Integrand size = 18, antiderivative size = 187 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {4 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}} \]
2*(a+b*arcsech(c*x))*(e*x+d)^(1/2)/e-4*b*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1 /2),2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(c*(e*x+d)/ (c*d+e))^(1/2)/c/(e*x+d)^(1/2)-4*b*d*EllipticPi(1/2*(-c*x+1)^(1/2)*2^(1/2) ,2,2^(1/2)*(e/(c*d+e))^(1/2))*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*(c*(e*x+d)/( c*d+e))^(1/2)/e/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 18.89 (sec) , antiderivative size = 1707, normalized size of antiderivative = 9.13 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\frac {2 a \sqrt {d+e x}}{e}+\frac {2 b \sqrt {d+e x} \text {sech}^{-1}(c x)}{e}-\frac {4 i b \sqrt {\frac {c d+e+\frac {c d (1-c x)}{1+c x}-\frac {e (1-c x)}{1+c x}}{c+\frac {c (1-c x)}{1+c x}}} \left (2 c d \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}+c d \sqrt {\frac {1-c x}{1+c x}}-e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (-i c d+\sqrt {-c d-e} \sqrt {c d-e}+i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}-c d \sqrt {\frac {1-c x}{1+c x}}+e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (i c d+\sqrt {-c d-e} \sqrt {c d-e}-i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \left (1+\frac {1-c x}{1+c x}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}}\right ),\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )+(c d-e) \sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \sqrt {1+\frac {1-c x}{1+c x}} \sqrt {\frac {e-\frac {e (1-c x)}{1+c x}+c d \left (1+\frac {1-c x}{1+c x}\right )}{c d+e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {1-c x}{1+c x}}\right ),\frac {c d-e}{c d+e}\right )+2 i c d \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}+c d \sqrt {\frac {1-c x}{1+c x}}-e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (-i c d+\sqrt {-c d-e} \sqrt {c d-e}+i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \sqrt {-\frac {i \left (\sqrt {-c d-e} \sqrt {c d-e}-c d \sqrt {\frac {1-c x}{1+c x}}+e \sqrt {\frac {1-c x}{1+c x}}\right )}{\left (i c d+\sqrt {-c d-e} \sqrt {c d-e}-i e\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \left (1+\frac {1-c x}{1+c x}\right ) \left (\operatorname {EllipticPi}\left (\frac {i \sqrt {-c d-e}-\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}},\arcsin \left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}}\right ),\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )-\operatorname {EllipticPi}\left (\frac {-i \sqrt {-c d-e}+\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}},\arcsin \left (\sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}}\right ),\frac {\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right )^2}{\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right )^2}\right )\right )\right )}{e \sqrt {\frac {\left (\sqrt {-c d-e}-i \sqrt {c d-e}\right ) \left (i+\sqrt {\frac {1-c x}{1+c x}}\right )}{\left (\sqrt {-c d-e}+i \sqrt {c d-e}\right ) \left (-i+\sqrt {\frac {1-c x}{1+c x}}\right )}} \left (e-\frac {e (1-c x)}{1+c x}+c d \left (1+\frac {1-c x}{1+c x}\right )\right )} \]
(2*a*Sqrt[d + e*x])/e + (2*b*Sqrt[d + e*x]*ArcSech[c*x])/e - ((4*I)*b*Sqrt [(c*d + e + (c*d*(1 - c*x))/(1 + c*x) - (e*(1 - c*x))/(1 + c*x))/(c + (c*( 1 - c*x))/(1 + c*x))]*(2*c*d*Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]* Sqrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*x)]))/((I*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e] - I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*Ellipt icF[ArcSin[Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])*(I + Sqrt[(1 - c*x)/ (1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]], (Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])^2/(Sqrt[-(c*d) - e] - I *Sqrt[c*d - e])^2] + (c*d - e)*Sqrt[((Sqrt[-(c*d) - e] - I*Sqrt[c*d - e])* (I + Sqrt[(1 - c*x)/(1 + c*x)]))/((Sqrt[-(c*d) - e] + I*Sqrt[c*d - e])*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[1 + (1 - c*x)/(1 + c*x)]*Sqrt[(e - (e *(1 - c*x))/(1 + c*x) + c*d*(1 + (1 - c*x)/(1 + c*x)))/(c*d + e)]*Elliptic F[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] + (2*I)*c*d*S qrt[((-I)*(Sqrt[-(c*d) - e]*Sqrt[c*d - e] + c*d*Sqrt[(1 - c*x)/(1 + c*x)] - e*Sqrt[(1 - c*x)/(1 + c*x)]))/(((-I)*c*d + Sqrt[-(c*d) - e]*Sqrt[c*d - e ] + I*e)*(-I + Sqrt[(1 - c*x)/(1 + c*x)]))]*Sqrt[((-I)*(Sqrt[-(c*d) - e]*S qrt[c*d - e] - c*d*Sqrt[(1 - c*x)/(1 + c*x)] + e*Sqrt[(1 - c*x)/(1 + c*...
Time = 0.47 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6842, 637, 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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 6842 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx}{e}+\frac {2 \sqrt {d+e x} \left (a+b \text {sech}^{-1}(c x)\right )}{e}\) |
| \(\Big \downarrow \) 637 |
| \(\displaystyle \frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx}{e}+\frac {2 \sqrt {d+e x} \left (a+b \text {sech}^{-1}(c x)\right )}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {d+e x} \left (a+b \text {sech}^{-1}(c x)\right )}{e}+\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )}{e}\) |
(2*Sqrt[d + e*x]*(a + b*ArcSech[c*x]))/e + (2*b*Sqrt[(1 + c*x)^(-1)]*Sqrt[ 1 + c*x]*((-2*e*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c* x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]) - (2*d*Sqrt[(c*(d + e*x)) /(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]) /Sqrt[d + e*x]))/e
3.1.83.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 /2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n + 1/2] && IntegerQ[m]
Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSech[c*x])/(e*(m + 1))), x] + Simp[ b*(Sqrt[1 + c*x]/(e*(m + 1)))*Sqrt[1/(1 + c*x)] Int[(d + e*x)^(m + 1)/(x* Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 11.28 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.53
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \operatorname {arcsech}\left (c x \right )-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right )\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{\left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right ) \sqrt {\frac {c}{c d +e}}}\right )}{e}\) | \(286\) |
| default | \(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \operatorname {arcsech}\left (c x \right )-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right )\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{\left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right ) \sqrt {\frac {c}{c d +e}}}\right )}{e}\) | \(286\) |
| parts | \(\frac {2 a \sqrt {e x +d}}{e}+\frac {2 b \left (\sqrt {e x +d}\, \operatorname {arcsech}\left (c x \right )-\frac {2 c \,e^{2} \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c e x}}\, x \sqrt {\frac {c \left (e x +d \right )-c d +e}{c e x}}\, \left (\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right )-\operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right )\right ) \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}}{\sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(291\) |
2/e*((e*x+d)^(1/2)*a+b*((e*x+d)^(1/2)*arcsech(c*x)-2*c*e^2*((-c*(e*x+d)+c* d+e)/c/e/x)^(1/2)*x*(-(-c*(e*x+d)+c*d-e)/c/e/x)^(1/2)*(EllipticF((e*x+d)^( 1/2)*(c/(c*d+e))^(1/2),((c*d+e)/(c*d-e))^(1/2))-EllipticPi((e*x+d)^(1/2)*( c/(c*d+e))^(1/2),1/c*(c*d+e)/d,(c/(c*d-e))^(1/2)/(c/(c*d+e))^(1/2)))*((-c* (e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)/(c^2*(e*x +d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)/(c/(c*d+e))^(1/2)))
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\sqrt {d + e x}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-e>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{\sqrt {d+e\,x}} \,d x \]