\(\int x \sqrt {d+e x^2} (a+b \text {csch}^{-1}(c x)) \, dx\) [120]

Optimal result

Integrand size = 21, antiderivative size = 203 \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b \left (3 c^2 d-e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 \sqrt {e} \sqrt {-c^2 x^2}}+\frac {b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 e \sqrt {-c^2 x^2}} \]

[Out]

1/3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/e+1/3*b*c*d^(3/2)*x*arctan((e*x^2+d)^(1/2)/d^(1/2)/(-c^2*x^2-1)^(1/2))/
e/(-c^2*x^2)^(1/2)+1/6*b*(3*c^2*d-e)*x*arctan(e^(1/2)*(-c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/c^2/e^(1/2)/(-c^2*
x^2)^(1/2)+1/6*b*x*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c/(-c^2*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6435, 457, 104, 163, 65, 223, 209, 95, 210} \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}+\frac {b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{3 e \sqrt {-c^2 x^2}}+\frac {b x \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 \sqrt {e} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{6 c \sqrt {-c^2 x^2}} \]

[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 65

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 95

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 104

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 163

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 209

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

Rule 210

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

Rule 223

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 457

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 6435

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]

Rubi steps \begin{align*} \text {integral}& = \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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 \arctan \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 ) \text {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 \arctan \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 \arctan \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 [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 2.00 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.15 \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {-2 b d^2 \sqrt {1+\frac {d}{e x^2}} \sqrt {1+c^2 x^2} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+b \left (3 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )+2 x \sqrt {1+c^2 x^2} \left (d+e x^2\right ) \left (b e \sqrt {1+\frac {1}{c^2 x^2}} x+2 a c \left (d+e x^2\right )+2 b c \left (d+e x^2\right ) \text {csch}^{-1}(c x)\right )}{12 c e x \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \]

[In]

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

[Out]

(-2*b*d^2*Sqrt[1 + d/(e*x^2)]*Sqrt[1 + c^2*x^2]*AppellF1[1, 1/2, 1/2, 2, -(1/(c^2*x^2)), -(d/(e*x^2))] + b*(3*
c^2*d - e)*e*Sqrt[1 + 1/(c^2*x^2)]*x^4*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, -(c^2*x^2), -((e*x^2)/d)]
+ 2*x*Sqrt[1 + c^2*x^2]*(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)*ArcCs
ch[c*x]))/(12*c*e*x*Sqrt[1 + c^2*x^2]*Sqrt[d + e*x^2])

Maple [F]

\[\int x \left (a +b \,\operatorname {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}d x\]

[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)

Fricas [A] (verification not implemented)

none

Time = 0.46 (sec) , antiderivative size = 1342, normalized size of antiderivative = 6.61 \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\text {Too large to display} \]

[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
)]

Sympy [F]

\[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x \,d x } \]

[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

Giac [F]

\[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x\,\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

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

[Out]

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