3.52 \(\int \frac {\cosh ^{-1}(a x)}{(c+d x^2)^{9/2}} \, dx\)

Optimal. Leaf size=369 \[ -\frac {16 \sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (1-a^2 x^2\right )}{35 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

[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]  time = 1.01, antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {192, 191, 5705, 12, 519, 6715, 1622, 949, 78, 63, 217, 206} \[ \frac {4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac {16 \sqrt {a^2 x^2-1} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a^2 x^2-1}}{a \sqrt {c+d x^2}}\right )}{35 c^4 \sqrt {d} \sqrt {a x-1} \sqrt {a x+1}}+\frac {a \left (1-a^2 x^2\right )}{35 c \sqrt {a x-1} \sqrt {a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[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 63

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 78

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] || LtQ
[p, n]))))

Rule 191

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 192

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

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 519

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)^FracP
art[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 949

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 1622

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 5705

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 6715

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*} \int \frac {\cosh ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx &=\frac {x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \operatorname {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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (2 a \sqrt {-1+a^2 x^2}\right ) \operatorname {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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (8 a \sqrt {-1+a^2 x^2}\right ) \operatorname {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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (16 \sqrt {-1+a^2 x^2}\right ) \operatorname {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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (16 \sqrt {-1+a^2 x^2}\right ) \operatorname {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 \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {16 \sqrt {-1+a^2 x^2} \tanh ^{-1}\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*}

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Mathematica [C]  time = 6.27, size = 844, normalized size = 2.29 \[ \frac {32 \sqrt {\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \left (1+\frac {2}{a x-1}\right )}{\sqrt {c} a+i \sqrt {d}}} \left (\left (\sqrt {c} a+i \sqrt {d}\right ) \sqrt {-\frac {i \left (\frac {c a^2}{a x-1}+i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}} \left (\frac {a \sqrt {c}}{a x-1}-i \sqrt {d} \left (1+\frac {1}{a x-1}\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}}}{\sqrt {2}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )+a \sqrt {c} \left (i \sqrt {d}-a \sqrt {c}\right ) \sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}} \sqrt {\frac {\left (c a^2+d\right ) \left (\frac {c a^2}{(a x-1)^2}+d \left (1+\frac {1}{a x-1}\right )^2\right )}{a^2 c d}} \Pi \left (\frac {2 a \sqrt {c}}{\sqrt {c} a+i \sqrt {d}};\sin ^{-1}\left (\frac {\sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}}}{\sqrt {2}}\right )|\frac {4 i a \sqrt {c} \sqrt {d}}{\left (\sqrt {c} a+i \sqrt {d}\right )^2}\right )\right ) (a x-1)^{3/2}}{35 a c^4 \left (c a^2+d\right ) \sqrt {a x+1} \sqrt {\frac {i \left (\frac {c a^2}{a x-1}-i \sqrt {c} \sqrt {d} a+d+\frac {d}{a x-1}\right )}{a \sqrt {c} \sqrt {d}}} \sqrt {\frac {d (a x-1)^2 \left (1+\frac {1}{a x-1}\right )^2}{a^2}+c}}+\sqrt {a x+1} \sqrt {d x^2+c} \left (-\frac {4 \left (11 c^2 a^4+15 c d a^2+6 d^2\right ) a}{105 c^3 \left (c a^2+d\right )^3 \left (d x^2+c\right )}-\frac {2 \left (5 c a^2+3 d\right ) a}{105 c^2 \left (c a^2+d\right )^2 \left (d x^2+c\right )^2}-\frac {a}{35 c \left (c a^2+d\right ) \left (d x^2+c\right )^3}\right ) \sqrt {a x-1}+\frac {x \left (16 d^3 x^6+56 c d^2 x^4+70 c^2 d x^2+35 c^3\right ) \cosh ^{-1}(a x)}{35 c^4 \left (d x^2+c\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + a*x]*Sqrt[1 + a*x]*Sqrt[c + d*x^2]*(-1/35*a/(c*(a^2*c + d)*(c + d*x^2)^3) - (2*a*(5*a^2*c + 3*d))/(1
05*c^2*(a^2*c + d)^2*(c + d*x^2)^2) - (4*a*(11*a^4*c^2 + 15*a^2*c*d + 6*d^2))/(105*c^3*(a^2*c + d)^3*(c + d*x^
2))) + (x*(35*c^3 + 70*c^2*d*x^2 + 56*c*d^2*x^4 + 16*d^3*x^6)*ArcCosh[a*x])/(35*c^4*(c + d*x^2)^(7/2)) + (32*(
-1 + a*x)^(3/2)*Sqrt[((a*Sqrt[c] - I*Sqrt[d])*(1 + 2/(-1 + a*x)))/(a*Sqrt[c] + I*Sqrt[d])]*((a*Sqrt[c] + I*Sqr
t[d])*Sqrt[((-I)*(I*a*Sqrt[c]*Sqrt[d] + d + (a^2*c)/(-1 + a*x) + d/(-1 + a*x)))/(a*Sqrt[c]*Sqrt[d])]*((a*Sqrt[
c])/(-1 + a*x) - I*Sqrt[d]*(1 + (-1 + a*x)^(-1)))*EllipticF[ArcSin[Sqrt[(I*((-I)*a*Sqrt[c]*Sqrt[d] + d + (a^2*
c)/(-1 + a*x) + d/(-1 + a*x)))/(a*Sqrt[c]*Sqrt[d])]/Sqrt[2]], ((4*I)*a*Sqrt[c]*Sqrt[d])/(a*Sqrt[c] + I*Sqrt[d]
)^2] + a*Sqrt[c]*(-(a*Sqrt[c]) + I*Sqrt[d])*Sqrt[(I*((-I)*a*Sqrt[c]*Sqrt[d] + d + (a^2*c)/(-1 + a*x) + d/(-1 +
a*x)))/(a*Sqrt[c]*Sqrt[d])]*Sqrt[((a^2*c + d)*((a^2*c)/(-1 + a*x)^2 + d*(1 + (-1 + a*x)^(-1))^2))/(a^2*c*d)]*
EllipticPi[(2*a*Sqrt[c])/(a*Sqrt[c] + I*Sqrt[d]), ArcSin[Sqrt[(I*((-I)*a*Sqrt[c]*Sqrt[d] + d + (a^2*c)/(-1 + a
*x) + d/(-1 + a*x)))/(a*Sqrt[c]*Sqrt[d])]/Sqrt[2]], ((4*I)*a*Sqrt[c]*Sqrt[d])/(a*Sqrt[c] + I*Sqrt[d])^2]))/(35
*a*c^4*(a^2*c + d)*Sqrt[1 + a*x]*Sqrt[(I*((-I)*a*Sqrt[c]*Sqrt[d] + d + (a^2*c)/(-1 + a*x) + d/(-1 + a*x)))/(a*
Sqrt[c]*Sqrt[d])]*Sqrt[c + (d*(-1 + a*x)^2*(1 + (-1 + a*x)^(-1))^2)/a^2])

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fricas [B]  time = 1.26, size = 1752, normalized size = 4.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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giac [B]  time = 1.19, size = 876, normalized size = 2.37 \[ \frac {8}{105} \, a {\left (\frac {3 \, {\left | d \right |} \log \left ({\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2}\right )}{c^{4} d^{\frac {3}{2}} {\left | a \right |}} - \frac {11 \, a^{10} c^{4} d^{\frac {9}{2}} {\left | d \right |} + 49 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{8} c^{3} d^{\frac {7}{2}} {\left | d \right |} + 37 \, a^{8} c^{3} d^{\frac {11}{2}} {\left | d \right |} + 77 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{6} c^{2} d^{\frac {5}{2}} {\left | d \right |} + 112 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{6} c^{2} d^{\frac {9}{2}} {\left | d \right |} + 47 \, a^{6} c^{2} d^{\frac {13}{2}} {\left | d \right |} + 33 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{6} a^{4} c d^{\frac {3}{2}} {\left | d \right |} + 93 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{4} c d^{\frac {7}{2}} {\left | d \right |} + 87 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{4} c d^{\frac {11}{2}} {\left | d \right |} + 27 \, a^{4} c d^{\frac {15}{2}} {\left | d \right |} + 6 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{8} a^{2} \sqrt {d} {\left | d \right |} + 24 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{6} a^{2} d^{\frac {5}{2}} {\left | d \right |} + 36 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{4} a^{2} d^{\frac {9}{2}} {\left | d \right |} + 24 \, {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} a^{2} d^{\frac {13}{2}} {\left | d \right |} + 6 \, a^{2} d^{\frac {17}{2}} {\left | d \right |}}{{\left (a^{2} c d + {\left (\sqrt {a^{2} d} \sqrt {d x^{2} + c} - \sqrt {{\left (d x^{2} + c\right )} a^{2} d - a^{2} c d - d^{2}}\right )}^{2} + d^{2}\right )}^{5} c^{3} d {\left | a \right |}}\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}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/105*a*(3*abs(d)*log((sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^2*d - a^2*c*d - d^2))^2)/(c^4*d^(3/2)*
abs(a)) - (11*a^10*c^4*d^(9/2)*abs(d) + 49*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^2*d - a^2*c*d - d
^2))^2*a^8*c^3*d^(7/2)*abs(d) + 37*a^8*c^3*d^(11/2)*abs(d) + 77*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c
)*a^2*d - a^2*c*d - d^2))^4*a^6*c^2*d^(5/2)*abs(d) + 112*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^2*d
- a^2*c*d - d^2))^2*a^6*c^2*d^(9/2)*abs(d) + 47*a^6*c^2*d^(13/2)*abs(d) + 33*(sqrt(a^2*d)*sqrt(d*x^2 + c) - s
qrt((d*x^2 + c)*a^2*d - a^2*c*d - d^2))^6*a^4*c*d^(3/2)*abs(d) + 93*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2
+ c)*a^2*d - a^2*c*d - d^2))^4*a^4*c*d^(7/2)*abs(d) + 87*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^2*
d - a^2*c*d - d^2))^2*a^4*c*d^(11/2)*abs(d) + 27*a^4*c*d^(15/2)*abs(d) + 6*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt
((d*x^2 + c)*a^2*d - a^2*c*d - d^2))^8*a^2*sqrt(d)*abs(d) + 24*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)
*a^2*d - a^2*c*d - d^2))^6*a^2*d^(5/2)*abs(d) + 36*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^2*d - a^2
*c*d - d^2))^4*a^2*d^(9/2)*abs(d) + 24*(sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^2*d - a^2*c*d - d^2))
^2*a^2*d^(13/2)*abs(d) + 6*a^2*d^(17/2)*abs(d))/((a^2*c*d + (sqrt(a^2*d)*sqrt(d*x^2 + c) - sqrt((d*x^2 + c)*a^
2*d - a^2*c*d - d^2))^2 + d^2)^5*c^3*d*abs(a))) + 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)

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccosh}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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 details)Is d-a^2*c zero or nonzero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acosh}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)/(c + d*x^2)^(9/2),x)

[Out]

int(acosh(a*x)/(c + d*x^2)^(9/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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