\(\int x \sqrt {d+e x^2} (a+b \text {csch}^{-1}(c x)) \, dx\) [121]

Optimal result

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

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

Mathematica [C] (warning: unable to verify)

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

Time = 1.23 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.15 \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {-2 b d^2 \sqrt {1+\frac {d}{e x^2}} \sqrt {1+c^2 x^2} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+b \left (3 c^2 d-e\right ) e \sqrt {1+\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )+2 x \sqrt {1+c^2 x^2} \left (d+e x^2\right ) \left (b e \sqrt {1+\frac {1}{c^2 x^2}} x+2 a c \left (d+e x^2\right )+2 b c \left (d+e x^2\right ) \text {csch}^{-1}(c x)\right )}{12 c e x \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \] Input:

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

Output:

(-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)*ArcCsch[c*x]))/(12*c*e*x*Sqrt[1 + c^2*x^2]*Sq 
rt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.92, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6854, 354, 113, 27, 175, 66, 104, 217, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6854

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \int \frac {\left (e x^2+d\right )^{3/2}}{x \sqrt {-c^2 x^2-1}}dx}{3 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \int \frac {\left (e x^2+d\right )^{3/2}}{x^2 \sqrt {-c^2 x^2-1}}dx^2}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 113

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (-\frac {\int -\frac {2 c^2 d^2+\left (3 c^2 d-e\right ) e x^2}{2 x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (\frac {\int \frac {2 c^2 d^2+\left (3 c^2 d-e\right ) e x^2}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (\frac {2 c^2 d^2 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2+e \left (3 c^2 d-e\right ) \int \frac {1}{\sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2}{2 c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (\frac {2 c^2 d^2 \int \frac {1}{x^2 \sqrt {-c^2 x^2-1} \sqrt {e x^2+d}}dx^2+2 e \left (3 c^2 d-e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (\frac {4 c^2 d^2 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {-c^2 x^2-1}}+2 e \left (3 c^2 d-e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (\frac {2 e \left (3 c^2 d-e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}-4 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{2 c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e}-\frac {b c x \left (\frac {-4 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )-\frac {2 \sqrt {e} \left (3 c^2 d-e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}-\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e \sqrt {-c^2 x^2}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

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

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 113

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1))   Int[(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 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(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 217

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 218

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 6854

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] - Simp[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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1342, normalized size of antiderivative = 6.61 \[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[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...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {\sqrt {e \,x^{2}+d}\, a d +\sqrt {e \,x^{2}+d}\, a e \,x^{2}+3 \left (\int \sqrt {e \,x^{2}+d}\, \mathit {acsch} \left (c x \right ) x d x \right ) b e}{3 e} \] Input:

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

Output:

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