Integrand size = 21, antiderivative size = 177 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{x}-\frac {b e^2 x \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{6 c^2}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {b e \left (12 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \arcsin (c x)}{6 c^3} \]
-d^2*(a+b*arcsech(c*x))/x+2*d*e*x*(a+b*arcsech(c*x))+1/3*e^2*x^3*(a+b*arcs ech(c*x))+1/6*b*e*(12*c^2*d+e)*arcsin(c*x)*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2) /c^3+b*d^2*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/x-1/6*b*e^2* x*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^2
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\frac {-b c \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (-6 c^2 d^2+e^2 x^2\right )+2 a c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right )+2 b c^3 \left (-3 d^2+6 d e x^2+e^2 x^4\right ) \text {sech}^{-1}(c x)+i b e \left (12 c^2 d+e\right ) x \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{6 c^3 x} \]
(-(b*c*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(-6*c^2*d^2 + e^2*x^2)) + 2*a*c ^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4) + 2*b*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4) *ArcSech[c*x] + I*b*e*(12*c^2*d + e)*x*Log[(-2*I)*c*x + 2*Sqrt[(1 - c*x)/( 1 + c*x)]*(1 + c*x)])/(6*c^3*x)
Time = 0.37 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6855, 27, 1588, 27, 299, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {-e^2 x^4-6 d e x^2+3 d^2}{3 x^2 \sqrt {1-c^2 x^2}}dx-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {-e^2 x^4-6 d e x^2+3 d^2}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 1588 |
| \(\displaystyle -\frac {1}{3} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\int \frac {e \left (e x^2+6 d\right )}{\sqrt {1-c^2 x^2}}dx-\frac {3 d^2 \sqrt {1-c^2 x^2}}{x}\right )-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-e \int \frac {e x^2+6 d}{\sqrt {1-c^2 x^2}}dx-\frac {3 d^2 \sqrt {1-c^2 x^2}}{x}\right )-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle -\frac {1}{3} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-e \left (\frac {\left (12 c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {e x \sqrt {1-c^2 x^2}}{2 c^2}\right )-\frac {3 d^2 \sqrt {1-c^2 x^2}}{x}\right )-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {1}{3} b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-e \left (\frac {\arcsin (c x) \left (12 c^2 d+e\right )}{2 c^3}-\frac {e x \sqrt {1-c^2 x^2}}{2 c^2}\right )-\frac {3 d^2 \sqrt {1-c^2 x^2}}{x}\right )\) |
-((d^2*(a + b*ArcSech[c*x]))/x) + 2*d*e*x*(a + b*ArcSech[c*x]) + (e^2*x^3* (a + b*ArcSech[c*x]))/3 - (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*((-3*d^2*S qrt[1 - c^2*x^2])/x - e*(-1/2*(e*x*Sqrt[1 - c^2*x^2])/c^2 + ((12*c^2*d + e )*ArcSin[c*x])/(2*c^3))))/3
3.2.2.3.1 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[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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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_.) + 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 = 0.62 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.04
| method | result | size |
| parts | \(a \left (\frac {e^{2} x^{3}}{3}+2 d e x -\frac {d^{2}}{x}\right )+b c \left (\frac {\operatorname {arcsech}\left (c x \right ) e^{2} x^{3}}{3 c}+\frac {2 \,\operatorname {arcsech}\left (c x \right ) d e x}{c}-\frac {\operatorname {arcsech}\left (c x \right ) d^{2}}{x c}-\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (-6 \sqrt {-c^{2} x^{2}+1}\, c^{4} d^{2}-12 \arcsin \left (c x \right ) c^{3} d e x +e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\arcsin \left (c x \right ) e^{2} c x \right )}{6 c^{4} \sqrt {-c^{2} x^{2}+1}}\right )\) | \(184\) |
| derivativedivides | \(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arcsech}\left (c x \right ) c^{3} d e x +\frac {\operatorname {arcsech}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\operatorname {arcsech}\left (c x \right ) c^{3} d^{2}}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (6 \sqrt {-c^{2} x^{2}+1}\, c^{4} d^{2}+12 \arcsin \left (c x \right ) c^{3} d e x -e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\arcsin \left (c x \right ) e^{2} c x \right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}\right )\) | \(197\) |
| default | \(c \left (\frac {a \left (2 c^{3} d e x +\frac {e^{2} c^{3} x^{3}}{3}-\frac {c^{3} d^{2}}{x}\right )}{c^{4}}+\frac {b \left (2 \,\operatorname {arcsech}\left (c x \right ) c^{3} d e x +\frac {\operatorname {arcsech}\left (c x \right ) e^{2} c^{3} x^{3}}{3}-\frac {\operatorname {arcsech}\left (c x \right ) c^{3} d^{2}}{x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (6 \sqrt {-c^{2} x^{2}+1}\, c^{4} d^{2}+12 \arcsin \left (c x \right ) c^{3} d e x -e^{2} \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\arcsin \left (c x \right ) e^{2} c x \right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{4}}\right )\) | \(197\) |
a*(1/3*e^2*x^3+2*d*e*x-d^2/x)+b*c*(1/3/c*arcsech(c*x)*e^2*x^3+2/c*arcsech( c*x)*d*e*x-arcsech(c*x)*d^2/x/c-1/6/c^4*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x) ^(1/2)*(-6*(-c^2*x^2+1)^(1/2)*c^4*d^2-12*arcsin(c*x)*c^3*d*e*x+e^2*(-c^2*x ^2+1)^(1/2)*c^2*x^2-arcsin(c*x)*e^2*c*x)/(-c^2*x^2+1)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (107) = 214\).
Time = 0.32 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.62 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\frac {2 \, a c^{3} e^{2} x^{4} + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} - 2 \, {\left (12 \, b c^{2} d e + b e^{2}\right )} x \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 2 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} d^{2} x - b c^{2} e^{2} x^{3}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{6 \, c^{3} x} \]
1/6*(2*a*c^3*e^2*x^4 + 12*a*c^3*d*e*x^2 - 6*a*c^3*d^2 - 2*(12*b*c^2*d*e + b*e^2)*x*arctan((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - 1)/(c*x)) + 2*(3*b*c ^3*d^2 - 6*b*c^3*d*e - b*c^3*e^2)*x*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2) ) - 1)/x) + 2*(b*c^3*e^2*x^4 + 6*b*c^3*d*e*x^2 - 3*b*c^3*d^2 + (3*b*c^3*d^ 2 - 6*b*c^3*d*e - b*c^3*e^2)*x)*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) + (6*b*c^4*d^2*x - b*c^2*e^2*x^3)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) )/(c^3*x)
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{2}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.86 \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\frac {1}{3} \, a e^{2} x^{3} + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b e^{2} + 2 \, a d e x + \frac {2 \, {\left (c x \operatorname {arsech}\left (c x\right ) - \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \]
1/3*a*e^2*x^3 + (c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*b*d^2 + 1/6*(2* x^3*arcsech(c*x) - (sqrt(1/(c^2*x^2) - 1)/(c^2*(1/(c^2*x^2) - 1) + c^2) + arctan(sqrt(1/(c^2*x^2) - 1))/c^2)/c)*b*e^2 + 2*a*d*e*x + 2*(c*x*arcsech(c *x) - arctan(sqrt(1/(c^2*x^2) - 1)))*b*d*e/c - a*d^2/x
\[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^2} \,d x \]