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

Optimal. Leaf size=149 \[ \frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}} \]

[Out]

(-2*(a + b*ArcCsch[c*x]))/(e*Sqrt[d + e*x]) + (4*b*Sqrt[(Sqrt[-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*Sqrt[1 + 1/(c^2*x
^2)]*x*Sqrt[d + e*x])

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Rubi [A]  time = 0.284051, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6290, 1574, 933, 168, 538, 537} \[ \frac{4 b \sqrt{c^2 x^2+1} \sqrt{\frac{\sqrt{-c^2} (d+e x)}{\sqrt{-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac{\sqrt{1-\sqrt{-c^2} x}}{\sqrt{2}}\right )|\frac{2 e}{\sqrt{-c^2} d+e}\right )}{c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{d+e x}}-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*ArcCsch[c*x]))/(e*Sqrt[d + e*x]) + (4*b*Sqrt[(Sqrt[-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*Sqrt[1 + 1/(c^2*x
^2)]*x*Sqrt[d + e*x])

Rule 6290

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

Rule 1574

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Dist[(x^(2*n*Fr
acPart[p])*(a + c/x^(2*n))^FracPart[p])/(c + a*x^(2*n))^FracPart[p], Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + a*x^
(2*n))^p, x], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[mn2, -2*n] &&  !IntegerQ[p] &&  !IntegerQ[q] &&
PosQ[n]

Rule 933

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[-(c
/a), 2]}, Dist[Sqrt[1 + (c*x^2)/a]/Sqrt[a + c*x^2], 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{csch}^{-1}(c x)}{(d+e x)^{3/2}} \, dx &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{(2 b) \int \frac{1}{\sqrt{1+\frac{1}{c^2 x^2}} x^2 \sqrt{d+e x}} \, dx}{c e}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{\left (2 b \sqrt{\frac{1}{c^2}+x^2}\right ) \int \frac{1}{x \sqrt{d+e x} \sqrt{\frac{1}{c^2}+x^2}} \, dx}{c e \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}}-\frac{\left (2 b \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 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{\left (4 b \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 \sqrt{1+\frac{1}{c^2 x^2}} x}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{\left (4 b \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 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ &=-\frac{2 \left (a+b \text{csch}^{-1}(c x)\right )}{e \sqrt{d+e x}}+\frac{4 b \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 \sqrt{1+\frac{1}{c^2 x^2}} x \sqrt{d+e x}}\\ \end{align*}

Mathematica [C]  time = 0.655283, size = 166, normalized size = 1.11 \[ \frac{-2 e \left (c^2 x^2+1\right ) \left (a+b \text{csch}^{-1}(c x)\right )+2 b c x \sqrt{\frac{2}{c^2 x^2}+2} \sqrt{1+i c x} (e+i c d) \sqrt{\frac{c e (c x+i) (d+e x)}{(e+i c d)^2}} \Pi \left (\frac{i c d}{e}+1;\sin ^{-1}\left (\sqrt{-\frac{e (c x+i)}{c d-i e}}\right )|\frac{i c d+e}{2 e}\right )}{e^2 \left (c^2 x^2+1\right ) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.299, size = 328, normalized size = 2.2 \begin{align*} 2\,{\frac{1}{e} \left ( -{\frac{a}{\sqrt{ex+d}}}+b \left ( -{\frac{{\rm arccsch} \left (cx\right )}{\sqrt{ex+d}}}+2\,{\frac{1}{cdx}\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}}}}{\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 ){\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((a+b*arccsch(c*x))/(e*x+d)^(3/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x + d)*(b*arccsch(c*x) + a)/(e^2*x^2 + 2*d*e*x + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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