Integrand size = 21, antiderivative size = 373 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\frac {b c d^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}-\frac {b e^2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}} x}{2 c}+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d^2 \text {sech}^{-1}(c x)-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} e^2 x^2 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {2 b d e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-2 d e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b d e \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \]
1/4*b*c^2*d^2*arcsech(c*x)-1/2*d^2*(a+b*arcsech(c*x))/x^2+1/2*e^2*x^2*(a+b *arcsech(c*x))-2*d*e*(a+b*arcsech(c*x))*ln(1/x)+I*b*d*e*arccsc(c*x)^2*(1-1 /c^2/x^2)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)-2*b*d*e*arccsc(c*x)*ln(1- (I/c/x+(1-1/c^2/x^2)^(1/2))^2)*(1-1/c^2/x^2)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c /x)^(1/2)+2*b*d*e*arccsc(c*x)*ln(1/x)*(1-1/c^2/x^2)^(1/2)/(-1+1/c/x)^(1/2) /(1+1/c/x)^(1/2)+I*b*d*e*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)*(1-1/c^2 /x^2)^(1/2)/(-1+1/c/x)^(1/2)/(1+1/c/x)^(1/2)+1/4*b*c*d^2*(-1+1/c/x)^(1/2)* (1+1/c/x)^(1/2)/x-1/2*b*e^2*x*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/c
Time = 1.11 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\frac {1}{4} \left (-\frac {2 a d^2}{x^2}+2 a e^2 x^2-\frac {2 b e^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c^2}-\frac {2 b d^2 \text {sech}^{-1}(c x)}{x^2}+2 b e^2 x^2 \text {sech}^{-1}(c x)+\frac {b d^2 \sqrt {\frac {1-c x}{1+c x}} \left (\sqrt {1-c x} (1+c x)+2 c^2 x^2 \sqrt {1+c x} \text {arctanh}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{x^2 \sqrt {1-c x}}-4 b d e \text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+8 a d e \log (x)+4 b d e \operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \]
((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*e^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c* x))/c^2 - (2*b*d^2*ArcSech[c*x])/x^2 + 2*b*e^2*x^2*ArcSech[c*x] + (b*d^2*S qrt[(1 - c*x)/(1 + c*x)]*(Sqrt[1 - c*x]*(1 + c*x) + 2*c^2*x^2*Sqrt[1 + c*x ]*ArcTanh[Sqrt[1 - c*x]/Sqrt[1 + c*x]]))/(x^2*Sqrt[1 - c*x]) - 4*b*d*e*Arc Sech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])]) + 8*a*d*e*Log[x] + 4*b*d*e*PolyLog[2, -E^(-2*ArcSech[c*x])])/4
Time = 1.36 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6857, 6373, 27, 7293, 2009}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \left (\frac {d}{x^2}+e\right )^2 x^3 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle \frac {b \int -\frac {-\frac {d^2}{x^2}-4 e \log \left (\frac {1}{x}\right ) d+e^2 x^2}{2 \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}d\frac {1}{x}}{c}-\frac {d^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \int \frac {-\frac {d^2}{x^2}-4 e \log \left (\frac {1}{x}\right ) d+e^2 x^2}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}d\frac {1}{x}}{2 c}-\frac {d^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {b \int \left (-\frac {d^2}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} x^2}-\frac {4 e \log \left (\frac {1}{x}\right ) d}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}+\frac {e^2 x^2}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}\right )d\frac {1}{x}}{2 c}-\frac {d^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-2 d e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )+\frac {1}{2} e^2 x^2 \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )-\frac {b \left (-\frac {1}{2} c^3 d^2 \text {arccosh}\left (\frac {1}{c x}\right )-\frac {2 i c d e \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {2 i c d e \sqrt {1-\frac {1}{c^2 x^2}} \arcsin \left (\frac {1}{c x}\right )^2}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {4 c d e \sqrt {1-\frac {1}{c^2 x^2}} \arcsin \left (\frac {1}{c x}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {4 c d e \sqrt {1-\frac {1}{c^2 x^2}} \log \left (\frac {1}{x}\right ) \arcsin \left (\frac {1}{c x}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {c^2 d^2 \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{2 x}+e^2 x \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}\right )}{2 c}\) |
-1/2*(d^2*(a + b*ArcCosh[1/(c*x)]))/x^2 + (e^2*x^2*(a + b*ArcCosh[1/(c*x)] ))/2 - 2*d*e*(a + b*ArcCosh[1/(c*x)])*Log[x^(-1)] - (b*(-1/2*(c^2*d^2*Sqrt [-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/x + e^2*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c *x)]*x - (c^3*d^2*ArcCosh[1/(c*x)])/2 - ((2*I)*c*d*e*Sqrt[1 - 1/(c^2*x^2)] *ArcSin[1/(c*x)]^2)/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) + (4*c*d*e*Sqrt [1 - 1/(c^2*x^2)]*ArcSin[1/(c*x)]*Log[1 - E^((2*I)*ArcSin[1/(c*x)])])/(Sqr t[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - (4*c*d*e*Sqrt[1 - 1/(c^2*x^2)]*ArcSin [1/(c*x)]*Log[x^(-1)])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - ((2*I)*c*d *e*Sqrt[1 - 1/(c^2*x^2)]*PolyLog[2, E^((2*I)*ArcSin[1/(c*x)])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])))/(2*c)
3.2.9.3.1 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[((a_.) + ArcCosh[(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*ArcCosh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && Le Q[m + p, 0]))
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcCosh[x/c])^n/x ^(m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0 ] && IntegersQ[m, p]
Time = 1.39 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.67
| method | result | size |
| parts | \(a \left (\frac {e^{2} x^{2}}{2}-\frac {d^{2}}{2 x^{2}}+2 d e \ln \left (x \right )\right )+b d e \operatorname {arcsech}\left (c x \right )^{2}+\frac {b \,c^{2} d^{2} \operatorname {arcsech}\left (c x \right )}{4}+\frac {b c \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 x}-\frac {b \,d^{2} \operatorname {arcsech}\left (c x \right )}{2 x^{2}}+\frac {b \,e^{2} x^{2} \operatorname {arcsech}\left (c x \right )}{2}-\frac {b \,e^{2} \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}}{2 c}+\frac {b \,e^{2}}{2 c^{2}}-2 b e d \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-b e d \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\) | \(250\) |
| derivativedivides | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b d e \operatorname {arcsech}\left (c x \right )^{2}}{c^{2}}+\frac {b \,d^{2} \operatorname {arcsech}\left (c x \right )}{4}+\frac {b \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\operatorname {arcsech}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {b \,\operatorname {arcsech}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e^{2} x}{2 c^{3}}+\frac {b \,e^{2}}{2 c^{4}}-\frac {2 b \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) d e \,\operatorname {arcsech}\left (c x \right )}{c^{2}}-\frac {b \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) d e}{c^{2}}\right )\) | \(279\) |
| default | \(c^{2} \left (\frac {a \,x^{2} e^{2}}{2 c^{2}}-\frac {a \,d^{2}}{2 c^{2} x^{2}}+\frac {2 a d e \ln \left (c x \right )}{c^{2}}+\frac {b d e \operatorname {arcsech}\left (c x \right )^{2}}{c^{2}}+\frac {b \,d^{2} \operatorname {arcsech}\left (c x \right )}{4}+\frac {b \,d^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\operatorname {arcsech}\left (c x \right ) d^{2}}{2 c^{2} x^{2}}+\frac {b \,\operatorname {arcsech}\left (c x \right ) x^{2} e^{2}}{2 c^{2}}-\frac {b \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, e^{2} x}{2 c^{3}}+\frac {b \,e^{2}}{2 c^{4}}-\frac {2 b \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) d e \,\operatorname {arcsech}\left (c x \right )}{c^{2}}-\frac {b \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right ) d e}{c^{2}}\right )\) | \(279\) |
a*(1/2*e^2*x^2-1/2*d^2/x^2+2*d*e*ln(x))+b*d*e*arcsech(c*x)^2+1/4*b*c^2*d^2 *arcsech(c*x)+1/4*b*c*d^2/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-1/2*b *d^2/x^2*arcsech(c*x)+1/2*b*e^2*x^2*arcsech(c*x)-1/2*b/c*e^2*(-(c*x-1)/c/x )^(1/2)*x*((c*x+1)/c/x)^(1/2)+1/2*b/c^2*e^2-2*b*e*d*arcsech(c*x)*ln(1+(1/c /x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-b*e*d*polylog(2,-(1/c/x+(-1+1/c/x) ^(1/2)*(1+1/c/x)^(1/2))^2)
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
integral((a*e^2*x^4 + 2*a*d*e*x^2 + a*d^2 + (b*e^2*x^4 + 2*b*d*e*x^2 + b*d ^2)*arcsech(c*x))/x^3, x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
1/2*a*e^2*x^2 - 1/8*b*d^2*((2*c^4*x*sqrt(1/(c^2*x^2) - 1)/(c^2*x^2*(1/(c^2 *x^2) - 1) - 1) - c^3*log(c*x*sqrt(1/(c^2*x^2) - 1) + 1) + c^3*log(c*x*sqr t(1/(c^2*x^2) - 1) - 1))/c + 4*arcsech(c*x)/x^2) + 2*a*d*e*log(x) - 1/2*a* d^2/x^2 + integrate(b*e^2*x*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c *x)) + 2*b*d*e*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/x, x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]