3.1.83 \(\int \frac {a+b \text {sech}^{-1}(c x)}{\sqrt {d+e x}} \, dx\) [83]

Optimal. Leaf size=187 \[ \frac {2 \sqrt {d+e x} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {4 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}} \]

[Out]

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)

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Rubi [A]
time = 0.17, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6423, 958, 733, 430, 946, 174, 552, 551} \begin {gather*} \frac {2 \sqrt {d+e x} \left (a+b \text {sech}^{-1}(c x)\right )}{e}-\frac {4 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\text {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}{c x+1}} \sqrt {c x+1} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(a + b*ArcSech[c*x]))/e - (4*b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*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]) - (4*b*d*Sqrt[(1 + c*x)^(-1)]*
Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(e*
Sqrt[d + e*x])

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-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]

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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]

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 946

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-c/
a, 2]}, Dist[1/Sqrt[a], Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, c, d
, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && GtQ[a, 0]

Rule 958

Int[Sqrt[(f_.) + (g_.)*(x_)]/(((d_.) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[g/e, Int[1/(S
qrt[f + g*x]*Sqrt[a + c*x^2]), x], x] + Dist[(e*f - d*g)/e, Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]), x
], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0]

Rule 6423

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a +
b*ArcSech[c*x])/(e*(m + 1))), x] + Dist[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]

Rubi steps

\begin {align*} \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 {\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \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}+\left (2 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx+\frac {\left (2 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {d+e 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}+\frac {\left (2 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x \sqrt {1-c x} \sqrt {1+c x} \sqrt {d+e x}} \, dx}{e}-\frac {\left (4 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {-\frac {c^2 (d+e x)}{-c^2 d-c e}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 c e x^2}{-c^2 d-c e}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{c \sqrt {d+e x}}\\ &=\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}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c x}\right )}{e}\\ &=\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}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {\frac {c (d+e x)}{c d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{e \sqrt {d+e x}}\\ &=\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}} F\left (\sin ^{-1}\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}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{e \sqrt {d+e x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 14.18, size = 1707, normalized size = 9.13 \begin {gather*} \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 ) F\left (\text {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}} F\left (i \sinh ^{-1}\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 (\Pi \left (\frac {i \sqrt {-c d-e}-\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}};\text {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 )-\Pi \left (\frac {-i \sqrt {-c d-e}+\sqrt {c d-e}}{\sqrt {-c d-e}-i \sqrt {c d-e}};\text {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 )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcSech[c*x])/Sqrt[d + e*x],x]

[Out]

(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))*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] + (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)
]*EllipticF[I*ArcSinh[Sqrt[(1 - c*x)/(1 + c*x)]], (c*d - e)/(c*d + e)] + (2*I)*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 + Sqr
t[(1 - c*x)/(1 + c*x)]))]*(1 + (1 - c*x)/(1 + c*x))*(EllipticPi[(I*Sqrt[-(c*d) - e] - Sqrt[c*d - e])/(Sqrt[-(c
*d) - e] - I*Sqrt[c*d - e]), 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] - EllipticPi[((-I)*Sqrt[-(c*d) - e] + Sqrt[c*d - e])/(Sqrt[-(c*d
) - e] - I*Sqrt[c*d - e]), 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])))/(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)]))]*(e - (e*(1 - c*x))/(1
+ c*x) + c*d*(1 + (1 - c*x)/(1 + c*x))))

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Maple [A]
time = 0.44, size = 286, normalized size = 1.53

method	result	size
derivativedivides	\(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \mathrm {arcsech}\left (c x \right )-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right )-\EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right )\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{\left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right ) \sqrt {\frac {c}{c d +e}}}\right )}{e}\)	\(286\)
default	\(\frac {2 \sqrt {e x +d}\, a +2 b \left (\sqrt {e x +d}\, \mathrm {arcsech}\left (c x \right )-\frac {2 c \,e^{2} \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c e x}}\, x \sqrt {-\frac {-c \left (e x +d \right )+c d -e}{c e x}}\, \left (\EllipticF \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \sqrt {\frac {c d +e}{c d -e}}\right )-\EllipticPi \left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d +e}}, \frac {c d +e}{c d}, \frac {\sqrt {\frac {c}{c d -e}}}{\sqrt {\frac {c}{c d +e}}}\right )\right ) \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}}{\left (c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}\right ) \sqrt {\frac {c}{c d +e}}}\right )}{e}\)	\(286\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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)))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(%e-c*d>0)', see `assume?` for
more details

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral((b*arcsech(c*x) + a)/sqrt(x*e + d), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{\sqrt {d + e x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))/(e*x+d)**(1/2),x)

[Out]

Integral((a + b*asech(c*x))/sqrt(d + e*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)/sqrt(e*x + d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))/(d + e*x)^(1/2),x)

[Out]

int((a + b*acosh(1/(c*x)))/(d + e*x)^(1/2), x)

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