Integrand size = 19, antiderivative size = 96 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{c} \]
-d*(a+b*arcsech(c*x))/x+e*x*(a+b*arcsech(c*x))+b*e*arcsin(c*x)*(1/(c*x+1)) ^(1/2)*(c*x+1)^(1/2)/c+b*d*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1 /2)/x
Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.32 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=-\frac {a d}{x}+a e x+b d \left (c+\frac {1}{x}\right ) \sqrt {\frac {1-c x}{1+c x}}-\frac {b d \text {sech}^{-1}(c x)}{x}+b e x \text {sech}^{-1}(c x)+\frac {2 b e \sqrt {\frac {1-c x}{1+c x}} \sqrt {1-c^2 x^2} \arctan \left (\frac {\sqrt {1-c^2 x^2}}{1-c x}\right )}{c-c^2 x} \]
-((a*d)/x) + a*e*x + b*d*(c + x^(-1))*Sqrt[(1 - c*x)/(1 + c*x)] - (b*d*Arc Sech[c*x])/x + b*e*x*ArcSech[c*x] + (2*b*e*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[ 1 - c^2*x^2]*ArcTan[Sqrt[1 - c^2*x^2]/(1 - c*x)])/(c - c^2*x)
Time = 0.27 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6855, 25, 358, 223}
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.
\(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 6855 |
\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {d-e x^2}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {d-e x^2}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 358 |
\(\displaystyle -b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-e \int \frac {1}{\sqrt {1-c^2 x^2}}dx-\frac {d \sqrt {1-c^2 x^2}}{x}\right )-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{x}+e x \left (a+b \text {sech}^{-1}(c x)\right )-b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {e \arcsin (c x)}{c}-\frac {d \sqrt {1-c^2 x^2}}{x}\right )\) |
-((d*(a + b*ArcSech[c*x]))/x) + e*x*(a + b*ArcSech[c*x]) - b*Sqrt[(1 + c*x )^(-1)]*Sqrt[1 + c*x]*(-((d*Sqrt[1 - c^2*x^2])/x) - (e*ArcSin[c*x])/c)
3.1.91.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
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] + S imp[d/e^2 Int[(e*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e , m, p}, x] && NeQ[b*c - a*d, 0] && EqQ[Simplify[m + 2*p + 3], 0] && NeQ[m, -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 = 0.35 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17
method | result | size |
parts | \(a \left (e x -\frac {d}{x}\right )+b c \left (\frac {\operatorname {arcsech}\left (c x \right ) e x}{c}-\frac {\operatorname {arcsech}\left (c x \right ) d}{x c}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\sqrt {-c^{2} x^{2}+1}\, c^{2} d +\arcsin \left (c x \right ) e c x \right )}{c^{2} \sqrt {-c^{2} x^{2}+1}}\right )\) | \(112\) |
derivativedivides | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (c \,\operatorname {arcsech}\left (c x \right ) e x -\frac {\operatorname {arcsech}\left (c x \right ) d c}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\sqrt {-c^{2} x^{2}+1}\, c^{2} d +\arcsin \left (c x \right ) e c x \right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(114\) |
default | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (c \,\operatorname {arcsech}\left (c x \right ) e x -\frac {\operatorname {arcsech}\left (c x \right ) d c}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (\sqrt {-c^{2} x^{2}+1}\, c^{2} d +\arcsin \left (c x \right ) e c x \right )}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(114\) |
a*(e*x-d/x)+b*c*(1/c*arcsech(c*x)*e*x-arcsech(c*x)*d/x/c+1/c^2*(-(c*x-1)/c /x)^(1/2)*((c*x+1)/c/x)^(1/2)*((-c^2*x^2+1)^(1/2)*c^2*d+arcsin(c*x)*e*c*x) /(-c^2*x^2+1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.90 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\frac {b c^{2} d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + a c e x^{2} - 2 \, b e x \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - a c d + {\left (b c d - b c e\right )} x \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{c x} \]
(b*c^2*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + a*c*e*x^2 - 2*b*e*x*arctan((c* x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) - a*c*d + (b*c*d - b*c*e)*x*l og((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/x) + (b*c*e*x^2 - b*c*d + (b*c *d - b*c*e)*x)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)))/(c*x)
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{2}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx={\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b e}{c} - \frac {a d}{x} \]
(c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*b*d + a*e*x + (c*x*arcsech(c*x) - arctan(sqrt(1/(c^2*x^2) - 1)))*b*e/c - a*d/x
\[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
Time = 4.40 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.02 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=a\,e\,x-\frac {a\,d}{x}+b\,c\,d\,\left (\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}-\frac {\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{c\,x}\right )+\frac {b\,e\,\mathrm {atan}\left (\frac {1}{\sqrt {\frac {1}{c\,x}-1}\,\sqrt {\frac {1}{c\,x}+1}}\right )}{c}+b\,e\,x\,\mathrm {acosh}\left (\frac {1}{c\,x}\right ) \]