Integrand size = 21, antiderivative size = 249 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=-\frac {2 b c^3 \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{49 x^6 \sqrt {-c^2 x^2}}-\frac {2 b c d \left (15 c^2 d-49 e\right ) \sqrt {-1-c^2 x^2}}{1225 x^4 \sqrt {-c^2 x^2}}+\frac {b c \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \sqrt {-1-c^2 x^2}}{11025 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3} \] Output:
-2/11025*b*c^3*(360*c^4*d^2-1176*c^2*d*e+1225*e^2)*(-c^2*x^2-1)^(1/2)/(-c^ 2*x^2)^(1/2)+1/49*b*c*d^2*(-c^2*x^2-1)^(1/2)/x^6/(-c^2*x^2)^(1/2)-2/1225*b *c*d*(15*c^2*d-49*e)*(-c^2*x^2-1)^(1/2)/x^4/(-c^2*x^2)^(1/2)+1/11025*b*c*( 360*c^4*d^2-1176*c^2*d*e+1225*e^2)*(-c^2*x^2-1)^(1/2)/x^2/(-c^2*x^2)^(1/2) -1/7*d^2*(a+b*arccsch(c*x))/x^7-2/5*d*e*(a+b*arccsch(c*x))/x^5-1/3*e^2*(a+ b*arccsch(c*x))/x^3
Time = 0.17 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.61 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\frac {-105 a \left (15 d^2+42 d e x^2+35 e^2 x^4\right )+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1225 e^2 x^4 \left (1-2 c^2 x^2\right )+294 d e x^2 \left (3-4 c^2 x^2+8 c^4 x^4\right )-45 d^2 \left (-5+6 c^2 x^2-8 c^4 x^4+16 c^6 x^6\right )\right )-105 b \left (15 d^2+42 d e x^2+35 e^2 x^4\right ) \text {csch}^{-1}(c x)}{11025 x^7} \] Input:
Integrate[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^8,x]
Output:
(-105*a*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4) + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*( 1225*e^2*x^4*(1 - 2*c^2*x^2) + 294*d*e*x^2*(3 - 4*c^2*x^2 + 8*c^4*x^4) - 4 5*d^2*(-5 + 6*c^2*x^2 - 8*c^4*x^4 + 16*c^6*x^6)) - 105*b*(15*d^2 + 42*d*e* x^2 + 35*e^2*x^4)*ArcCsch[c*x])/(11025*x^7)
Time = 0.44 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.83, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6856, 27, 1588, 25, 359, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 6856 |
| \(\displaystyle -\frac {b c x \int -\frac {35 e^2 x^4+42 d e x^2+15 d^2}{105 x^8 \sqrt {-c^2 x^2-1}}dx}{\sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b c x \int \frac {35 e^2 x^4+42 d e x^2+15 d^2}{x^8 \sqrt {-c^2 x^2-1}}dx}{105 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \int -\frac {6 d \left (15 c^2 d-49 e\right )-245 e^2 x^2}{x^6 \sqrt {-c^2 x^2-1}}dx+\frac {15 d^2 \sqrt {-c^2 x^2-1}}{7 x^7}\right )}{105 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c x \left (\frac {15 d^2 \sqrt {-c^2 x^2-1}}{7 x^7}-\frac {1}{7} \int \frac {6 d \left (15 c^2 d-49 e\right )-245 e^2 x^2}{x^6 \sqrt {-c^2 x^2-1}}dx\right )}{105 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\) |
| \(\Big \downarrow \) 359 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \int \frac {1}{x^4 \sqrt {-c^2 x^2-1}}dx-\frac {6 d \sqrt {-c^2 x^2-1} \left (15 c^2 d-49 e\right )}{5 x^5}\right )+\frac {15 d^2 \sqrt {-c^2 x^2-1}}{7 x^7}\right )}{105 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {b c x \left (\frac {1}{7} \left (\frac {1}{5} \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right ) \left (\frac {\sqrt {-c^2 x^2-1}}{3 x^3}-\frac {2}{3} c^2 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1}}dx\right )-\frac {6 d \sqrt {-c^2 x^2-1} \left (15 c^2 d-49 e\right )}{5 x^5}\right )+\frac {15 d^2 \sqrt {-c^2 x^2-1}}{7 x^7}\right )}{105 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{7 x^7}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{5 x^5}-\frac {e^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}+\frac {b c x \left (\frac {15 d^2 \sqrt {-c^2 x^2-1}}{7 x^7}+\frac {1}{7} \left (\frac {1}{5} \left (\frac {\sqrt {-c^2 x^2-1}}{3 x^3}-\frac {2 c^2 \sqrt {-c^2 x^2-1}}{3 x}\right ) \left (360 c^4 d^2-1176 c^2 d e+1225 e^2\right )-\frac {6 d \sqrt {-c^2 x^2-1} \left (15 c^2 d-49 e\right )}{5 x^5}\right )\right )}{105 \sqrt {-c^2 x^2}}\) |
Input:
Int[((d + e*x^2)^2*(a + b*ArcCsch[c*x]))/x^8,x]
Output:
(b*c*x*((15*d^2*Sqrt[-1 - c^2*x^2])/(7*x^7) + ((-6*d*(15*c^2*d - 49*e)*Sqr t[-1 - c^2*x^2])/(5*x^5) + ((360*c^4*d^2 - 1176*c^2*d*e + 1225*e^2)*(Sqrt[ -1 - c^2*x^2]/(3*x^3) - (2*c^2*Sqrt[-1 - c^2*x^2])/(3*x)))/5)/7))/(105*Sqr t[-(c^2*x^2)]) - (d^2*(a + b*ArcCsch[c*x]))/(7*x^7) - (2*d*e*(a + b*ArcCsc h[c*x]))/(5*x^5) - (e^2*(a + b*ArcCsch[c*x]))/(3*x^3)
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[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c _.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, f*x, x], R = PolynomialRemainder[(a + b*x^2 + c*x^4)^p, f*x, x]}, Simp[R*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(d*f*(m + 1))), x] + Simp[1/(d*f ^2*(m + 1)) Int[(f*x)^(m + 2)*(d + e*x^2)^q*ExpandToSum[d*f*(m + 1)*(Qx/x ) - e*R*(m + 2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && Ne Q[b^2 - 4*a*c, 0] && IGtQ[p, 0] && LtQ[m, -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 = 0.36 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.83
| method | result | size |
| parts | \(a \left (-\frac {d^{2}}{7 x^{7}}-\frac {e^{2}}{3 x^{3}}-\frac {2 d e}{5 x^{5}}\right )+b \,c^{7} \left (-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{7 c^{7} x^{7}}-\frac {\operatorname {arccsch}\left (c x \right ) e^{2}}{3 c^{7} x^{3}}-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{5 c^{7} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (720 c^{10} d^{2} x^{6}-2352 c^{8} d e \,x^{6}-360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}-1225 c^{4} e^{2} x^{4}-882 c^{4} d e \,x^{2}-225 c^{4} d^{2}\right )}{11025 c^{12} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x^{8}}\right )\) | \(207\) |
| derivativedivides | \(c^{7} \left (\frac {a \left (-\frac {e^{2}}{3 c^{3} x^{3}}-\frac {d^{2}}{7 c^{3} x^{7}}-\frac {2 d e}{5 c^{3} x^{5}}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) e^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{7 c^{3} x^{7}}-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{5 c^{3} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (720 c^{10} d^{2} x^{6}-2352 c^{8} d e \,x^{6}-360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}-1225 c^{4} e^{2} x^{4}-882 c^{4} d e \,x^{2}-225 c^{4} d^{2}\right )}{11025 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{4}}\right )\) | \(223\) |
| default | \(c^{7} \left (\frac {a \left (-\frac {e^{2}}{3 c^{3} x^{3}}-\frac {d^{2}}{7 c^{3} x^{7}}-\frac {2 d e}{5 c^{3} x^{5}}\right )}{c^{4}}+\frac {b \left (-\frac {\operatorname {arccsch}\left (c x \right ) e^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arccsch}\left (c x \right ) d^{2}}{7 c^{3} x^{7}}-\frac {2 \,\operatorname {arccsch}\left (c x \right ) d e}{5 c^{3} x^{5}}-\frac {\left (c^{2} x^{2}+1\right ) \left (720 c^{10} d^{2} x^{6}-2352 c^{8} d e \,x^{6}-360 c^{8} d^{2} x^{4}+2450 c^{6} e^{2} x^{6}+1176 c^{6} d e \,x^{4}+270 c^{6} d^{2} x^{2}-1225 c^{4} e^{2} x^{4}-882 c^{4} d e \,x^{2}-225 c^{4} d^{2}\right )}{11025 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{8} x^{8}}\right )}{c^{4}}\right )\) | \(223\) |
Input:
int((e*x^2+d)^2*(a+b*arccsch(c*x))/x^8,x,method=_RETURNVERBOSE)
Output:
a*(-1/7*d^2/x^7-1/3/x^3*e^2-2/5*d*e/x^5)+b*c^7*(-1/7*arccsch(c*x)*d^2/c^7/ x^7-1/3/c^7*arccsch(c*x)/x^3*e^2-2/5/c^7*arccsch(c*x)*d*e/x^5-1/11025/c^12 *(c^2*x^2+1)*(720*c^10*d^2*x^6-2352*c^8*d*e*x^6-360*c^8*d^2*x^4+2450*c^6*e ^2*x^6+1176*c^6*d*e*x^4+270*c^6*d^2*x^2-1225*c^4*e^2*x^4-882*c^4*d*e*x^2-2 25*c^4*d^2)/((c^2*x^2+1)/c^2/x^2)^(1/2)/x^8)
Time = 0.09 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.79 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=-\frac {3675 \, a e^{2} x^{4} + 4410 \, a d e x^{2} + 1575 \, a d^{2} + 105 \, {\left (35 \, b e^{2} x^{4} + 42 \, b d e x^{2} + 15 \, b d^{2}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (2 \, {\left (360 \, b c^{7} d^{2} - 1176 \, b c^{5} d e + 1225 \, b c^{3} e^{2}\right )} x^{7} - {\left (360 \, b c^{5} d^{2} - 1176 \, b c^{3} d e + 1225 \, b c e^{2}\right )} x^{5} - 225 \, b c d^{2} x + 18 \, {\left (15 \, b c^{3} d^{2} - 49 \, b c d e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{11025 \, x^{7}} \] Input:
integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^8,x, algorithm="fricas")
Output:
-1/11025*(3675*a*e^2*x^4 + 4410*a*d*e*x^2 + 1575*a*d^2 + 105*(35*b*e^2*x^4 + 42*b*d*e*x^2 + 15*b*d^2)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c *x)) + (2*(360*b*c^7*d^2 - 1176*b*c^5*d*e + 1225*b*c^3*e^2)*x^7 - (360*b*c ^5*d^2 - 1176*b*c^3*d*e + 1225*b*c*e^2)*x^5 - 225*b*c*d^2*x + 18*(15*b*c^3 *d^2 - 49*b*c*d*e)*x^3)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/x^7
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{8}}\, dx \] Input:
integrate((e*x**2+d)**2*(a+b*acsch(c*x))/x**8,x)
Output:
Integral((a + b*acsch(c*x))*(d + e*x**2)**2/x**8, x)
Time = 0.03 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.93 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\frac {1}{245} \, b d^{2} {\left (\frac {5 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {7}{2}} - 21 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} + 35 \, c^{8} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 35 \, c^{8} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {35 \, \operatorname {arcsch}\left (c x\right )}{x^{7}}\right )} + \frac {2}{75} \, b d e {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {15 \, \operatorname {arcsch}\left (c x\right )}{x^{5}}\right )} + \frac {1}{9} \, b e^{2} {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} - \frac {a e^{2}}{3 \, x^{3}} - \frac {2 \, a d e}{5 \, x^{5}} - \frac {a d^{2}}{7 \, x^{7}} \] Input:
integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^8,x, algorithm="maxima")
Output:
1/245*b*d^2*((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 *arccsch(c*x)/x^7) + 2/75*b*d*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) + 1/9*b*e^2*((c^4*(1/(c^2*x^2) + 1)^(3/2) - 3*c^4*sqrt(1/(c^2*x^2) + 1))/c - 3*arccsch(c*x)/x^3) - 1/3*a*e^2/x^3 - 2/5*a*d*e/x^5 - 1/7*a*d^2/x ^7
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{8}} \,d x } \] Input:
integrate((e*x^2+d)^2*(a+b*arccsch(c*x))/x^8,x, algorithm="giac")
Output:
integrate((e*x^2 + d)^2*(b*arccsch(c*x) + a)/x^8, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^8} \,d x \] Input:
int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^8,x)
Output:
int(((d + e*x^2)^2*(a + b*asinh(1/(c*x))))/x^8, x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^8} \, dx=\frac {105 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{8}}d x \right ) b \,d^{2} x^{7}+210 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{6}}d x \right ) b d e \,x^{7}+105 \left (\int \frac {\mathit {acsch} \left (c x \right )}{x^{4}}d x \right ) b \,e^{2} x^{7}-15 a \,d^{2}-42 a d e \,x^{2}-35 a \,e^{2} x^{4}}{105 x^{7}} \] Input:
int((e*x^2+d)^2*(a+b*acsch(c*x))/x^8,x)
Output:
(105*int(acsch(c*x)/x**8,x)*b*d**2*x**7 + 210*int(acsch(c*x)/x**6,x)*b*d*e *x**7 + 105*int(acsch(c*x)/x**4,x)*b*e**2*x**7 - 15*a*d**2 - 42*a*d*e*x**2 - 35*a*e**2*x**4)/(105*x**7)