Integrand size = 20, antiderivative size = 92 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c d \sqrt {d+e x^2}} \]
x*(a+b*arcsech(c*x))/d/(e*x^2+d)^(1/2)+b*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 = 50.61 (sec) , antiderivative size = 334, normalized size of antiderivative = 3.63 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {2 i b \sqrt {\frac {1-c x}{1+c x}} \sqrt {\frac {\left (c \sqrt {d}+i \sqrt {e}\right ) (1+c x)}{\left (c \sqrt {d}-i \sqrt {e}\right ) (-1+c x)}} \left (-i \sqrt {d}+\sqrt {e} x\right ) \sqrt {-\frac {-1+\frac {i \sqrt {e} x}{\sqrt {d}}+c \left (\frac {i \sqrt {d}}{\sqrt {e}}+x\right )}{1-c x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{2-2 c x}}\right ),-\frac {4 i c \sqrt {d} \sqrt {e}}{\left (c \sqrt {d}-i \sqrt {e}\right )^2}\right )}{d \left (c \sqrt {d}+i \sqrt {e}\right ) \sqrt {\frac {1+\frac {i c \sqrt {d}}{\sqrt {e}}-c x+\frac {i \sqrt {e} x}{\sqrt {d}}}{1-c x}} \sqrt {d+e x^2}} \]
(x*(a + b*ArcSech[c*x]))/(d*Sqrt[d + e*x^2]) + ((2*I)*b*Sqrt[(1 - c*x)/(1 + c*x)]*Sqrt[((c*Sqrt[d] + I*Sqrt[e])*(1 + c*x))/((c*Sqrt[d] - I*Sqrt[e])* (-1 + c*x))]*((-I)*Sqrt[d] + Sqrt[e]*x)*Sqrt[-((-1 + (I*Sqrt[e]*x)/Sqrt[d] + c*((I*Sqrt[d])/Sqrt[e] + x))/(1 - c*x))]*EllipticF[ArcSin[Sqrt[(1 + (I* c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(2 - 2*c*x)]], ((-4*I)*c *Sqrt[d]*Sqrt[e])/(c*Sqrt[d] - I*Sqrt[e])^2])/(d*(c*Sqrt[d] + I*Sqrt[e])*S qrt[(1 + (I*c*Sqrt[d])/Sqrt[e] - c*x + (I*Sqrt[e]*x)/Sqrt[d])/(1 - c*x)]*S qrt[d + e*x^2])
Time = 0.28 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6845, 27, 323, 321}
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.
\(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6845 |
\(\displaystyle b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{d \sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {e x^2+d}}dx}{d}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 323 |
\(\displaystyle \frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {e x^2}{d}+1} \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {\frac {e x^2}{d}+1}}dx}{d \sqrt {d+e x^2}}+\frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {x \left (a+b \text {sech}^{-1}(c x)\right )}{d \sqrt {d+e x^2}}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {e x^2}{d}+1} \operatorname {EllipticF}\left (\arcsin (c x),-\frac {e}{c^2 d}\right )}{c d \sqrt {d+e x^2}}\) |
(x*(a + b*ArcSech[c*x]))/(d*Sqrt[d + e*x^2]) + (b*Sqrt[(1 + c*x)^(-1)]*Sqr t[1 + c*x]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(c*d* Sqrt[d + e*x^2])
3.2.66.3.1 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[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_)^2)^(p_.), x_Sym bol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(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]] /; FreeQ[{a, b, c, d, e}, x] && ( IGtQ[p, 0] || ILtQ[p + 1/2, 0])
\[\int \frac {a +b \,\operatorname {arcsech}\left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {3}{2}}}d x\]
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\sqrt {e x^{2} + d} b c d x \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + \sqrt {e x^{2} + d} a c d x + {\left (b e x^{2} + b d\right )} \sqrt {d} F(\arcsin \left (c x\right )\,|\,-\frac {e}{c^{2} d})}{c d^{2} e x^{2} + c d^{3}} \]
(sqrt(e*x^2 + d)*b*c*d*x*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x )) + sqrt(e*x^2 + d)*a*c*d*x + (b*e*x^2 + b*d)*sqrt(d)*elliptic_f(arcsin(c *x), -e/(c^2*d)))/(c*d^2*e*x^2 + c*d^3)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
b*integrate(log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))/(e*x^2 + d) ^(3/2), x) + a*x/(sqrt(e*x^2 + d)*d)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\left (d+e x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (e\,x^2+d\right )}^{3/2}} \,d x \]