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

Optimal. Leaf size=294 \[ \frac{b e x \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{d x}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{d \sqrt{-c^2 x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

[Out]

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

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Rubi [A]  time = 0.250901, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {264, 6302, 12, 475, 21, 422, 418, 492, 411} \[ -\frac{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{d x}+\frac{b e x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}+\frac{b c^3 x^2 \sqrt{d+e x^2}}{d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{d \sqrt{-c^2 x^2}}-\frac{b c^2 x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 264

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

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 475

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 422

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

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

Mathematica [A]  time = 0.1997, size = 139, normalized size = 0.47 \[ \frac{\sqrt{d+e x^2} \left (-a+b c x \sqrt{\frac{1}{c^2 x^2}+1}-b \text{csch}^{-1}(c x)\right )}{d x}-\frac{b c e x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} E\left (\sin ^{-1}\left (\sqrt{-\frac{e}{d}} x\right )|\frac{c^2 d}{e}\right )}{d \sqrt{c^2 x^2+1} \sqrt{-\frac{e}{d}} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.455, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{2}}{\frac{1}{\sqrt{e{x}^{2}+d}}}}\, 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)^(1/2),x)

[Out]

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

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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))/x^2/(e*x^2+d)^(1/2),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^{2} + d}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}}{e x^{4} + d 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)^(1/2),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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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}{\sqrt{e x^{2} + d} 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)^(1/2),x, algorithm="giac")

[Out]

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