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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.246547, antiderivative size = 187, normalized size of antiderivative = 1., 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 = {6288, 944, 719, 419, 932, 168, 538, 537} \[ \frac{2 \sqrt{d+e x} \left (a+b \text{sech}^{-1}(c x)\right )}{e}-\frac{4 b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{c \sqrt{d+e x}}-\frac{4 b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{2}}\right )|\frac{2 e}{c d+e}\right )}{e \sqrt{d+e x}} \]

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 6288

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]*Sqrt[1/(1 + c*x)])/(e*(m + 1)), 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]

Rule 944

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 719

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 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 932

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 168

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 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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)
])

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [C]  time = 10.7838, size = 1707, normalized size = 9.13 \[ \frac{2 \sqrt{d+e x} a}{e}+\frac{2 b \sqrt{d+e x} \text{sech}^{-1}(c x)}{e}-\frac{4 i b \sqrt{\frac{c d+\frac{c (1-c x) d}{c x+1}+e-\frac{e (1-c x)}{c x+1}}{\frac{(1-c x) c}{c x+1}+c}} \left (2 c d \sqrt{-\frac{i \left (c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \sqrt{-\frac{i \left (-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \left (\frac{1-c x}{c x+1}+1\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\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 (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \sqrt{\frac{1-c x}{c x+1}+1} \sqrt{\frac{-\frac{(1-c x) e}{c x+1}+e+c d \left (\frac{1-c x}{c x+1}+1\right )}{c d+e}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{1-c x}{c x+1}}\right ),\frac{c d-e}{c d+e}\right )+2 i c d \sqrt{-\frac{i \left (c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}-e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (-i c d+i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \sqrt{-\frac{i \left (-c \sqrt{\frac{1-c x}{c x+1}} d+\sqrt{-c d-e} \sqrt{c d-e}+e \sqrt{\frac{1-c x}{c x+1}}\right )}{\left (i c d-i e+\sqrt{-c d-e} \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \left (\frac{1-c x}{c x+1}+1\right ) \left (\Pi \left (\frac{i \sqrt{-c d-e}-\sqrt{c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\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{\sqrt{c d-e}-i \sqrt{-c d-e}}{\sqrt{-c d-e}-i \sqrt{c d-e}};\sin ^{-1}\left (\sqrt{\frac{\left (\sqrt{-c d-e}-i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\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 (\sqrt{\frac{1-c x}{c x+1}}+i\right )}{\left (\sqrt{-c d-e}+i \sqrt{c d-e}\right ) \left (\sqrt{\frac{1-c x}{c x+1}}-i\right )}} \left (-\frac{(1-c x) e}{c x+1}+e+c d \left (\frac{1-c x}{c x+1}+1\right )\right )} \]

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 = 0.275, size = 288, normalized size = 1.5 \begin{align*} 2\,{\frac{1}{e} \left ( a\sqrt{ex+d}+b \left ( \sqrt{ex+d}{\rm arcsech} \left (cx\right )-2\,{\frac{c{e}^{2}x}{ \left ( ex+d \right ) ^{2}{c}^{2}-2\, \left ( ex+d \right ){c}^{2}d+{c}^{2}{d}^{2}-{e}^{2}}\sqrt{-{\frac{ \left ( ex+d \right ) c-cd-e}{cxe}}}\sqrt{{\frac{ \left ( ex+d \right ) c-cd+e}{cxe}}} \left ({\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{cd+e}}},\sqrt{{\frac{cd+e}{cd-e}}} \right ) -{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{c}{cd+e}}},{\frac{cd+e}{cd}},{\sqrt{{\frac{c}{cd-e}}}{\frac{1}{\sqrt{{\frac{c}{cd+e}}}}}} \right ) \right ) \sqrt{-{\frac{ \left ( ex+d \right ) c-cd+e}{cd-e}}}\sqrt{-{\frac{ \left ( ex+d \right ) c-cd-e}{cd+e}}}{\frac{1}{\sqrt{{\frac{c}{cd+e}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/e*(a*(e*x+d)^(1/2)+b*((e*x+d)^(1/2)*arcsech(c*x)-2*c*e^2*(-((e*x+d)*c-c*d-e)/c/x/e)^(1/2)*x*(((e*x+d)*c-c*d+
e)/c/x/e)^(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)))*(-((e*x+d)*c-c*d+e)/(c*d-e))^(1/2)*(-((e*
x+d)*c-c*d-e)/(c*d+e))^(1/2)/(c/(c*d+e))^(1/2)/((e*x+d)^2*c^2-2*(e*x+d)*c^2*d+c^2*d^2-e^2)))

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

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arsech}\left (c x\right ) + a}{\sqrt{e x + d}}\,{d x} \end{align*}

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)