3.375 \(\int \frac {x^2 (d-c^2 d x^2)^2}{(a+b \cosh ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=462 \[ \frac {5 \sqrt {\pi } d^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 \sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {7 \pi } d^2 e^{\frac {7 a}{b}} \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 \sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {7 \pi } d^2 e^{-\frac {7 a}{b}} \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {2 d^2 x^2 (c x-1)^{5/2} (c x+1)^{5/2}}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]

[Out]

5/64*d^2*exp(a/b)*erf((a+b*arccosh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3+5/64*d^2*erfi((a+b*arccosh(c*x))^
(1/2)/b^(1/2))*Pi^(1/2)/b^(3/2)/c^3/exp(a/b)+1/64*d^2*exp(3*a/b)*erf(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))
*3^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/64*d^2*erfi(3^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/b^(3/2)
/c^3/exp(3*a/b)-3/64*d^2*exp(5*a/b)*erf(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3/2)/c^3
-3/64*d^2*erfi(5^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*5^(1/2)*Pi^(1/2)/b^(3/2)/c^3/exp(5*a/b)+1/64*d^2*exp(
7*a/b)*erf(7^(1/2)*(a+b*arccosh(c*x))^(1/2)/b^(1/2))*7^(1/2)*Pi^(1/2)/b^(3/2)/c^3+1/64*d^2*erfi(7^(1/2)*(a+b*a
rccosh(c*x))^(1/2)/b^(1/2))*7^(1/2)*Pi^(1/2)/b^(3/2)/c^3/exp(7*a/b)-2*d^2*x^2*(c*x-1)^(5/2)*(c*x+1)^(5/2)/b/c/
(a+b*arccosh(c*x))^(1/2)

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Rubi [A]  time = 2.27, antiderivative size = 474, normalized size of antiderivative = 1.03, number of steps used = 42, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {5776, 5781, 5448, 3307, 2180, 2204, 2205} \[ \frac {5 \sqrt {\pi } d^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 \sqrt {5 \pi } d^2 e^{\frac {5 a}{b}} \text {Erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {7 \pi } d^2 e^{\frac {7 a}{b}} \text {Erf}\left (\frac {\sqrt {7} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {5 \sqrt {\pi } d^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 \sqrt {5 \pi } d^2 e^{-\frac {5 a}{b}} \text {Erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {\sqrt {7 \pi } d^2 e^{-\frac {7 a}{b}} \text {Erfi}\left (\frac {\sqrt {7} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {2 d^2 x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^2)/(b*c*Sqrt[a + b*ArcCosh[c*x]]) + (5*d^2*E^(a/b)*Sqrt
[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3) + (d^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a
+ b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) - (3*d^2*E^((5*a)/b)*Sqrt[5*Pi]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh
[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3) + (d^2*E^((7*a)/b)*Sqrt[7*Pi]*Erf[(Sqrt[7]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b
]])/(64*b^(3/2)*c^3) + (5*d^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(64*b^(3/2)*c^3*E^(a/b)) + (d^2
*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((3*a)/b)) - (3*d^2*Sqrt[5*Pi]
*Erfi[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((5*a)/b)) + (d^2*Sqrt[7*Pi]*Erfi[(Sqrt[7
]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(64*b^(3/2)*c^3*E^((7*a)/b))

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5776

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(
f*m*(-d)^p)/(b*c*(n + 1)), Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n
+ 1), x], x] - Dist[(c*(-d)^p*(m + 2*p + 1))/(b*f*(n + 1)), Int[(f*x)^(m + 1)*(1 + c*x)^(p - 1/2)*(-1 + c*x)^(
p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && IGtQ[m, -3] && IGtQ[p, 0]

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rubi steps

\begin {align*} \int \frac {x^2 \left (d-c^2 d x^2\right )^2}{\left (a+b \cosh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (4 d^2\right ) \int \frac {x (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b c}+\frac {\left (14 c d^2\right ) \int \frac {x^3 (-1+c x)^{3/2} (1+c x)^{3/2}}{\sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (4 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}+\frac {\left (14 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^3(x) \sinh ^4(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (4 d^2\right ) \operatorname {Subst}\left (\int \left (\frac {\cosh (x)}{8 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{16 \sqrt {a+b x}}+\frac {\cosh (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}+\frac {\left (14 d^2\right ) \operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{64 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{64 \sqrt {a+b x}}-\frac {\cosh (5 x)}{64 \sqrt {a+b x}}+\frac {\cosh (7 x)}{64 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^3}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (7 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c^3}+\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^3}-\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{32 b c^3}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-7 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{7 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {e^{-5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {e^{5 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {d^2 \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^3}-\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}+\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}+\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}-\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{64 b c^3}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {7 a}{b}-\frac {7 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}-\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}+\frac {\left (7 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {7 a}{b}+\frac {7 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}-\frac {d^2 \operatorname {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^3}-\frac {d^2 \operatorname {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^3}-\frac {d^2 \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^3}-\frac {d^2 \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{2 b^2 c^3}-\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}+\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}+\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}-\frac {\left (21 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{32 b^2 c^3}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^3}+\frac {\left (3 d^2\right ) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{4 b^2 c^3}\\ &=-\frac {2 d^2 x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^2}{b c \sqrt {a+b \cosh ^{-1}(c x)}}+\frac {5 d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{\frac {7 a}{b}} \sqrt {7 \pi } \text {erf}\left (\frac {\sqrt {7} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}-\frac {3 d^2 e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}+\frac {d^2 e^{-\frac {7 a}{b}} \sqrt {7 \pi } \text {erfi}\left (\frac {\sqrt {7} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 b^{3/2} c^3}\\ \end {align*}

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Mathematica [A]  time = 2.99, size = 498, normalized size = 1.08 \[ -\frac {d^2 e^{-\frac {7 a}{b}} \left (10 c x e^{\frac {7 a}{b}} \sqrt {\frac {c x-1}{c x+1}}+10 e^{\frac {7 a}{b}} \sqrt {\frac {c x-1}{c x+1}}+2 e^{\frac {7 a}{b}} \sinh \left (3 \cosh ^{-1}(c x)\right )-6 e^{\frac {7 a}{b}} \sinh \left (5 \cosh ^{-1}(c x)\right )+2 e^{\frac {7 a}{b}} \sinh \left (7 \cosh ^{-1}(c x)\right )+5 e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )-\sqrt {7} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3 \sqrt {5} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-\sqrt {3} e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-5 e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+\sqrt {3} e^{\frac {10 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-3 \sqrt {5} e^{\frac {12 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+\sqrt {7} e^{\frac {14 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{64 b c^3 \sqrt {a+b \cosh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/64*(d^2*(10*E^((7*a)/b)*Sqrt[(-1 + c*x)/(1 + c*x)] + 10*c*E^((7*a)/b)*x*Sqrt[(-1 + c*x)/(1 + c*x)] + 5*E^((
8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] - Sqrt[7]*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamm
a[1/2, (-7*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[5]*E^((2*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5*(a
+ b*ArcCosh[c*x]))/b] - Sqrt[3]*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]
))/b] - 5*E^((6*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + Sqrt[3]*E^((10*a
)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] - 3*Sqrt[5]*E^((12*a)/b)*Sqrt[a/b + ArcCo
sh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b] + Sqrt[7]*E^((14*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (7*
(a + b*ArcCosh[c*x]))/b] + 2*E^((7*a)/b)*Sinh[3*ArcCosh[c*x]] - 6*E^((7*a)/b)*Sinh[5*ArcCosh[c*x]] + 2*E^((7*a
)/b)*Sinh[7*ArcCosh[c*x]]))/(b*c^3*E^((7*a)/b)*Sqrt[a + b*ArcCosh[c*x]])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c^2*d*x^2 - d)^2*x^2/(b*arccosh(c*x) + a)^(3/2), x)

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \left (-c^{2} d \,x^{2}+d \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(-c^2*d*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x)

[Out]

int(x^2*(-c^2*d*x^2+d)^2/(a+b*arccosh(c*x))^(3/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c^2*d*x^2 - d)^2*x^2/(b*arccosh(c*x) + a)^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(d - c^2*d*x^2)^2)/(a + b*acosh(c*x))^(3/2),x)

[Out]

int((x^2*(d - c^2*d*x^2)^2)/(a + b*acosh(c*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ d^{2} \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx + \int \left (- \frac {2 c^{2} x^{4}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{6}}{a \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c x \right )}} \operatorname {acosh}{\left (c x \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(-c**2*d*x**2+d)**2/(a+b*acosh(c*x))**(3/2),x)

[Out]

d**2*(Integral(x**2/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(-2*c**2*x*
*4/(a*sqrt(a + b*acosh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x) + Integral(c**4*x**6/(a*sqrt(a + b*aco
sh(c*x)) + b*sqrt(a + b*acosh(c*x))*acosh(c*x)), x))

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