3.2.19 \(\int x^3 \sqrt {d+e x^2} (a+b \text {csch}^{-1}(c x)) \, dx\) [119]

Optimal. Leaf size=302 \[ \frac {b \left (c^2 d-9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {b \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) x \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt {-c^2 x^2}}-\frac {2 b c d^{5/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{15 e^2 \sqrt {-c^2 x^2}} \]

[Out]

-1/3*d*(e*x^2+d)^(3/2)*(a+b*arccsch(c*x))/e^2+1/5*(e*x^2+d)^(5/2)*(a+b*arccsch(c*x))/e^2-1/120*b*(15*c^4*d^2+1
0*c^2*d*e-9*e^2)*x*arctan(e^(1/2)*(-c^2*x^2-1)^(1/2)/c/(e*x^2+d)^(1/2))/c^4/e^(3/2)/(-c^2*x^2)^(1/2)-2/15*b*c*
d^(5/2)*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)+1/20*b*x*(e*x^2+d)^(3/2)*(-c
^2*x^2-1)^(1/2)/c/e/(-c^2*x^2)^(1/2)+1/120*b*(c^2*d-9*e)*x*(-c^2*x^2-1)^(1/2)*(e*x^2+d)^(1/2)/c^3/e/(-c^2*x^2)
^(1/2)

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Rubi [A]
time = 0.28, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps used = 11, 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 )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 b c d^{5/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-c^2 x^2-1}}\right )}{15 e^2 \sqrt {-c^2 x^2}}-\frac {b x \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {-c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-c^2 x^2-1} \left (c^2 d-9 e\right ) \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(c^2*d - 9*e)*x*Sqrt[-1 - c^2*x^2]*Sqrt[d + e*x^2])/(120*c^3*e*Sqrt[-(c^2*x^2)]) + (b*x*Sqrt[-1 - c^2*x^2]*
(d + e*x^2)^(3/2))/(20*c*e*Sqrt[-(c^2*x^2)]) - (d*(d + e*x^2)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + ((d + e*x^
2)^(5/2)*(a + b*ArcCsch[c*x]))/(5*e^2) - (b*(15*c^4*d^2 + 10*c^2*d*e - 9*e^2)*x*ArcTan[(Sqrt[e]*Sqrt[-1 - c^2*
x^2])/(c*Sqrt[d + e*x^2])])/(120*c^4*e^(3/2)*Sqrt[-(c^2*x^2)]) - (2*b*c*d^(5/2)*x*ArcTan[Sqrt[d + e*x^2]/(Sqrt
[d]*Sqrt[-1 - c^2*x^2])])/(15*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 \sqrt {d+e x^2} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{15 e^2 x \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2} \left (-2 d+3 e x^2\right )}{x \sqrt {-1-c^2 x^2}} \, dx}{15 e^2 \sqrt {-c^2 x^2}}\\ &=-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^{3/2} (-2 d+3 e x)}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{30 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {(b x) \text {Subst}\left (\int \frac {\sqrt {d+e x} \left (4 c^2 d^2-\frac {1}{2} \left (c^2 d-9 e\right ) e x\right )}{x \sqrt {-1-c^2 x}} \, dx,x,x^2\right )}{60 c e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (c^2 d-9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {(b x) \text {Subst}\left (\int \frac {-4 c^4 d^3-\frac {1}{4} e \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) x}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{60 c^3 e^2 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (c^2 d-9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (b c d^3 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{15 e^2 \sqrt {-c^2 x^2}}+\frac {\left (b \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1-c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{240 c^3 e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (c^2 d-9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {\left (2 b c d^3 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 )}{15 e^2 \sqrt {-c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e-9 e^2\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 )}{120 c^5 e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (c^2 d-9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{15 e^2 \sqrt {-c^2 x^2}}-\frac {\left (b \left (15 c^4 d^2+10 c^2 d e-9 e^2\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 )}{120 c^5 e \sqrt {-c^2 x^2}}\\ &=\frac {b \left (c^2 d-9 e\right ) x \sqrt {-1-c^2 x^2} \sqrt {d+e x^2}}{120 c^3 e \sqrt {-c^2 x^2}}+\frac {b x \sqrt {-1-c^2 x^2} \left (d+e x^2\right )^{3/2}}{20 c e \sqrt {-c^2 x^2}}-\frac {d \left (d+e x^2\right )^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {\left (d+e x^2\right )^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {b \left (15 c^4 d^2+10 c^2 d e-9 e^2\right ) x \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {-1-c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{120 c^4 e^{3/2} \sqrt {-c^2 x^2}}-\frac {2 b c d^{5/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1-c^2 x^2}}\right )}{15 e^2 \sqrt {-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 2.36, size = 255, normalized size = 0.84 \begin {gather*} \frac {\sqrt {d+e x^2} \left (8 a c^3 \left (-2 d^2+d e x^2+3 e^2 x^4\right )+b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (-9 e+c^2 \left (7 d+6 e x^2\right )\right )+8 b c^3 \left (-2 d^2+d e x^2+3 e^2 x^4\right ) \text {csch}^{-1}(c x)\right )}{120 c^3 e^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (16 c^5 d^{5/2} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (-15 c^4 d^2-10 c^2 d e+9 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{120 c^4 e^2 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d + e*x^2]*(8*a*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 + 1/(c^2*x^2)]*x*(-9*e + c^2*(7*d + 6*e*
x^2)) + 8*b*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4)*ArcCsch[c*x]))/(120*c^3*e^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x*(16*c
^5*d^(5/2)*ArcTanh[(Sqrt[d]*Sqrt[1 + c^2*x^2])/Sqrt[d + e*x^2]] + Sqrt[e]*(-15*c^4*d^2 - 10*c^2*d*e + 9*e^2)*A
rcTanh[(Sqrt[e]*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + e*x^2])]))/(120*c^4*e^2*Sqrt[1 + c^2*x^2])

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/15*(3*(x^2*e + d)^(3/2)*x^2*e^(-1) - 2*(x^2*e + d)^(3/2)*d*e^(-2))*a + 1/15*((3*x^4*e^2 + d*x^2*e - 2*d^2)*s
qrt(x^2*e + d)*e^(-2)*log(sqrt(c^2*x^2 + 1) + 1) + 15*integrate(1/15*(3*c^2*x^5*e^2 + c^2*d*x^3*e - 2*c^2*d^2*
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) - 15*integrate(1/15*(3*c^2*
x^5*(5*log(c) + 1)*e^2 - 2*c^2*d^2*x + (c^2*d*e + 15*e^2*log(c))*x^3 + 15*(c^2*x^5*e^2 + 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. 743 vs. \(2 (254) = 508\).
time = 0.85, size = 1524, normalized size = 5.05 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/480*(16*b*c^5*d^(5/2)*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) - (15*b*
c^4*d^2 + 10*b*c^2*d*cosh(1) - 9*b*cosh(1)^2 - 9*b*sinh(1)^2 + 2*(5*b*c^2*d - 9*b*cosh(1))*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)) + 32*(3*b*c^5*x^4*cosh(1)^2 + 3*b*c^5*x^4*sinh(1)^2 + b*c^5
*d*x^2*cosh(1) - 2*b*c^5*d^2 + (6*b*c^5*x^4*cosh(1) + b*c^5*d*x^2)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d
)*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + 4*(24*a*c^5*x^4*cosh(1)^2 + 24*a*c^5*x^4*sinh(1)^2 + 8*
a*c^5*d*x^2*cosh(1) - 16*a*c^5*d^2 + 8*(6*a*c^5*x^4*cosh(1) + a*c^5*d*x^2)*sinh(1) + (7*b*c^4*d*x*cosh(1) + 3*
(2*b*c^4*x^3 - 3*b*c^2*x)*cosh(1)^2 + 3*(2*b*c^4*x^3 - 3*b*c^2*x)*sinh(1)^2 + (7*b*c^4*d*x + 6*(2*b*c^4*x^3 -
3*b*c^2*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^5*cosh(1)^
2 + 2*c^5*cosh(1)*sinh(1) + c^5*sinh(1)^2), -1/480*(32*b*c^5*sqrt(-d)*d^2*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))) + (15*b*c^4*d^2 + 10*b*c^2*d*cosh(1
) - 9*b*cosh(1)^2 - 9*b*sinh(1)^2 + 2*(5*b*c^2*d - 9*b*cosh(1))*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))*s
qrt(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)) - 32*(3*b*c^5*x^4*cosh(1)^2 + 3*b*c^5*x^4*sinh(1)^2 + b*c^5*d*x^2*cosh(1) - 2*b*c^5*d^
2 + (6*b*c^5*x^4*cosh(1) + b*c^5*d*x^2)*sinh(1))*sqrt(x^2*cosh(1) + x^2*sinh(1) + d)*log((c*x*sqrt((c^2*x^2 +
1)/(c^2*x^2)) + 1)/(c*x)) - 4*(24*a*c^5*x^4*cosh(1)^2 + 24*a*c^5*x^4*sinh(1)^2 + 8*a*c^5*d*x^2*cosh(1) - 16*a*
c^5*d^2 + 8*(6*a*c^5*x^4*cosh(1) + a*c^5*d*x^2)*sinh(1) + (7*b*c^4*d*x*cosh(1) + 3*(2*b*c^4*x^3 - 3*b*c^2*x)*c
osh(1)^2 + 3*(2*b*c^4*x^3 - 3*b*c^2*x)*sinh(1)^2 + (7*b*c^4*d*x + 6*(2*b*c^4*x^3 - 3*b*c^2*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^5*cosh(1)^2 + 2*c^5*cosh(1)*sinh(1) +
c^5*sinh(1)^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*(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^3, x)

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

[Out]

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

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