Integrand size = 23, antiderivative size = 221 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{d x}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}+\frac {b c \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{d \sqrt {1+\frac {e x^2}{d}}}-\frac {b \left (c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c d \sqrt {d+e x^2}} \]
-(a+b*arcsech(c*x))*(e*x^2+d)^(1/2)/d/x+b*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)* (-c^2*x^2+1)^(1/2)*(e*x^2+d)^(1/2)/d/x+b*c*EllipticE(c*x,(-e/c^2/d)^(1/2)) *(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(e*x^2+d)^(1/2)/d/(1+e*x^2/d)^(1/2)-b*(c^ 2*d+e)*EllipticF(c*x,(-e/c^2/d)^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(1+ e*x^2/d)^(1/2)/c/d/(e*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 25.41 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.27 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {a \left (\frac {d}{x}+e x\right )+b c \sqrt {\frac {1-c x}{1+c x}} \left (d+e x^2\right )-\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+e x^2\right )}{x}+\frac {b \left (d+e x^2\right ) \text {sech}^{-1}(c x)}{x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} \sqrt {\frac {c \left (\sqrt {d}+i \sqrt {e} x\right )}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \left (i \sqrt {d}+\sqrt {e} x\right ) \left (\left (c \sqrt {d}-i \sqrt {e}\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right )|\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )+2 i \sqrt {e} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {\left (c^2 d+e\right ) (1-c x)}{\left (c \sqrt {d}+i \sqrt {e}\right )^2 (1+c x)}}\right ),\frac {\left (c \sqrt {d}+i \sqrt {e}\right )^2}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )\right )}{\sqrt {-\frac {\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}{\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}} \sqrt {\frac {c \left (\sqrt {d}-i \sqrt {e} x\right )}{\left (c \sqrt {d}-i \sqrt {e}\right ) (1+c x)}}}}{d \sqrt {d+e x^2}} \]
-((a*(d/x + e*x) + b*c*Sqrt[(1 - c*x)/(1 + c*x)]*(d + e*x^2) - (b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(d + e*x^2))/x + (b*(d + e*x^2)*ArcSech[c*x])/ x + (b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[(c*(Sqrt[d] + I*Sqrt[e]*x))/((c*Sqrt [d] + I*Sqrt[e])*(1 + c*x))]*(I*Sqrt[d] + Sqrt[e]*x)*((c*Sqrt[d] - I*Sqrt[ e])*EllipticE[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d] + I*Sqrt[ e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sqrt[e])^2] + (2*I)*Sqrt[e]*EllipticF[I*ArcSinh[Sqrt[((c^2*d + e)*(1 - c*x))/((c*Sqrt[d ] + I*Sqrt[e])^2*(1 + c*x))]], (c*Sqrt[d] + I*Sqrt[e])^2/(c*Sqrt[d] - I*Sq rt[e])^2]))/(Sqrt[-(((c*Sqrt[d] - I*Sqrt[e])*(-1 + c*x))/((c*Sqrt[d] + I*S qrt[e])*(1 + c*x)))]*Sqrt[(c*(Sqrt[d] - I*Sqrt[e]*x))/((c*Sqrt[d] - I*Sqrt [e])*(1 + c*x))]))/(d*Sqrt[d + e*x^2]))
Time = 0.46 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.84, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6855, 25, 27, 377, 27, 326, 323, 321, 330, 327}
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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx\) |
| \(\Big \downarrow \) 6855 |
| \(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int -\frac {\sqrt {e x^2+d}}{d x^2 \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\sqrt {e x^2+d}}{d x^2 \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\sqrt {e x^2+d}}{x^2 \sqrt {1-c^2 x^2}}dx}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (\int \frac {e \sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}dx-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (e \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {e x^2+d}}dx-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 326 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (e \left (\frac {\left (c^2 d+e\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{e}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (e \left (\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{e \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (e \left (\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \int \frac {\sqrt {e x^2+d}}{\sqrt {1-c^2 x^2}}dx}{e}\right )-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (e \left (\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c^2 \sqrt {d+e x^2} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-c^2 x^2}}dx}{e \sqrt {\frac {e x^2}{d}+1}}\right )-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}-\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {\sqrt {d+e x^2} \left (a+b \text {sech}^{-1}(c x)\right )}{d x}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (e \left (\frac {\left (c^2 d+e\right ) \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c e \sqrt {d+e x^2}}-\frac {c \sqrt {d+e x^2} E\left (\arcsin (c x)\left |-\frac {e}{c^2 d}\right .\right )}{e \sqrt {\frac {e x^2}{d}+1}}\right )-\frac {\sqrt {1-c^2 x^2} \sqrt {d+e x^2}}{x}\right )}{d}\) |
-((Sqrt[d + e*x^2]*(a + b*ArcSech[c*x]))/(d*x)) - (b*Sqrt[(1 + c*x)^(-1)]* Sqrt[1 + c*x]*(-((Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])/x) + e*(-((c*Sqrt[d + e*x^2]*EllipticE[ArcSin[c*x], -(e/(c^2*d))])/(e*Sqrt[1 + (e*x^2)/d])) + ( (c^2*d + e)*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*e *Sqrt[d + e*x^2]))))/d
3.2.57.3.1 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]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ b/d Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Simp[(b*c - a*d)/d In t[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && NegQ[b/a]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
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]))
\[\int \frac {a +b \,\operatorname {arcsech}\left (c x \right )}{x^{2} \sqrt {e \,x^{2}+d}}d x\]
Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=-\frac {\sqrt {e x^{2} + d} b c d \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} d x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a c d\right )} \sqrt {e x^{2} + d} - {\left (b c^{4} d x E(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d}) - {\left (b c^{4} d + b e\right )} x F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})\right )} \sqrt {d}}{c d^{2} x} \]
-(sqrt(e*x^2 + d)*b*c*d*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x) ) - (b*c^2*d*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) - a*c*d)*sqrt(e*x^2 + d) - ( b*c^4*d*x*elliptic_e(arcsin(c*x), -e/(c^2*d)) - (b*c^4*d + b*e)*x*elliptic _f(arcsin(c*x), -e/(c^2*d)))*sqrt(d))/(c*d^2*x)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x^{2} \sqrt {d + e x^{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{\sqrt {e x^{2} + d} x^{2}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{x^2 \sqrt {d+e x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x^2\,\sqrt {e\,x^2+d}} \,d x \]