\(\int \frac {\text {arccosh}(a x)}{(c+d x^2)^{9/2}} \, dx\) [52]

Optimal result

Integrand size = 16, antiderivative size = 369 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {16 \sqrt {-1+a^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-1+a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

1/7*x*arccosh(a*x)/c/(d*x^2+c)^(7/2)+6/35*x*arccosh(a*x)/c^2/(d*x^2+c)^(5/2)+8/35*x*arccosh(a*x)/c^3/(d*x^2+c)
^(3/2)+1/35*a*(-a^2*x^2+1)/c/(a^2*c+d)/(d*x^2+c)^(5/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+2/105*a*(5*a^2*c+3*d)*(-a^2
*x^2+1)/c^2/(a^2*c+d)^2/(d*x^2+c)^(3/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)-16/35*arctanh(d^(1/2)*(a^2*x^2-1)^(1/2)/a/
(d*x^2+c)^(1/2))*(a^2*x^2-1)^(1/2)/c^4/d^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+16/35*x*arccosh(a*x)/c^4/(d*x^2+c)^
(1/2)+4/105*a*(11*a^4*c^2+15*a^2*c*d+6*d^2)*(-a^2*x^2+1)/c^3/(a^2*c+d)^3/(a*x-1)^(1/2)/(a*x+1)^(1/2)/(d*x^2+c)
^(1/2)

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {198, 197, 5908, 12, 533, 6847, 1636, 963, 79, 65, 223, 212} \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=-\frac {16 \sqrt {a^2 x^2-1} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (1-a^2 x^2\right )}{35 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

[In]

Int[ArcCosh[a*x]/(c + d*x^2)^(9/2),x]

[Out]

(a*(1 - a^2*x^2))/(35*c*(a^2*c + d)*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2)^(5/2)) + (2*a*(5*a^2*c + 3*d)*(1
- a^2*x^2))/(105*c^2*(a^2*c + d)^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2)^(3/2)) + (4*a*(11*a^4*c^2 + 15*a^2
*c*d + 6*d^2)*(1 - a^2*x^2))/(105*c^3*(a^2*c + d)^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c + d*x^2]) + (x*ArcCosh
[a*x])/(7*c*(c + d*x^2)^(7/2)) + (6*x*ArcCosh[a*x])/(35*c^2*(c + d*x^2)^(5/2)) + (8*x*ArcCosh[a*x])/(35*c^3*(c
+ d*x^2)^(3/2)) + (16*x*ArcCosh[a*x])/(35*c^4*Sqrt[c + d*x^2]) - (16*Sqrt[-1 + a^2*x^2]*ArcTanh[(Sqrt[d]*Sqrt
[-1 + a^2*x^2])/(a*Sqrt[c + d*x^2])])/(35*c^4*Sqrt[d]*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Rule 12

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

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 79

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 533

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPa
rt[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

Rule 963

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{Qx = PolynomialQuotient[(a + b*x + c*x^2)^p, d + e*x, x], R = PolynomialRemainder[(a + b*x + c*x^2)^p,
d + e*x, x]}, Simp[R*(d + e*x)^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Dist[1/((m + 1)*(e*f -
d*g)), Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d*g)*Qx - g*R*(m + n + 2), x], x], x]] /;
FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&& IGtQ[p, 0] && LtQ[m, -1]

Rule 1636

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px,
a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(
b*c - a*d))), x] + Dist[1/((m + 1)*(b*c - a*d)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*ExpandToSum[(m + 1)*(b*c -
a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; FreeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && GtQ[Expon
[Px, x], 2]

Rule 5908

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{35 c^4 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{7/2}} \, dx \\ & = \frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4} \\ & = \frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\sqrt {-1+a^2 x^2} \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\sqrt {-1+a^2 x} (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {5 c^2 \left (17 a^2 c+15 d\right )+100 c d \left (a^2 c+d\right ) x+40 d^2 \left (a^2 c+d\right ) x^2}{\sqrt {-1+a^2 x} (c+d x)^{5/2}} \, dx,x,x^2\right )}{175 c^4 \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (2 a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {5 c \left (23 a^4 c^2+39 a^2 c d+18 d^2\right )+60 d \left (a^2 c+d\right )^2 x}{\sqrt {-1+a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{525 c^4 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (8 a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{35 c^4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (16 \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d}{a^2}+\frac {d x^2}{a^2}}} \, dx,x,\sqrt {-1+a^2 x^2}\right )}{35 a c^4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (16 \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{a^2}} \, dx,x,\frac {\sqrt {-1+a^2 x^2}}{\sqrt {c+d x^2}}\right )}{35 a c^4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{5/2}}+\frac {2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right )^{3/2}}+\frac {4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt {-1+a x} \sqrt {1+a x} \sqrt {c+d x^2}}+\frac {x \text {arccosh}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \text {arccosh}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \text {arccosh}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \text {arccosh}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {16 \sqrt {-1+a^2 x^2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {-1+a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 4.48 (sec) , antiderivative size = 723, normalized size of antiderivative = 1.96 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\frac {-\frac {a \sqrt {-1+a x} \sqrt {1+a x} \left (c+d x^2\right ) \left (3 d^2 \left (11 c^2+18 c d x^2+8 d^2 x^4\right )+2 a^2 c d \left (41 c^2+68 c d x^2+30 d^2 x^4\right )+a^4 c^2 \left (57 c^2+98 c d x^2+44 d^2 x^4\right )\right )}{3 c^3 \left (a^2 c+d\right )^3}+\frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right ) \text {arccosh}(a x)}{c^4}+\frac {32 (-1+a x)^{3/2} \sqrt {\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) (1+a x)}{\left (a \sqrt {c}+i \sqrt {d}\right ) (-1+a x)}} \left (c+d x^2\right )^3 \left (\frac {a \left (-i a \sqrt {c}+\sqrt {d}\right ) \left (i \sqrt {c}+\sqrt {d} x\right ) \sqrt {\frac {1+\frac {i a \sqrt {c}}{\sqrt {d}}-a x+\frac {i \sqrt {d} x}{\sqrt {c}}}{1-a x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{2-2 a x}}\right ),\frac {4 i a \sqrt {c} \sqrt {d}}{\left (a \sqrt {c}+i \sqrt {d}\right )^2}\right )}{-1+a x}+a \sqrt {c} \left (-a \sqrt {c}+i \sqrt {d}\right ) \sqrt {\frac {\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (-1+a x)^2}} \sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{1-a x}} \operatorname {EllipticPi}\left (\frac {2 a \sqrt {c}}{a \sqrt {c}+i \sqrt {d}},\arcsin \left (\sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{2-2 a x}}\right ),\frac {4 i a \sqrt {c} \sqrt {d}}{\left (a \sqrt {c}+i \sqrt {d}\right )^2}\right )\right )}{a c^4 \left (a^2 c+d\right ) \sqrt {1+a x} \sqrt {-\frac {-1+\frac {i \sqrt {d} x}{\sqrt {c}}+a \left (\frac {i \sqrt {c}}{\sqrt {d}}+x\right )}{1-a x}}}}{35 \left (c+d x^2\right )^{7/2}} \]

[In]

Integrate[ArcCosh[a*x]/(c + d*x^2)^(9/2),x]

[Out]

(-1/3*(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(c + d*x^2)*(3*d^2*(11*c^2 + 18*c*d*x^2 + 8*d^2*x^4) + 2*a^2*c*d*(41*c^2
+ 68*c*d*x^2 + 30*d^2*x^4) + a^4*c^2*(57*c^2 + 98*c*d*x^2 + 44*d^2*x^4)))/(c^3*(a^2*c + d)^3) + (x*(35*c^3 +
70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x^6)*ArcCosh[a*x])/c^4 + (32*(-1 + a*x)^(3/2)*Sqrt[((a*Sqrt[c] - I*Sqrt[d
])*(1 + a*x))/((a*Sqrt[c] + I*Sqrt[d])*(-1 + a*x))]*(c + d*x^2)^3*((a*((-I)*a*Sqrt[c] + Sqrt[d])*(I*Sqrt[c] +
Sqrt[d]*x)*Sqrt[(1 + (I*a*Sqrt[c])/Sqrt[d] - a*x + (I*Sqrt[d]*x)/Sqrt[c])/(1 - a*x)]*EllipticF[ArcSin[Sqrt[-((
-1 + (I*Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(2 - 2*a*x))]], ((4*I)*a*Sqrt[c]*Sqrt[d])/(a*Sqrt[c]
+ I*Sqrt[d])^2])/(-1 + a*x) + a*Sqrt[c]*(-(a*Sqrt[c]) + I*Sqrt[d])*Sqrt[((a^2*c + d)*(c + d*x^2))/(c*d*(-1 +
a*x)^2)]*Sqrt[-((-1 + (I*Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(1 - a*x))]*EllipticPi[(2*a*Sqrt[c]
)/(a*Sqrt[c] + I*Sqrt[d]), ArcSin[Sqrt[-((-1 + (I*Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(2 - 2*a*x
))]], ((4*I)*a*Sqrt[c]*Sqrt[d])/(a*Sqrt[c] + I*Sqrt[d])^2]))/(a*c^4*(a^2*c + d)*Sqrt[1 + a*x]*Sqrt[-((-1 + (I*
Sqrt[d]*x)/Sqrt[c] + a*((I*Sqrt[c])/Sqrt[d] + x))/(1 - a*x))]))/(35*(c + d*x^2)^(7/2))

Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}d x\]

[In]

int(arccosh(a*x)/(d*x^2+c)^(9/2),x)

[Out]

int(arccosh(a*x)/(d*x^2+c)^(9/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 870 vs. \(2 (310) = 620\).

Time = 0.48 (sec) , antiderivative size = 1752, normalized size of antiderivative = 4.75 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Too large to display} \]

[In]

integrate(arccosh(a*x)/(d*x^2+c)^(9/2),x, algorithm="fricas")

[Out]

[1/105*(12*(a^6*c^7 + 3*a^4*c^6*d + 3*a^2*c^5*d^2 + (a^6*c^3*d^4 + 3*a^4*c^2*d^5 + 3*a^2*c*d^6 + d^7)*x^8 + c^
4*d^3 + 4*(a^6*c^4*d^3 + 3*a^4*c^3*d^4 + 3*a^2*c^2*d^5 + c*d^6)*x^6 + 6*(a^6*c^5*d^2 + 3*a^4*c^4*d^3 + 3*a^2*c
^3*d^4 + c^2*d^5)*x^4 + 4*(a^6*c^6*d + 3*a^4*c^5*d^2 + 3*a^2*c^4*d^3 + c^3*d^4)*x^2)*sqrt(d)*log(8*a^4*d^2*x^4
+ a^4*c^2 - 6*a^2*c*d + 8*(a^4*c*d - a^2*d^2)*x^2 - 4*(2*a^3*d*x^2 + a^3*c - a*d)*sqrt(a^2*x^2 - 1)*sqrt(d*x^
2 + c)*sqrt(d) + d^2) + 3*(16*(a^6*c^3*d^4 + 3*a^4*c^2*d^5 + 3*a^2*c*d^6 + d^7)*x^7 + 56*(a^6*c^4*d^3 + 3*a^4*
c^3*d^4 + 3*a^2*c^2*d^5 + c*d^6)*x^5 + 70*(a^6*c^5*d^2 + 3*a^4*c^4*d^3 + 3*a^2*c^3*d^4 + c^2*d^5)*x^3 + 35*(a^
6*c^6*d + 3*a^4*c^5*d^2 + 3*a^2*c^4*d^3 + c^3*d^4)*x)*sqrt(d*x^2 + c)*log(a*x + sqrt(a^2*x^2 - 1)) - (57*a^5*c
^6*d + 82*a^3*c^5*d^2 + 33*a*c^4*d^3 + 4*(11*a^5*c^3*d^4 + 15*a^3*c^2*d^5 + 6*a*c*d^6)*x^6 + 2*(71*a^5*c^4*d^3
+ 98*a^3*c^3*d^4 + 39*a*c^2*d^5)*x^4 + (155*a^5*c^5*d^2 + 218*a^3*c^4*d^3 + 87*a*c^3*d^4)*x^2)*sqrt(a^2*x^2 -
1)*sqrt(d*x^2 + c))/(a^6*c^11*d + 3*a^4*c^10*d^2 + 3*a^2*c^9*d^3 + c^8*d^4 + (a^6*c^7*d^5 + 3*a^4*c^6*d^6 + 3
*a^2*c^5*d^7 + c^4*d^8)*x^8 + 4*(a^6*c^8*d^4 + 3*a^4*c^7*d^5 + 3*a^2*c^6*d^6 + c^5*d^7)*x^6 + 6*(a^6*c^9*d^3 +
3*a^4*c^8*d^4 + 3*a^2*c^7*d^5 + c^6*d^6)*x^4 + 4*(a^6*c^10*d^2 + 3*a^4*c^9*d^3 + 3*a^2*c^8*d^4 + c^7*d^5)*x^2
), 1/105*(24*(a^6*c^7 + 3*a^4*c^6*d + 3*a^2*c^5*d^2 + (a^6*c^3*d^4 + 3*a^4*c^2*d^5 + 3*a^2*c*d^6 + d^7)*x^8 +
c^4*d^3 + 4*(a^6*c^4*d^3 + 3*a^4*c^3*d^4 + 3*a^2*c^2*d^5 + c*d^6)*x^6 + 6*(a^6*c^5*d^2 + 3*a^4*c^4*d^3 + 3*a^2
*c^3*d^4 + c^2*d^5)*x^4 + 4*(a^6*c^6*d + 3*a^4*c^5*d^2 + 3*a^2*c^4*d^3 + c^3*d^4)*x^2)*sqrt(-d)*arctan(1/2*(2*
a^2*d*x^2 + a^2*c - d)*sqrt(a^2*x^2 - 1)*sqrt(d*x^2 + c)*sqrt(-d)/(a^3*d^2*x^4 - a*c*d + (a^3*c*d - a*d^2)*x^2
)) + 3*(16*(a^6*c^3*d^4 + 3*a^4*c^2*d^5 + 3*a^2*c*d^6 + d^7)*x^7 + 56*(a^6*c^4*d^3 + 3*a^4*c^3*d^4 + 3*a^2*c^2
*d^5 + c*d^6)*x^5 + 70*(a^6*c^5*d^2 + 3*a^4*c^4*d^3 + 3*a^2*c^3*d^4 + c^2*d^5)*x^3 + 35*(a^6*c^6*d + 3*a^4*c^5
*d^2 + 3*a^2*c^4*d^3 + c^3*d^4)*x)*sqrt(d*x^2 + c)*log(a*x + sqrt(a^2*x^2 - 1)) - (57*a^5*c^6*d + 82*a^3*c^5*d
^2 + 33*a*c^4*d^3 + 4*(11*a^5*c^3*d^4 + 15*a^3*c^2*d^5 + 6*a*c*d^6)*x^6 + 2*(71*a^5*c^4*d^3 + 98*a^3*c^3*d^4 +
39*a*c^2*d^5)*x^4 + (155*a^5*c^5*d^2 + 218*a^3*c^4*d^3 + 87*a*c^3*d^4)*x^2)*sqrt(a^2*x^2 - 1)*sqrt(d*x^2 + c)
)/(a^6*c^11*d + 3*a^4*c^10*d^2 + 3*a^2*c^9*d^3 + c^8*d^4 + (a^6*c^7*d^5 + 3*a^4*c^6*d^6 + 3*a^2*c^5*d^7 + c^4*
d^8)*x^8 + 4*(a^6*c^8*d^4 + 3*a^4*c^7*d^5 + 3*a^2*c^6*d^6 + c^5*d^7)*x^6 + 6*(a^6*c^9*d^3 + 3*a^4*c^8*d^4 + 3*
a^2*c^7*d^5 + c^6*d^6)*x^4 + 4*(a^6*c^10*d^2 + 3*a^4*c^9*d^3 + 3*a^2*c^8*d^4 + c^7*d^5)*x^2)]

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate(acosh(a*x)/(d*x**2+c)**(9/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(arccosh(a*x)/(d*x^2+c)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d-a^2*c>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 481, normalized size of antiderivative = 1.30 \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=-\frac {1}{105} \, a {\left (\frac {\sqrt {a^{2} x^{2} - 1} {\left (2 \, {\left (a^{2} x^{2} - 1\right )} {\left (\frac {2 \, {\left (11 \, a^{8} c^{15} d^{4} {\left | a \right |} + 15 \, a^{6} c^{14} d^{5} {\left | a \right |} + 6 \, a^{4} c^{13} d^{6} {\left | a \right |}\right )} {\left (a^{2} x^{2} - 1\right )}}{a^{10} c^{19} d^{2} + 3 \, a^{8} c^{18} d^{3} + 3 \, a^{6} c^{17} d^{4} + a^{4} c^{16} d^{5}} + \frac {49 \, a^{10} c^{16} d^{3} {\left | a \right |} + 112 \, a^{8} c^{15} d^{4} {\left | a \right |} + 87 \, a^{6} c^{14} d^{5} {\left | a \right |} + 24 \, a^{4} c^{13} d^{6} {\left | a \right |}}{a^{10} c^{19} d^{2} + 3 \, a^{8} c^{18} d^{3} + 3 \, a^{6} c^{17} d^{4} + a^{4} c^{16} d^{5}}\right )} + \frac {3 \, {\left (19 \, a^{12} c^{17} d^{2} {\left | a \right |} + 60 \, a^{10} c^{16} d^{3} {\left | a \right |} + 71 \, a^{8} c^{15} d^{4} {\left | a \right |} + 38 \, a^{6} c^{14} d^{5} {\left | a \right |} + 8 \, a^{4} c^{13} d^{6} {\left | a \right |}\right )}}{a^{10} c^{19} d^{2} + 3 \, a^{8} c^{18} d^{3} + 3 \, a^{6} c^{17} d^{4} + a^{4} c^{16} d^{5}}\right )}}{{\left (a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d\right )}^{\frac {5}{2}}} - \frac {48 \, {\left | a \right |} \log \left ({\left | -\sqrt {a^{2} x^{2} - 1} \sqrt {d} + \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d} \right |}\right )}{a^{2} c^{4} \sqrt {d}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{35 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \]

[In]

integrate(arccosh(a*x)/(d*x^2+c)^(9/2),x, algorithm="giac")

[Out]

-1/105*a*(sqrt(a^2*x^2 - 1)*(2*(a^2*x^2 - 1)*(2*(11*a^8*c^15*d^4*abs(a) + 15*a^6*c^14*d^5*abs(a) + 6*a^4*c^13*
d^6*abs(a))*(a^2*x^2 - 1)/(a^10*c^19*d^2 + 3*a^8*c^18*d^3 + 3*a^6*c^17*d^4 + a^4*c^16*d^5) + (49*a^10*c^16*d^3
*abs(a) + 112*a^8*c^15*d^4*abs(a) + 87*a^6*c^14*d^5*abs(a) + 24*a^4*c^13*d^6*abs(a))/(a^10*c^19*d^2 + 3*a^8*c^
18*d^3 + 3*a^6*c^17*d^4 + a^4*c^16*d^5)) + 3*(19*a^12*c^17*d^2*abs(a) + 60*a^10*c^16*d^3*abs(a) + 71*a^8*c^15*
d^4*abs(a) + 38*a^6*c^14*d^5*abs(a) + 8*a^4*c^13*d^6*abs(a))/(a^10*c^19*d^2 + 3*a^8*c^18*d^3 + 3*a^6*c^17*d^4
+ a^4*c^16*d^5))/(a^2*c + (a^2*x^2 - 1)*d + d)^(5/2) - 48*abs(a)*log(abs(-sqrt(a^2*x^2 - 1)*sqrt(d) + sqrt(a^2
*c + (a^2*x^2 - 1)*d + d)))/(a^2*c^4*sqrt(d))) + 1/35*(2*(4*x^2*(2*d^3*x^2/c^4 + 7*d^2/c^3) + 35*d/c^2)*x^2 +
35/c)*x*log(a*x + sqrt(a^2*x^2 - 1))/(d*x^2 + c)^(7/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]

[In]

int(acosh(a*x)/(c + d*x^2)^(9/2),x)

[Out]

int(acosh(a*x)/(c + d*x^2)^(9/2), x)