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

Optimal. Leaf size=321 \[ \frac{2 b e x \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

[Out]

(b*c^3*x^2*Sqrt[d + e*x^2])/(d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2
])/(d^2*Sqrt[-(c^2*x^2)]) - (a + b*ArcCsch[c*x])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*ArcCsch[c*x]))/(d^2*Sqr
t[d + e*x^2]) - (b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1
- c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (2*b*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d
)])/(d^3*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])

________________________________________________________________________________________

Rubi [A]  time = 0.319709, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {271, 191, 6302, 12, 583, 531, 418, 492, 411} \[ -\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}+\frac{2 b e x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^3 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c^3*x^2*Sqrt[d + e*x^2])/(d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2
])/(d^2*Sqrt[-(c^2*x^2)]) - (a + b*ArcCsch[c*x])/(d*x*Sqrt[d + e*x^2]) - (2*e*x*(a + b*ArcCsch[c*x]))/(d^2*Sqr
t[d + e*x^2]) - (b*c^2*x*Sqrt[d + e*x^2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(d^2*Sqrt[-(c^2*x^2)]*Sqrt[-1
- c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))]) + (2*b*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d
)])/(d^3*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^2*x^2))])

Rule 271

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 6302

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u
= IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCsch[c*x], u, x] - Dist[(b*c*x)/Sqrt[-(c^2*x^2)], Int[Simp
lifyIntegrand[u/(x*Sqrt[-1 - c^2*x^2]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] &&
!(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) || (IGtQ[(m + 1)/2, 0] &&  !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0]))
|| (ILtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))

Rule 12

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

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 418

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

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{a+b \text{csch}^{-1}(c x)}{x^2 \left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-d-2 e x^2}{d^2 x^2 \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-d-2 e x^2}{x^2 \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^2 \sqrt{-c^2 x^2}}\\ &=\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{(b c x) \int \frac{-2 d e-c^2 d e x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^3 \sqrt{-c^2 x^2}}\\ &=\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{(2 b c e x) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^2 \sqrt{-c^2 x^2}}+\frac{\left (b c^3 e x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{d^2 \sqrt{-c^2 x^2}}\\ &=\frac{b c^3 x^2 \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}+\frac{2 b e x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^3 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac{\left (b c^3 x\right ) \int \frac{\sqrt{d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt{-c^2 x^2}}\\ &=\frac{b c^3 x^2 \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{d^2 \sqrt{-c^2 x^2}}-\frac{a+b \text{csch}^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{2 e x \left (a+b \text{csch}^{-1}(c x)\right )}{d^2 \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}+\frac{2 b e x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^3 \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2} \sqrt{\frac{d+e x^2}{d \left (1+c^2 x^2\right )}}}\\ \end{align*}

Mathematica [C]  time = 0.435116, size = 201, normalized size = 0.63 \[ \frac{-a \left (d+2 e x^2\right )+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (d+e x^2\right )-b \text{csch}^{-1}(c x) \left (d+2 e x^2\right )}{d^2 x \sqrt{d+e x^2}}+\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left (\left (2 e-c^2 d\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right ),\frac{e}{c^2 d}\right )+c^2 d E\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right )|\frac{e}{c^2 d}\right )\right )}{\sqrt{c^2} d^2 \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.455, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{2}} \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{e x^{2} + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{e^{2} x^{6} + 2 \, d e x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arccsch(c*x) + a)/(e^2*x^6 + 2*d*e*x^4 + d^2*x^2), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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