Integrand size = 18, antiderivative size = 105 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}} \]
-2*(a+b*arcsech(c*x))/e/(e*x+d)^(1/2)+4*b*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 = 18.02 (sec) , antiderivative size = 1675, normalized size of antiderivative = 15.95 \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=-\frac {2 a}{e \sqrt {d+e x}}-\frac {2 b \text {sech}^{-1}(c x)}{e \sqrt {d+e x}}+\frac {4 i b \left (2 \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 )+\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 \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 (1+\frac {1-c x}{1+c x}\right ) \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}}}} \]
(-2*a)/(e*Sqrt[d + e*x]) - (2*b*ArcSech[c*x])/(e*Sqrt[d + e*x]) + ((4*I)*b *(2*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))*EllipticF[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] + 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)]))]*Sqr t[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)]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x )]], (c*d - e)/(c*d + e)] + (2*I)*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)...
Time = 0.36 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6842, 632, 186, 413, 412}
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.
| \(\displaystyle \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6842 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{e}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{e \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{e \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {2 \left (a+b \text {sech}^{-1}(c x)\right )}{e \sqrt {d+e x}}\) |
(-2*(a + b*ArcSech[c*x]))/(e*Sqrt[d + e*x]) + (4*b*Sqrt[(1 + c*x)^(-1)]*Sq rt[1 + c*x]*Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(e*Sqrt[d + e/c - (e*(1 - c*x))/c])
3.1.84.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(250\) vs. \(2(96)=192\).
Time = 11.35 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.39
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(251\) |
| default | \(\frac {-\frac {2 a}{\sqrt {e x +d}}+2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(251\) |
| parts | \(-\frac {2 a}{\sqrt {e x +d}\, e}+\frac {2 b \left (-\frac {\operatorname {arcsech}\left (c x \right )}{\sqrt {e x +d}}-\frac {2 c \,e^{2} \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c e x}}\, x \sqrt {\frac {c \left (e x +d \right )-c d +e}{c e x}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right ) \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}}{d \sqrt {\frac {c}{c d +e}}\, \left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right )}\right )}{e}\) | \(255\) |
2/e*(-a/(e*x+d)^(1/2)+b*(-1/(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)*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)/ d/(c/(c*d+e))^(1/2)/(c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+c^2*d^2-e^2)))
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arsech}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {sech}^{-1}(c x)}{(d+e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \]