3.65 \(\int \frac{x (a+b \text{csch}^{-1}(c x))}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=318 \[ \frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right ),-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{\left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{8 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]

[Out]

(2*d*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2 + (4*b*c*Sqrt[(c^2
*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sq
rt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/((-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (8*b*d*Sqrt[(Sqr
t[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]],
(2*e)/(Sqrt[-c^2]*d + e)])/(c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

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Rubi [A]  time = 1.69131, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.579, Rules used = {43, 6310, 12, 6721, 6742, 719, 419, 932, 168, 538, 537} \[ \frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}-\frac{8 b d \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e^2 x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}+\frac{4 b c \sqrt{c^2 x^2+1} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{\left (-c^2\right )^{3/2} e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

Int[(x*(a + b*ArcCsch[c*x]))/(d + e*x)^(3/2),x]

[Out]

(2*d*(a + b*ArcCsch[c*x]))/(e^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2 + (4*b*c*Sqrt[(c^2
*(d + e*x))/(c^2*d - Sqrt[-c^2]*e)]*Sqrt[1 + c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]], (-2*Sq
rt[-c^2]*e)/(c^2*d - Sqrt[-c^2]*e)])/((-c^2)^(3/2)*e*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x]) - (8*b*d*Sqrt[(Sqr
t[-c^2]*(d + e*x))/(Sqrt[-c^2]*d + e)]*Sqrt[1 + c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - Sqrt[-c^2]*x]/Sqrt[2]],
(2*e)/(Sqrt[-c^2]*d + e)])/(c*e^2*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6310

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide[u, x]}, Dist[a + b*ArcCsch[c*x],
v, x] + Dist[b/c, Int[SimplifyIntegrand[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x
]] /; FreeQ[{a, b, c}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6721

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(a + b*x^n)^FracPart[p])/(x^(n*FracP
art[p])*(1 + a/(x^n*b))^FracPart[p]), Int[u*x^(n*p)*(1 + a/(x^n*b))^p, x], x] /; FreeQ[{a, b, p}, x] &&  !Inte
gerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

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{x \left (a+b \text{csch}^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{b \int \frac{2 (2 d+e x)}{e^2 \sqrt{1+\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{(2 b) \int \frac{2 d+e x}{\sqrt{1+\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c e^2}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{2 d+e x}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \left (\frac{e}{\sqrt{d+e x} \sqrt{1+c^2 x^2}}+\frac{2 d}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}}\right ) \, dx}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (2 b \sqrt{1+c^2 x^2}\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{1+c^2 x^2}} \, dx}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{\left (4 b d \sqrt{1+c^2 x^2}\right ) \int \frac{1}{x \sqrt{1-\sqrt{-c^2} x} \sqrt{1+\sqrt{-c^2} x} \sqrt{d+e x}} \, dx}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}+\frac{\left (4 b \sqrt{-c^2} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{-c^2} e x^2}{c^2 d-\sqrt{-c^2} e}}} \, dx,x,\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{4 b \sqrt{-c^2} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (8 b d \sqrt{1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{d+\frac{e}{\sqrt{-c^2}}-\frac{e x^2}{\sqrt{-c^2}}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{4 b \sqrt{-c^2} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{\left (8 b d \sqrt{1+c^2 x^2} \sqrt{1+\frac{e \left (-1+\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2} \sqrt{1-\frac{e x^2}{\sqrt{-c^2} \left (d+\frac{e}{\sqrt{-c^2}}\right )}}} \, dx,x,\sqrt{1-\sqrt{-c^2} x}\right )}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=\frac{2 d \left (a+b \text{csch}^{-1}(c x)\right )}{e^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a+b \text{csch}^{-1}(c x)\right )}{e^2}+\frac{4 b \sqrt{-c^2} \sqrt{\frac{c^2 (d+e x)}{c^2 d-\sqrt{-c^2} e}} \sqrt{1+c^2 x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|-\frac{2 \sqrt{-c^2} e}{c^2 d-\sqrt{-c^2} e}\right )}{c^3 e \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}-\frac{8 b d \sqrt{1+c^2 x^2} \sqrt{1-\frac{e \left (1-\sqrt{-c^2} x\right )}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e^2 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 1.55801, size = 264, normalized size = 0.83 \[ \frac{2 \left (-\frac{2 i b \sqrt{-\frac{e (c x-i)}{c d+i e}} \sqrt{-\frac{e (c x+i)}{c d-i e}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right ),\frac{c d-i e}{c d+i e}\right )-2 \Pi \left (1-\frac{i e}{c d};i \sinh ^{-1}\left (\sqrt{-\frac{c}{c d-i e}} \sqrt{d+e x}\right )|\frac{c d-i e}{c d+i e}\right )\right )}{c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{-\frac{c}{c d-i e}}}+\frac{a (2 d+e x)}{\sqrt{d+e x}}+\frac{b \text{csch}^{-1}(c x) (2 d+e x)}{\sqrt{d+e x}}\right )}{e^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(a + b*ArcCsch[c*x]))/(d + e*x)^(3/2),x]

[Out]

(2*((a*(2*d + e*x))/Sqrt[d + e*x] + (b*(2*d + e*x)*ArcCsch[c*x])/Sqrt[d + e*x] - ((2*I)*b*Sqrt[-((e*(-I + c*x)
)/(c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]],
(c*d - I*e)/(c*d + I*e)] - 2*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*
d - I*e)/(c*d + I*e)]))/(c*Sqrt[-(c/(c*d - I*e))]*Sqrt[1 + 1/(c^2*x^2)]*x)))/e^2

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Maple [C]  time = 0.323, size = 418, normalized size = 1.3 \begin{align*} 2\,{\frac{1}{{e}^{2}} \left ( a \left ( \sqrt{ex+d}+{\frac{d}{\sqrt{ex+d}}} \right ) +b \left ( \sqrt{ex+d}{\rm arccsch} \left (cx\right )+{\frac{{\rm arccsch} \left (cx\right )d}{\sqrt{ex+d}}}+2\,{\frac{1}{cx}\sqrt{-{\frac{i \left ( ex+d \right ) ce+ \left ( ex+d \right ){c}^{2}d-{c}^{2}{d}^{2}-{e}^{2}}{{c}^{2}{d}^{2}+{e}^{2}}}}\sqrt{{\frac{i \left ( ex+d \right ) ce- \left ( ex+d \right ){c}^{2}d+{c}^{2}{d}^{2}+{e}^{2}}{{c}^{2}{d}^{2}+{e}^{2}}}} \left ({\it EllipticF} \left ( \sqrt{ex+d}\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}},\sqrt{-{\frac{2\,icde-{c}^{2}{d}^{2}+{e}^{2}}{{c}^{2}{d}^{2}+{e}^{2}}}} \right ) -2\,{\it EllipticPi} \left ( \sqrt{ex+d}\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}},{\frac{{c}^{2}{d}^{2}+{e}^{2}}{ \left ( ie+cd \right ) cd}},{\sqrt{-{\frac{ \left ( ie-cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}}{\frac{1}{\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ \left ( ex+d \right ) ^{2}{c}^{2}-2\, \left ( ex+d \right ){c}^{2}d+{c}^{2}{d}^{2}+{e}^{2}}{{c}^{2}{x}^{2}{e}^{2}}}}}}{\frac{1}{\sqrt{{\frac{ \left ( ie+cd \right ) c}{{c}^{2}{d}^{2}+{e}^{2}}}}}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccsch(c*x))/(e*x+d)^(3/2),x)

[Out]

2/e^2*(a*((e*x+d)^(1/2)+d/(e*x+d)^(1/2))+b*((e*x+d)^(1/2)*arccsch(c*x)+arccsch(c*x)*d/(e*x+d)^(1/2)+2/c*(-(I*(
e*x+d)*c*e+(e*x+d)*c^2*d-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*(e*x+d)*c*e-(e*x+d)*c^2*d+c^2*d^2+e^2)/(c^2*d^2
+e^2))^(1/2)*(EllipticF((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2
))^(1/2))-2*EllipticPi((e*x+d)^(1/2)*((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2),1/(I*e+c*d)/c*(c^2*d^2+e^2)/d,(-(I*e-c*
d)*c/(c^2*d^2+e^2))^(1/2)/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)))/(((e*x+d)^2*c^2-2*(e*x+d)*c^2*d+c^2*d^2+e^2)/c^2
/x^2/e^2)^(1/2)/x/((I*e+c*d)*c/(c^2*d^2+e^2))^(1/2)))

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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(x*(a+b*arccsch(c*x))/(e*x+d)^(3/2),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(x*(a+b*arccsch(c*x))/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [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(x*(a+b*acsch(c*x))/(e*x+d)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccsch(c*x))/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)*x/(e*x + d)^(3/2), x)