3.126 \(\int \frac{\sqrt{d+e x^2} (a+b \text{csch}^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=389 \[ -\frac{b e x \left (c^2 d-3 e\right ) \sqrt{d+e x^2} \text{EllipticF}\left (\tan ^{-1}(c x),1-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{2 b c^3 x^2 \left (c^2 d-2 e\right ) \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}-\frac{2 b c \sqrt{-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}+\frac{2 b c^2 x \left (c^2 d-2 e\right ) \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{9 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

[Out]

(-2*b*c^3*(c^2*d - 2*e)*x^2*Sqrt[d + e*x^2])/(9*d*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) - (2*b*c*(c^2*d - 2*e)*
Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*x^2*S
qrt[-(c^2*x^2)]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*d*x^3) + (2*b*c^2*(c^2*d - 2*e)*x*Sqrt[d + e*x^
2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^
2*x^2))]) - (b*(c^2*d - 3*e)*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(9*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 = 0.427077, antiderivative size = 389, 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, 474, 583, 531, 418, 492, 411} \[ -\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{b e x \left (c^2 d-3 e\right ) \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{9 d^2 \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}}-\frac{2 b c^3 x^2 \left (c^2 d-2 e\right ) \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1}}-\frac{2 b c \sqrt{-c^2 x^2-1} \left (c^2 d-2 e\right ) \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}+\frac{2 b c^2 x \left (c^2 d-2 e\right ) \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{9 d \sqrt{-c^2 x^2} \sqrt{-c^2 x^2-1} \sqrt{\frac{d+e x^2}{d \left (c^2 x^2+1\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*c^3*(c^2*d - 2*e)*x^2*Sqrt[d + e*x^2])/(9*d*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]) - (2*b*c*(c^2*d - 2*e)*
Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*d*Sqrt[-(c^2*x^2)]) + (b*c*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(9*x^2*S
qrt[-(c^2*x^2)]) - ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*d*x^3) + (2*b*c^2*(c^2*d - 2*e)*x*Sqrt[d + e*x^
2]*EllipticE[ArcTan[c*x], 1 - e/(c^2*d)])/(9*d*Sqrt[-(c^2*x^2)]*Sqrt[-1 - c^2*x^2]*Sqrt[(d + e*x^2)/(d*(1 + c^
2*x^2))]) - (b*(c^2*d - 3*e)*e*x*Sqrt[d + e*x^2]*EllipticF[ArcTan[c*x], 1 - e/(c^2*d)])/(9*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 474

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(c*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(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 - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, 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{\sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right )}{x^4} \, dx &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{(b c x) \int -\frac{\left (d+e x^2\right )^{3/2}}{3 d x^4 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}+\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2}}{x^4 \sqrt{-1-c^2 x^2}} \, dx}{3 d \sqrt{-c^2 x^2}}\\ &=\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{(b c x) \int \frac{2 d \left (c^2 d-2 e\right )+\left (c^2 d-3 e\right ) e x^2}{x^2 \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d \sqrt{-c^2 x^2}}\\ &=-\frac{2 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{(b c x) \int \frac{d \left (c^2 d-3 e\right ) e+2 c^2 d \left (c^2 d-2 e\right ) e x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d^2 \sqrt{-c^2 x^2}}\\ &=-\frac{2 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{\left (b c \left (c^2 d-3 e\right ) e x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d \sqrt{-c^2 x^2}}-\frac{\left (2 b c^3 \left (c^2 d-2 e\right ) e x\right ) \int \frac{x^2}{\sqrt{-1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{9 d \sqrt{-c^2 x^2}}\\ &=-\frac{2 b c^3 \left (c^2 d-2 e\right ) x^2 \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}-\frac{2 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}-\frac{b \left (c^2 d-3 e\right ) e x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{9 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 (2 b c^3 \left (c^2 d-2 e\right ) x\right ) \int \frac{\sqrt{d+e x^2}}{\left (-1-c^2 x^2\right )^{3/2}} \, dx}{9 d \sqrt{-c^2 x^2}}\\ &=-\frac{2 b c^3 \left (c^2 d-2 e\right ) x^2 \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2} \sqrt{-1-c^2 x^2}}-\frac{2 b c \left (c^2 d-2 e\right ) \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 d \sqrt{-c^2 x^2}}+\frac{b c \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{9 x^2 \sqrt{-c^2 x^2}}-\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 d x^3}+\frac{2 b c^2 \left (c^2 d-2 e\right ) x \sqrt{d+e x^2} E\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{9 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 \left (c^2 d-3 e\right ) e x \sqrt{d+e x^2} F\left (\tan ^{-1}(c x)|1-\frac{e}{c^2 d}\right )}{9 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 [C]  time = 0.597136, size = 237, normalized size = 0.61 \[ -\frac{\sqrt{d+e x^2} \left (3 a \left (d+e x^2\right )+b c x \sqrt{\frac{1}{c^2 x^2}+1} \left (2 c^2 d x^2-d-4 e x^2\right )+3 b \text{csch}^{-1}(c x) \left (d+e x^2\right )\right )}{9 d x^3}-\frac{i b c x \sqrt{\frac{1}{c^2 x^2}+1} \sqrt{\frac{e x^2}{d}+1} \left (\left (-2 c^4 d^2+5 c^2 d e-3 e^2\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right ),\frac{e}{c^2 d}\right )+2 c^2 d \left (c^2 d-2 e\right ) E\left (i \sinh ^{-1}\left (\sqrt{c^2} x\right )|\frac{e}{c^2 d}\right )\right )}{9 \sqrt{c^2} d \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d + e*x^2]*(b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-d + 2*c^2*d*x^2 - 4*e*x^2) + 3*a*(d + e*x^2) + 3*b*(d + e*x^2
)*ArcCsch[c*x]))/(9*d*x^3) - ((I/9)*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d - 2*e)*Ell
ipticE[I*ArcSinh[Sqrt[c^2]*x], e/(c^2*d)] + (-2*c^4*d^2 + 5*c^2*d*e - 3*e^2)*EllipticF[I*ArcSinh[Sqrt[c^2]*x],
e/(c^2*d)]))/(Sqrt[c^2]*d*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]  time = 0.539, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{{x}^{4}}\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))*(e*x^2+d)^(1/2)/x^4,x)

[Out]

int((a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/x^4,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))*(e*x^2+d)^(1/2)/x^4,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 )}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(e*x^2 + d)*(b*arccsch(c*x) + a)/x^4, 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))*(e*x**2+d)**(1/2)/x**4,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(c*x))*(e*x^2+d)^(1/2)/x^4,x, algorithm="giac")

[Out]

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