Integrand size = 19, antiderivative size = 309 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\frac {b c d \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}{4 x}+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x)^2}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {1}{4} b c^2 d \text {sech}^{-1}(c x)-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{2 x^2}-\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}+\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )}{\sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}}-e \left (a+b \text {sech}^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {i b e \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 \sqrt {-1+\frac {1}{c x}} \sqrt {1+\frac {1}{c x}}} \]
1/4*b*c^2*d*arcsech(c*x)-1/2*d*(a+b*arcsech(c*x))/x^2-e*(a+b*arcsech(c*x)) *ln(1/x)+1/2*I*b*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)-b*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)+b*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)+1/2*I*b*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*(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2)/x
Time = 0.25 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=-\frac {a d}{2 x^2}+b d \left (\frac {1}{4 x^2}+\frac {c}{4 x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b d \text {sech}^{-1}(c x)}{2 x^2}-\frac {1}{4} b c^2 d \log (x)+a e \log (x)+\frac {1}{4} b c^2 d \log \left (1+\sqrt {\frac {1-c x}{1+c x}}+c x \sqrt {\frac {1-c x}{1+c x}}\right )+\frac {1}{2} b e \left (-\text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \]
-1/2*(a*d)/x^2 + b*d*(1/(4*x^2) + c/(4*x))*Sqrt[(1 - c*x)/(1 + c*x)] - (b* d*ArcSech[c*x])/(2*x^2) - (b*c^2*d*Log[x])/4 + a*e*Log[x] + (b*c^2*d*Log[1 + Sqrt[(1 - c*x)/(1 + c*x)] + c*x*Sqrt[(1 - c*x)/(1 + c*x)]])/4 + (b*e*(- (ArcSech[c*x]*(ArcSech[c*x] + 2*Log[1 + E^(-2*ArcSech[c*x])])) + PolyLog[2 , -E^(-2*ArcSech[c*x])]))/2
Time = 1.12 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 6857 |
| \(\displaystyle -\int \left (\frac {d}{x^2}+e\right ) x \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 6373 |
| \(\displaystyle \frac {b \int \frac {\frac {d}{x^2}+2 e \log \left (\frac {1}{x}\right )}{2 \sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}d\frac {1}{x}}{c}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b \int \frac {\frac {d}{x^2}+2 e \log \left (\frac {1}{x}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}d\frac {1}{x}}{2 c}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {b \int \left (\frac {d}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}} x^2}+\frac {2 e \log \left (\frac {1}{x}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {1+\frac {1}{c x}}}\right )d\frac {1}{x}}{2 c}-\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \text {arccosh}\left (\frac {1}{c x}\right )\right )+\frac {b \left (\frac {1}{2} c^3 d \text {arccosh}\left (\frac {1}{c x}\right )+\frac {i c e \sqrt {1-\frac {1}{c^2 x^2}} \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {i c e \sqrt {1-\frac {1}{c^2 x^2}} \arcsin \left (\frac {1}{c x}\right )^2}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}-\frac {2 c e \sqrt {1-\frac {1}{c^2 x^2}} \arcsin \left (\frac {1}{c x}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {2 c e \sqrt {1-\frac {1}{c^2 x^2}} \log \left (\frac {1}{x}\right ) \arcsin \left (\frac {1}{c x}\right )}{\sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}+\frac {c^2 d \sqrt {\frac {1}{c x}-1} \sqrt {\frac {1}{c x}+1}}{2 x}\right )}{2 c}\) |
-1/2*(d*(a + b*ArcCosh[1/(c*x)]))/x^2 - e*(a + b*ArcCosh[1/(c*x)])*Log[x^( -1)] + (b*((c^2*d*Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)])/(2*x) + (c^3*d*Arc Cosh[1/(c*x)])/2 + (I*c*e*Sqrt[1 - 1/(c^2*x^2)]*ArcSin[1/(c*x)]^2)/(Sqrt[- 1 + 1/(c*x)]*Sqrt[1 + 1/(c*x)]) - (2*c*e*Sqrt[1 - 1/(c^2*x^2)]*ArcSin[1/(c *x)]*Log[1 - E^((2*I)*ArcSin[1/(c*x)])])/(Sqrt[-1 + 1/(c*x)]*Sqrt[1 + 1/(c *x)]) + (2*c*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)]) + (I*c*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)
3.1.99.3.1 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 = 1.05 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.54
| method | result | size |
| parts | \(-\frac {a d}{2 x^{2}}+a e \ln \left (x \right )+\frac {b e \operatorname {arcsech}\left (c x \right )^{2}}{2}+\frac {b \,c^{2} d \,\operatorname {arcsech}\left (c x \right )}{4}+\frac {b c d \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 x}-\frac {b d \,\operatorname {arcsech}\left (c x \right )}{2 x^{2}}-b e \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {b e \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}\) | \(168\) |
| derivativedivides | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {b e \operatorname {arcsech}\left (c x \right )^{2}}{2 c^{2}}+\frac {b d \,\operatorname {arcsech}\left (c x \right )}{4}+\frac {b d \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\operatorname {arcsech}\left (c x \right ) d}{2 c^{2} x^{2}}-\frac {b e \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b e \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2 c^{2}}\right )\) | \(191\) |
| default | \(c^{2} \left (-\frac {a d}{2 c^{2} x^{2}}+\frac {a e \ln \left (c x \right )}{c^{2}}+\frac {b e \operatorname {arcsech}\left (c x \right )^{2}}{2 c^{2}}+\frac {b d \,\operatorname {arcsech}\left (c x \right )}{4}+\frac {b d \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{4 c x}-\frac {b \,\operatorname {arcsech}\left (c x \right ) d}{2 c^{2} x^{2}}-\frac {b e \,\operatorname {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{c^{2}}-\frac {b e \operatorname {polylog}\left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2 c^{2}}\right )\) | \(191\) |
-1/2*a*d/x^2+a*e*ln(x)+1/2*b*e*arcsech(c*x)^2+1/4*b*c^2*d*arcsech(c*x)+1/4 *b*c*d/x*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-1/2*b*d/x^2*arcsech(c*x) -b*e*arcsech(c*x)*ln(1+(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)-1/2*b*e *polylog(2,-(1/c/x+(-1+1/c/x)^(1/2)*(1+1/c/x)^(1/2))^2)
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \]
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
-1/8*b*d*((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*sqrt(1/(c^2*x^2) - 1 ) - 1))/c + 4*arcsech(c*x)/x^2) + b*e*integrate(log(sqrt(1/(c*x) + 1)*sqrt (1/(c*x) - 1) + 1/(c*x))/x, x) + a*e*log(x) - 1/2*a*d/x^2
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]