3.2.28 \(\int x^3 (d+e x^2)^{3/2} (a+b \text {csch}^{-1}(c x)) \, dx\) [128]

Optimal. Leaf size=384 \[ -\frac {b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}}+\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {-c^2 x^2}}-\frac {2 b c d^{7/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{35 e^2 \sqrt {-c^2 x^2}} \]

[Out]

-1/5*d*(e*x^2+d)^(5/2)*(a+b*arccsch(c*x))/e^2+1/7*(e*x^2+d)^(7/2)*(a+b*arccsch(c*x))/e^2-1/560*b*(35*c^6*d^3+3
5*c^4*d^2*e-63*c^2*d*e^2+25*e^3)*x*arctan(e^(1/2)*(-c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/c^6/e^(3/2)/(-c^2*x^2)
^(1/2)-2/35*b*c*d^(7/2)*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)+1/840*b*(13*
c^2*d-25*e)*x*(e*x^2+d)^(3/2)*(-c^2*x^2-1)^(1/2)/c^3/e/(-c^2*x^2)^(1/2)+1/42*b*x*(e*x^2+d)^(5/2)*(-c^2*x^2-1)^
(1/2)/c/e/(-c^2*x^2)^(1/2)-1/560*b*(3*c^4*d^2+38*c^2*d*e-25*e^2)*x*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c^5/e/(-
c^2*x^2)^(1/2)

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Rubi [A]
time = 0.36, antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {272, 45, 6437, 12, 587, 159, 163, 65, 223, 209, 95, 210} \begin {gather*} -\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {2 b c d^{7/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{35 e^2 \sqrt {-c^2 x^2}}-\frac {b x \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (13 c^2 d-25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}-\frac {b x \sqrt {-c^2 x^2-1} \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/560*(b*(3*c^4*d^2 + 38*c^2*d*e - 25*e^2)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(c^5*e*Sqrt[-(c^2*x^2)]) + (
b*(13*c^2*d - 25*e)*x*Sqrt[-1 - c^2*x^2]*(d + e*x^2)^(3/2))/(840*c^3*e*Sqrt[-(c^2*x^2)]) + (b*x*Sqrt[-1 - c^2*
x^2]*(d + e*x^2)^(5/2))/(42*c*e*Sqrt[-(c^2*x^2)]) - (d*(d + e*x^2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) + ((d +
e*x^2)^(7/2)*(a + b*ArcCsch[c*x]))/(7*e^2) - (b*(35*c^6*d^3 + 35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*x*ArcTan[
(Sqrt[e]*Sqrt[-1 - c^2*x^2])/(c*Sqrt[d + e*x^2])])/(560*c^6*e^(3/2)*Sqrt[-(c^2*x^2)]) - (2*b*c*d^(7/2)*x*ArcTa
n[Sqrt[d + e*x^2]/(Sqrt[d]*Sqrt[-1 - c^2*x^2])])/(35*e^2*Sqrt[-(c^2*x^2)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 159

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Dist[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] /; FreeQ[{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 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 272

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

Rule 587

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

Rule 6437

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]))

Rubi steps

\begin {align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 x \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{x \sqrt {-1-c^2 x^2}} \, dx}{35 e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^{5/2} (-2 d+5 e x)}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{70 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}+\frac {(b x) \text {Subst}\left (\int \frac {(d+e x)^{3/2} \left (6 c^2 d^2-\frac {1}{2} \left (13 c^2 d-25 e\right ) e x\right )}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{210 c e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {(b x) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (-12 c^4 d^3-\frac {3}{4} e \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x\right )}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{420 c^3 e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}}+\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}+\frac {(b x) \text {Subst}\left (\int \frac {12 c^6 d^4+\frac {3}{8} e \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{420 c^5 e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}}+\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}+\frac {\left (b c d^4 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{35 e^2 \sqrt {-c^2 x^2}}+\frac {\left (b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{1120 c^5 e \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}}+\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}+\frac {\left (2 b c d^4 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 )}{35 e^2 \sqrt {-c^2 x^2}}-\frac {\left (b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\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 )}{560 c^7 e \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}}+\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {2 b c d^{7/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{35 e^2 \sqrt {-c^2 x^2}}-\frac {\left (b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\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 )}{560 c^7 e \sqrt {-c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2+38 c^2 d e-25 e^2\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {-c^2 x^2}}+\frac {b \left (13 c^2 d-25 e\right ) x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \text {csch}^{-1}(c x)\right )}{7 e^2}-\frac {b \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {-c^2 x^2}}-\frac {2 b c d^{7/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{35 e^2 \sqrt {-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 3.28, size = 293, normalized size = 0.76 \begin {gather*} \frac {\sqrt {d+e x^2} \left (-48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (75 e^2-2 c^2 e \left (82 d+25 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )-48 b c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2 \text {csch}^{-1}(c x)\right )}{1680 c^5 e^2}-\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (-32 c^7 d^{7/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (35 c^6 d^3+35 c^4 d^2 e-63 c^2 d e^2+25 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{560 c^6 e^2 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(-48*a*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(75*e^2 - 2*c^2*e*(82*
d + 25*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4)) - 48*b*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2*ArcCsch[c*x]
))/(1680*c^5*e^2) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(-32*c^7*d^(7/2)*ArcTanh[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[d + e
*x^2]] + Sqrt[e]*(35*c^6*d^3 + 35*c^4*d^2*e - 63*c^2*d*e^2 + 25*e^3)*ArcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sq
rt[d + e*x^2])]))/(560*c^6*e^2*Sqrt[1 + c^2*x^2])

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(5*(x^2*e + d)^(5/2)*x^2*e^(-1) - 2*(x^2*e + d)^(5/2)*d*e^(-2))*a + 1/35*((5*x^6*e^3 + 8*d*x^4*e^2 + d^2*
x^2*e - 2*d^3)*sqrt(x^2*e + d)*e^(-2)*log(sqrt(c^2*x^2 + 1) + 1) + 35*integrate(1/35*(5*c^2*x^7*e^3 + 8*c^2*d*
x^5*e^2 + c^2*d^2*x^3*e - 2*c^2*d^3*x)*sqrt(x^2*e + d)/(c^2*x^2*e^2 + (c^2*x^2*e^2 + e^2)*sqrt(c^2*x^2 + 1) +
e^2), x) - 35*integrate(1/35*(5*c^2*x^7*(7*log(c) + 1)*e^3 - 2*c^2*d^3*x + ((35*d*log(c) + 8*d)*c^2*e^2 + 35*e
^3*log(c))*x^5 + (c^2*d^2*e + 35*d*e^2*log(c))*x^3 + 35*(c^2*x^7*e^3 + (c^2*d*e^2 + e^3)*x^5 + d*x^3*e^2)*log(
x))*sqrt(x^2*e + d)/(c^2*x^2*e^2 + e^2), x))*b

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (327) = 654\).
time = 1.64, size = 2121, normalized size = 5.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6720*(96*b*c^7*d^(7/2)*log((c^4*d^2*x^4 + 8*c^2*d^2*x^2 + x^4*cosh(1)^2 + x^4*sinh(1)^2 + 4*(c^3*d*x^3 + c*
x^3*cosh(1) + c*x^3*sinh(1) + 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(d)*sqrt((c^2*x^2 + 1)/(c^2*x^2
)) + 8*d^2 + 2*(3*c^2*d*x^4 + 4*d*x^2)*cosh(1) + 2*(3*c^2*d*x^4 + x^4*cosh(1) + 4*d*x^2)*sinh(1))/x^4) + 3*(35
*b*c^6*d^3 + 35*b*c^4*d^2*cosh(1) - 63*b*c^2*d*cosh(1)^2 + 25*b*cosh(1)^3 + 25*b*sinh(1)^3 - 3*(21*b*c^2*d - 2
5*b*cosh(1))*sinh(1)^2 + (35*b*c^4*d^2 - 126*b*c^2*d*cosh(1) + 75*b*cosh(1)^2)*sinh(1))*sqrt(cosh(1) + sinh(1)
)*log(c^4*d^2 + (8*c^4*x^4 + 8*c^2*x^2 + 1)*cosh(1)^2 + (8*c^4*x^4 + 8*c^2*x^2 + 1)*sinh(1)^2 - 4*(c^4*d*x + (
2*c^4*x^3 + c^2*x)*cosh(1) + (2*c^4*x^3 + c^2*x)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt((c^2*x^2 +
1)/(c^2*x^2))*sqrt(cosh(1) + sinh(1)) + 2*(4*c^4*d*x^2 + 3*c^2*d)*cosh(1) + 2*(4*c^4*d*x^2 + 3*c^2*d + (8*c^4*
x^4 + 8*c^2*x^2 + 1)*cosh(1))*sinh(1)) + 192*(5*b*c^7*x^6*cosh(1)^3 + 5*b*c^7*x^6*sinh(1)^3 + 8*b*c^7*d*x^4*co
sh(1)^2 + b*c^7*d^2*x^2*cosh(1) - 2*b*c^7*d^3 + (15*b*c^7*x^6*cosh(1) + 8*b*c^7*d*x^4)*sinh(1)^2 + (15*b*c^7*x
^6*cosh(1)^2 + 16*b*c^7*d*x^4*cosh(1) + b*c^7*d^2*x^2)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log((c*x*s
qrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(240*a*c^7*x^6*cosh(1)^3 + 240*a*c^7*x^6*sinh(1)^3 + 384*a*c^7*d*
x^4*cosh(1)^2 + 48*a*c^7*d^2*x^2*cosh(1) - 96*a*c^7*d^3 + 48*(15*a*c^7*x^6*cosh(1) + 8*a*c^7*d*x^4)*sinh(1)^2
+ 48*(15*a*c^7*x^6*cosh(1)^2 + 16*a*c^7*d*x^4*cosh(1) + a*c^7*d^2*x^2)*sinh(1) + (57*b*c^6*d^2*x*cosh(1) + 5*(
8*b*c^6*x^5 - 10*b*c^4*x^3 + 15*b*c^2*x)*cosh(1)^3 + 5*(8*b*c^6*x^5 - 10*b*c^4*x^3 + 15*b*c^2*x)*sinh(1)^3 + 2
*(53*b*c^6*d*x^3 - 82*b*c^4*d*x)*cosh(1)^2 + (106*b*c^6*d*x^3 - 164*b*c^4*d*x + 15*(8*b*c^6*x^5 - 10*b*c^4*x^3
+ 15*b*c^2*x)*cosh(1))*sinh(1)^2 + (57*b*c^6*d^2*x + 15*(8*b*c^6*x^5 - 10*b*c^4*x^3 + 15*b*c^2*x)*cosh(1)^2 +
4*(53*b*c^6*d*x^3 - 82*b*c^4*d*x)*cosh(1))*sinh(1))*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(x^2*cosh(1) + x^2*sin
h(1) + d))/(c^7*cosh(1)^2 + 2*c^7*cosh(1)*sinh(1) + c^7*sinh(1)^2), -1/6720*(192*b*c^7*sqrt(-d)*d^3*arctan(1/2
*(c^3*d*x^3 + c*x^3*cosh(1) + c*x^3*sinh(1) + 2*c*d*x)*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt(-d)*sqrt((c^2*
x^2 + 1)/(c^2*x^2))/(c^2*d^2*x^2 + d^2 + (c^2*d*x^4 + d*x^2)*cosh(1) + (c^2*d*x^4 + d*x^2)*sinh(1))) - 3*(35*b
*c^6*d^3 + 35*b*c^4*d^2*cosh(1) - 63*b*c^2*d*cosh(1)^2 + 25*b*cosh(1)^3 + 25*b*sinh(1)^3 - 3*(21*b*c^2*d - 25*
b*cosh(1))*sinh(1)^2 + (35*b*c^4*d^2 - 126*b*c^2*d*cosh(1) + 75*b*cosh(1)^2)*sinh(1))*sqrt(cosh(1) + sinh(1))*
log(c^4*d^2 + (8*c^4*x^4 + 8*c^2*x^2 + 1)*cosh(1)^2 + (8*c^4*x^4 + 8*c^2*x^2 + 1)*sinh(1)^2 - 4*(c^4*d*x + (2*
c^4*x^3 + c^2*x)*cosh(1) + (2*c^4*x^3 + c^2*x)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*sqrt((c^2*x^2 + 1)
/(c^2*x^2))*sqrt(cosh(1) + sinh(1)) + 2*(4*c^4*d*x^2 + 3*c^2*d)*cosh(1) + 2*(4*c^4*d*x^2 + 3*c^2*d + (8*c^4*x^
4 + 8*c^2*x^2 + 1)*cosh(1))*sinh(1)) - 192*(5*b*c^7*x^6*cosh(1)^3 + 5*b*c^7*x^6*sinh(1)^3 + 8*b*c^7*d*x^4*cosh
(1)^2 + b*c^7*d^2*x^2*cosh(1) - 2*b*c^7*d^3 + (15*b*c^7*x^6*cosh(1) + 8*b*c^7*d*x^4)*sinh(1)^2 + (15*b*c^7*x^6
*cosh(1)^2 + 16*b*c^7*d*x^4*cosh(1) + b*c^7*d^2*x^2)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log((c*x*sqr
t((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) - 4*(240*a*c^7*x^6*cosh(1)^3 + 240*a*c^7*x^6*sinh(1)^3 + 384*a*c^7*d*x^
4*cosh(1)^2 + 48*a*c^7*d^2*x^2*cosh(1) - 96*a*c^7*d^3 + 48*(15*a*c^7*x^6*cosh(1) + 8*a*c^7*d*x^4)*sinh(1)^2 +
48*(15*a*c^7*x^6*cosh(1)^2 + 16*a*c^7*d*x^4*cosh(1) + a*c^7*d^2*x^2)*sinh(1) + (57*b*c^6*d^2*x*cosh(1) + 5*(8*
b*c^6*x^5 - 10*b*c^4*x^3 + 15*b*c^2*x)*cosh(1)^3 + 5*(8*b*c^6*x^5 - 10*b*c^4*x^3 + 15*b*c^2*x)*sinh(1)^3 + 2*(
53*b*c^6*d*x^3 - 82*b*c^4*d*x)*cosh(1)^2 + (106*b*c^6*d*x^3 - 164*b*c^4*d*x + 15*(8*b*c^6*x^5 - 10*b*c^4*x^3 +
15*b*c^2*x)*cosh(1))*sinh(1)^2 + (57*b*c^6*d^2*x + 15*(8*b*c^6*x^5 - 10*b*c^4*x^3 + 15*b*c^2*x)*cosh(1)^2 + 4
*(53*b*c^6*d*x^3 - 82*b*c^4*d*x)*cosh(1))*sinh(1))*sqrt((c^2*x^2 + 1)/(c^2*x^2)))*sqrt(x^2*cosh(1) + x^2*sinh(
1) + d))/(c^7*cosh(1)^2 + 2*c^7*cosh(1)*sinh(1) + c^7*sinh(1)^2)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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