - (d*(a + b*ArcCsch[c*x]))/( 7*x^7) - (e*(a + b*ArcCsch[c*x]))/(5*x^5)
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_.) + ArcCsch[(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*ArcCsch[c*x]) u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2]) Int[Simpl ifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), 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]))
Time = 0.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.71
| method | result | size |
| parts | \(a \left (-\frac {d}{7 x^{7}}-\frac {e}{5 x^{5}}\right )+b \,c^{7} \left (-\frac {\operatorname {arccsch}\left (c x \right ) d}{7 c^{7} x^{7}}-\frac {\operatorname {arccsch}\left (c x \right ) e}{5 c^{7} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (240 c^{8} d \,x^{6}-392 c^{6} e \,x^{6}-120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}-147 e \,c^{2} x^{2}-75 c^{2} d \right )}{3675 c^{10} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x^{8}}\right )\) | \(145\) |
| derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arccsch}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (240 c^{8} d \,x^{6}-392 c^{6} e \,x^{6}-120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}-147 e \,c^{2} x^{2}-75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\) | \(158\) |
| default | \(c^{7} \left (\frac {a \left (-\frac {d}{7 c^{5} x^{7}}-\frac {e}{5 c^{5} x^{5}}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) d}{7 c^{5} x^{7}}-\frac {\operatorname {arccsch}\left (c x \right ) e}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (240 c^{8} d \,x^{6}-392 c^{6} e \,x^{6}-120 c^{6} d \,x^{4}+196 c^{4} e \,x^{4}+90 c^{4} d \,x^{2}-147 e \,c^{2} x^{2}-75 c^{2} d \right )}{3675 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{2}}\right )\) | \(158\) |
Input:
int((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x,method=_RETURNVERBOSE)
Output:
a*(-1/7*d/x^7-1/5*e/x^5)+b*c^7*(-1/7*arccsch(c*x)*d/c^7/x^7-1/5/c^7*arccsc h(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 = 0.11 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=-\frac {735 \, a e x^{2} + 525 \, a d + 105 \, {\left (7 \, b e x^{2} + 5 \, b d\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (8 \, {\left (30 \, b c^{7} d - 49 \, b c^{5} e\right )} x^{7} - 4 \, {\left (30 \, b c^{5} d - 49 \, b c^{3} e\right )} x^{5} - 75 \, b c d x + 3 \, {\left (30 \, b c^{3} d - 49 \, b c e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{3675 \, x^{7}} \] Input:
integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x, algorithm="fricas")
Output:
-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
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{8}}\, dx \] Input:
integrate((e*x**2+d)*(a+b*acsch(c*x))/x**8,x)
Output:
Integral((a + b*acsch(c*x))*(d + e*x**2)/x**8, x)
Time = 0.03 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\frac {1}{245} \, b d {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {35 \, \operatorname {arcsch}\left (c x\right )}{x^{7}}\right )} + \frac {1}{75} \, b e {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} - \frac {a e}{5 \, x^{5}} - \frac {a d}{7 \, x^{7}} \] Input:
integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x, algorithm="maxima")
Output:
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 rccsch(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*arccsch(c*x)/x^5 ) - 1/5*a*e/x^5 - 1/7*a*d/x^7
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \] Input:
integrate((e*x^2+d)*(a+b*arccsch(c*x))/x^8,x, algorithm="giac")
Output:
integrate((e*x^2 + d)*(b*arccsch(c*x) + a)/x^8, x)
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \] Input:
int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^8,x)
Output:
int(((d + e*x^2)*(a + b*asinh(1/(c*x))))/x^8, x)
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\frac {35 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{8}}d x \right ) b d \,x^{7}+35 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{6}}d x \right ) b e \,x^{7}-5 a d -7 a e \,x^{2}}{35 x^{7}} \] Input:
int((e*x^2+d)*(a+b*acsch(c*x))/x^8,x)
Output:
(35*int(acsch(c*x)/x**8,x)*b*d*x**7 + 35*int(acsch(c*x)/x**6,x)*b*e*x**7 - 5*a*d - 7*a*e*x**2)/(35*x**7)