\(\int \frac {x^3 (a+b \text {csch}^{-1}(c x))}{\sqrt {d+e x^2}} \, dx\) [140]

Optimal result

Integrand size = 23, antiderivative size = 229 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {b x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{6 c e \sqrt {-c^2 x^2}}-\frac {d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {b \left (3 c^2 d+e\right ) x \arctan \left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 e^{3/2} \sqrt {-c^2 x^2}}-\frac {2 b c d^{3/2} x \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{3 e^2 \sqrt {-c^2 x^2}} \] Output:

1/6*b*x*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c/e/(-c^2*x^2)^(1/2)-d*(e*x^2+d 
)^(1/2)*(a+b*arccsch(c*x))/e^2+1/3*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/e^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^(3/2)/(-c^2*x^2)^(1/2)-2/3*b*c*d^(3/2)*x*arctan((e*x^2+d)^(1/2)/d^(1/ 
2)/(-c^2*x^2-1)^(1/2))/e^2/(-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 = 1.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.03 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\frac {4 b d^2 \sqrt {1+\frac {d}{e x^2}} \sqrt {1+c^2 x^2} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )-b e \left (3 c^2 d+e\right ) \sqrt {1+\frac {1}{c^2 x^2}} x^4 \sqrt {1+\frac {e x^2}{d}} \operatorname {AppellF1}\left (1,\frac {1}{2},\frac {1}{2},2,-c^2 x^2,-\frac {e x^2}{d}\right )+2 x \sqrt {1+c^2 x^2} \left (d+e x^2\right ) \left (-4 a c d+b e \sqrt {1+\frac {1}{c^2 x^2}} x+2 a c e x^2+2 b c \left (-2 d+e x^2\right ) \text {csch}^{-1}(c x)\right )}{12 c e^2 x \sqrt {1+c^2 x^2} \sqrt {d+e x^2}} \] Input:

Integrate[(x^3*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]
 

Output:

(4*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*e*(3*c^2*d + 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)*(-4*a*c*d + b*e*Sqrt[1 + 1/(c^2*x^2)]*x + 
 2*a*c*e*x^2 + 2*b*c*(-2*d + e*x^2)*ArcCsch[c*x]))/(12*c*e^2*x*Sqrt[1 + c^ 
2*x^2]*Sqrt[d + e*x^2])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {6856, 27, 435, 171, 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.

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

\(\Big \downarrow \) 6856

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 435

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

\(\displaystyle \frac {b c x \left (\frac {8 c^2 d^2 \int \frac {1}{-x^4-d}d\frac {\sqrt {e x^2+d}}{\sqrt {-c^2 x^2-1}}+2 e \left (3 c^2 d+e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^2 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {b c x \left (\frac {2 e \left (3 c^2 d+e\right ) \int \frac {1}{-e x^4-c^2}d\frac {\sqrt {-c^2 x^2-1}}{\sqrt {e x^2+d}}-8 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^2 \sqrt {-c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {d \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {b c x \left (\frac {-8 c^2 d^{3/2} \arctan \left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )-\frac {2 \sqrt {e} \left (3 c^2 d+e\right ) \arctan \left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{c}}{2 c^2}+\frac {e \sqrt {-c^2 x^2-1} \sqrt {d+e x^2}}{c^2}\right )}{6 e^2 \sqrt {-c^2 x^2}}\)

Input:

Int[(x^3*(a + b*ArcCsch[c*x]))/Sqrt[d + e*x^2],x]
 

Output:

-((d*Sqrt[d + e*x^2]*(a + b*ArcCsch[c*x]))/e^2) + ((d + e*x^2)^(3/2)*(a + 
b*ArcCsch[c*x]))/(3*e^2) + (b*c*x*((e*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/ 
c^2 + ((-2*Sqrt[e]*(3*c^2*d + e)*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sq 
rt[d + e*x^2])])/c - 8*c^2*d^(3/2)*ArcTan[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 
 - c^2*x^2])])/(2*c^2)))/(6*e^2*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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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 435

Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( 
e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2) 
*(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, 
 e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
 

rule 6856

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]}, Si 
mp[(a + b*ArcCsch[c*x])   u, x] - Simp[b*c*(x/Sqrt[(-c^2)*x^2])   Int[Simpl 
ifyIntegrand[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])) || (I 
LtQ[(m + 2*p + 1)/2, 0] &&  !ILtQ[(m - 1)/2, 0]))
 

Maple [F]

\[\int \frac {x^{3} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )}{\sqrt {x^{2} e +d}}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1341, normalized size of antiderivative = 5.86 \[ \int \frac {x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x^2}} \, dx=\text {Too large to display} \] Input:

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

Output:

[1/24*(4*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 - 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)) - 4*a*c^3*d)*sqrt(e*x^2 + d))/(c^3*e^2), 1/12*(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)*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 - 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)) - 4*a*c^3*d)*sqrt(e 
*x^2 + d))/(c^3*e^2), -1/24*(8*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^...
 

Sympy [F]

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

integrate(x**3*(a+b*acsch(c*x))/(e*x**2+d)**(1/2),x)
 

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate(x^3*(a+b*arccsch(c*x))/(e*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((x^3*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(1/2),x)
 

Output:

int((x^3*(a + b*asinh(1/(c*x))))/(d + e*x^2)^(1/2), x)
 

Reduce [F]

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

int(x^3*(a+b*acsch(c*x))/(e*x^2+d)^(1/2),x)
 

Output:

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