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

Optimal. Leaf size=203 \[ \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{b x \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{6 c^2 \sqrt{e} \sqrt{-c^2 x^2}} \]

[Out]

(b*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c*Sqrt[-(c^2*x^2)]) + ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*
e) + (b*(3*c^2*d - e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*Sqrt[e]*Sqrt[-(c^2*x^
2)]) + (b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e*Sqrt[-(c^2*x^2)])

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Rubi [A]  time = 0.205643, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6300, 446, 102, 157, 63, 217, 203, 93, 204} \[ \frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-c^2 x^2-1}}\right )}{3 e \sqrt{-c^2 x^2}}+\frac{b x \sqrt{-c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{b x \left (3 c^2 d-e\right ) \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-c^2 x^2-1}}{c \sqrt{d+e x^2}}\right )}{6 c^2 \sqrt{e} \sqrt{-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(6*c*Sqrt[-(c^2*x^2)]) + ((d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*
e) + (b*(3*c^2*d - e)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(6*c^2*Sqrt[e]*Sqrt[-(c^2*x^
2)]) + (b*c*d^(3/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(3*e*Sqrt[-(c^2*x^2)])

Rule 6300

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

Rule 446

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

Rule 102

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

Rule 157

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x \sqrt{d+e x^2} \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{(b c x) \int \frac{\left (d+e x^2\right )^{3/2}}{x \sqrt{-1-c^2 x^2}} \, dx}{3 e \sqrt{-c^2 x^2}}\\ &=\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{(b c x) \operatorname{Subst}\left (\int \frac{(d+e x)^{3/2}}{x \sqrt{-1-c^2 x}} \, dx,x,x^2\right )}{6 e \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{(b x) \operatorname{Subst}\left (\int \frac{-c^2 d^2-\frac{1}{2} \left (3 c^2 d-e\right ) e x}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 c e \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}-\frac{\left (b \left (3 c^2 d-e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{12 c \sqrt{-c^2 x^2}}-\frac{\left (b c d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{-1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{6 e \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{\left (b \left (3 c^2 d-e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{d-\frac{e}{c^2}-\frac{e x^2}{c^2}}} \, dx,x,\sqrt{-1-c^2 x^2}\right )}{6 c^3 \sqrt{-c^2 x^2}}-\frac{\left (b c d^2 x\right ) \operatorname{Subst}\left (\int \frac{1}{-d-x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{-1-c^2 x^2}}\right )}{3 e \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{3 e \sqrt{-c^2 x^2}}+\frac{\left (b \left (3 c^2 d-e\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{e x^2}{c^2}} \, dx,x,\frac{\sqrt{-1-c^2 x^2}}{\sqrt{d+e x^2}}\right )}{6 c^3 \sqrt{-c^2 x^2}}\\ &=\frac{b x \sqrt{-1-c^2 x^2} \sqrt{d+e x^2}}{6 c \sqrt{-c^2 x^2}}+\frac{\left (d+e x^2\right )^{3/2} \left (a+b \text{csch}^{-1}(c x)\right )}{3 e}+\frac{b \left (3 c^2 d-e\right ) x \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{-1-c^2 x^2}}{c \sqrt{d+e x^2}}\right )}{6 c^2 \sqrt{e} \sqrt{-c^2 x^2}}+\frac{b c d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{-1-c^2 x^2}}\right )}{3 e \sqrt{-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.506957, size = 278, normalized size = 1.37 \[ \frac{\sqrt{d+e x^2} \left (2 a c \left (d+e x^2\right )+b e x \sqrt{\frac{1}{c^2 x^2}+1}+2 b c \text{csch}^{-1}(c x) \left (d+e x^2\right )\right )}{6 c e}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (2 c^5 d^{3/2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{c^2 x^2+1}}{\sqrt{-d-e x^2}}\right )-\sqrt{c^2} \sqrt{e} \sqrt{c^2 d-e} \left (3 c^2 d-e\right ) \sqrt{\frac{c^2 \left (d+e x^2\right )}{c^2 d-e}} \sinh ^{-1}\left (\frac{c \sqrt{e} \sqrt{c^2 x^2+1}}{\sqrt{c^2} \sqrt{c^2 d-e}}\right )\right )}{6 c^4 e \sqrt{c^2 x^2+1} \sqrt{d+e x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(b*e*Sqrt[1 + 1/(c^2*x^2)]*x + 2*a*c*(d + e*x^2) + 2*b*c*(d + e*x^2)*ArcCsch[c*x]))/(6*c*e) -
(b*Sqrt[1 + 1/(c^2*x^2)]*x*(-(Sqrt[c^2]*Sqrt[c^2*d - e]*(3*c^2*d - e)*Sqrt[e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d -
e)]*ArcSinh[(c*Sqrt[e]*Sqrt[1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d - e])]) + 2*c^5*d^(3/2)*Sqrt[-d - e*x^2]*ArcT
an[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[-d - e*x^2]]))/(6*c^4*e*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

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Maple [F]  time = 0.466, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{\rm arccsch} \left (cx\right ) \right ) \sqrt{e{x}^{2}+d}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \,{\left (\frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right )}{e} + 3 \, \int \frac{{\left (c^{2} e x^{3} + c^{2} d x\right )} \sqrt{e x^{2} + d}}{3 \,{\left (c^{2} e x^{2} +{\left (c^{2} e x^{2} + e\right )} \sqrt{c^{2} x^{2} + 1} + e\right )}}\,{d x} - 3 \, \int \frac{{\left ({\left (3 \, e \log \left (c\right ) + e\right )} c^{2} x^{3} +{\left (c^{2} d + 3 \, e \log \left (c\right )\right )} x + 3 \,{\left (c^{2} e x^{3} + e x\right )} \log \left (x\right )\right )} \sqrt{e x^{2} + d}}{3 \,{\left (c^{2} e x^{2} + e\right )}}\,{d x}\right )} b + \frac{{\left (e x^{2} + d\right )}^{\frac{3}{2}} a}{3 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*((e*x^2 + d)^(3/2)*log(sqrt(c^2*x^2 + 1) + 1)/e + 3*integrate(1/3*(c^2*e*x^3 + c^2*d*x)*sqrt(e*x^2 + d)/(c
^2*e*x^2 + (c^2*e*x^2 + e)*sqrt(c^2*x^2 + 1) + e), x) - 3*integrate(1/3*((3*e*log(c) + e)*c^2*x^3 + (c^2*d + 3
*e*log(c))*x + 3*(c^2*e*x^3 + e*x)*log(x))*sqrt(e*x^2 + d)/(c^2*e*x^2 + e), x))*b + 1/3*(e*x^2 + d)^(3/2)*a/e

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Fricas [A]  time = 5.43913, size = 2984, normalized size = 14.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(2*b*c^3*d^(3/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4*((c^3*d + c*e)*x^3 + 2
*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) - (3*b*c^2*d - b*e)*sqrt(e)*log(8*
c^4*e^2*x^4 + c^4*d^2 + 6*c^2*d*e + 8*(c^4*d*e + c^2*e^2)*x^2 - 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2
+ d)*sqrt(e)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + e^2) + 8*(b*c^3*e*x^2 + b*c^3*d)*sqrt(e*x^2 + d)*log((c*x*sqrt((
c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(2*a*c^3*e*x^2 + b*c^2*e*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*a*c^3*d)*
sqrt(e*x^2 + d))/(c^3*e), 1/12*(b*c^3*d^(3/2)*log(((c^4*d^2 + 6*c^2*d*e + e^2)*x^4 + 8*(c^2*d^2 + d*e)*x^2 - 4
*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 8*d^2)/x^4) - (3*b*c^2*
d - b*e)*sqrt(-e)*arctan(1/2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^
2))/(c^2*e^2*x^4 + (c^2*d*e + e^2)*x^2 + d*e)) + 4*(b*c^3*e*x^2 + b*c^3*d)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*
x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 2*(2*a*c^3*e*x^2 + b*c^2*e*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*a*c^3*d)*sqrt
(e*x^2 + d))/(c^3*e), 1/24*(4*b*c^3*sqrt(-d)*d*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sqrt(-
d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) - (3*b*c^2*d - b*e)*sqrt(e)*log(8*
c^4*e^2*x^4 + c^4*d^2 + 6*c^2*d*e + 8*(c^4*d*e + c^2*e^2)*x^2 - 4*(2*c^4*e*x^3 + (c^4*d + c^2*e)*x)*sqrt(e*x^2
+ d)*sqrt(e)*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + e^2) + 8*(b*c^3*e*x^2 + b*c^3*d)*sqrt(e*x^2 + d)*log((c*x*sqrt((
c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(2*a*c^3*e*x^2 + b*c^2*e*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*a*c^3*d)*
sqrt(e*x^2 + d))/(c^3*e), 1/12*(2*b*c^3*sqrt(-d)*d*arctan(1/2*((c^3*d + c*e)*x^3 + 2*c*d*x)*sqrt(e*x^2 + d)*sq
rt(-d)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*d*e*x^4 + (c^2*d^2 + d*e)*x^2 + d^2)) - (3*b*c^2*d - b*e)*sqrt(-e)*a
rctan(1/2*(2*c^2*e*x^3 + (c^2*d + e)*x)*sqrt(e*x^2 + d)*sqrt(-e)*sqrt((c^2*x^2 + 1)/(c^2*x^2))/(c^2*e^2*x^4 +
(c^2*d*e + e^2)*x^2 + d*e)) + 4*(b*c^3*e*x^2 + b*c^3*d)*sqrt(e*x^2 + d)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2))
+ 1)/(c*x)) + 2*(2*a*c^3*e*x^2 + b*c^2*e*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 2*a*c^3*d)*sqrt(e*x^2 + d))/(c^3*e
)]

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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**2+d)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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