3.87 \(\int x^4 (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=454 \[ \frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

1/16*d*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/10*x^5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))-3/256*d^2*
x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/128*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/32*d
^2*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+3/512*b*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^
(1/2)+1/512*b*d^2*x^4*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)-31/960*b*c*d^2*x^6*(-c^2*d*x^2+d)^(1/
2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+21/640*b*c^3*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/100*b*c
^5*d^2*x^10*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/512*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/
2)/b/c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]  time = 1.38, antiderivative size = 485, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 5745, 5743, 5759, 5676, 30, 14, 266, 43} \[ \frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {c x-1} \sqrt {c x+1}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {c x-1} \sqrt {c x+1}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(3*b*d^2*x^2*Sqrt[d - c^2*d*x^2])/(512*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*x^4*Sqrt[d - c^2*d*x^2])/(51
2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (31*b*c*d^2*x^6*Sqrt[d - c^2*d*x^2])/(960*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) +
(21*b*c^3*d^2*x^8*Sqrt[d - c^2*d*x^2])/(640*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d^2*x^10*Sqrt[d - c^2*d*x^2
])/(100*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(256*c^4) - (d^2*x^
3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(128*c^2) + (d^2*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/32
+ (d^2*x^5*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/16 + (d^2*x^5*(1 - c*x)^2*(1 + c*x)^2
*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/10 - (3*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(512*b*c^5*
Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 43

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 266

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 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

Rule 5745

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(f*(m + 2*p + 1)
), x] + (Dist[(2*d1*d2*p)/(m + 2*p + 1), Int[(f*x)^m*(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*
x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2*p + 1)*Sqrt[1 + c*
x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[
{a, b, c, d1, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !L
tQ[m, -1] && IntegerQ[p - 1/2] && (RationalQ[m] || EqQ[n, 1])

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)
^(m - 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5798

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

Rubi steps

\begin {align*} \int x^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right )^2 \, dx}{10 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (3 d^2 \sqrt {d-c^2 d x^2}\right ) \int x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int x^2 \left (-1+c^2 x\right )^2 \, dx,x,x^2\right )}{20 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{32 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (x^2-2 c^2 x^3+c^4 x^4\right ) \, dx,x,x^2\right )}{20 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{128 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d^2 x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{10} d^2 x^5 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{512 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 6.59, size = 581, normalized size = 1.28 \[ -\frac {3 a d^{5/2} \tan ^{-1}\left (\frac {c x \sqrt {-d \left (c^2 x^2-1\right )}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )}{256 c^5}+\sqrt {-d \left (c^2 x^2-1\right )} \left (\frac {1}{10} a c^4 d^2 x^9-\frac {3 a d^2 x}{256 c^4}-\frac {21}{80} a c^2 d^2 x^7-\frac {a d^2 x^3}{128 c^2}+\frac {31}{160} a d^2 x^5\right )+\frac {b d^2 \sqrt {-d (c x-1) (c x+1)} \left (18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \left (36 \cosh ^{-1}(c x)^2+\cosh \left (6 \cosh ^{-1}(c x)\right )+18 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-18 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )-6 \cosh ^{-1}(c x) \sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{2304 c^5 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {b d^2 \sqrt {-d (c x-1) (c x+1)} \left (1440 \cosh ^{-1}(c x)^2-576 \cosh \left (2 \cosh ^{-1}(c x)\right )+144 \cosh \left (4 \cosh ^{-1}(c x)\right )+64 \cosh \left (6 \cosh ^{-1}(c x)\right )+9 \cosh \left (8 \cosh ^{-1}(c x)\right )+1152 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-576 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )-384 \cosh ^{-1}(c x) \sinh \left (6 \cosh ^{-1}(c x)\right )-72 \cosh ^{-1}(c x) \sinh \left (8 \cosh ^{-1}(c x)\right )\right )}{36864 c^5 \sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {b d^2 \sqrt {-d (c x-1) (c x+1)} \left (50400 \cosh ^{-1}(c x)^2-25200 \cosh \left (2 \cosh ^{-1}(c x)\right )+3600 \cosh \left (4 \cosh ^{-1}(c x)\right )+2600 \cosh \left (6 \cosh ^{-1}(c x)\right )+675 \cosh \left (8 \cosh ^{-1}(c x)\right )+72 \cosh \left (10 \cosh ^{-1}(c x)\right )+50400 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-14400 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )-15600 \cosh ^{-1}(c x) \sinh \left (6 \cosh ^{-1}(c x)\right )-5400 \cosh ^{-1}(c x) \sinh \left (8 \cosh ^{-1}(c x)\right )-720 \cosh ^{-1}(c x) \sinh \left (10 \cosh ^{-1}(c x)\right )\right )}{3686400 c^5 \sqrt {\frac {c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((-3*a*d^2*x)/(256*c^4) - (a*d^2*x^3)/(128*c^2) + (31*a*d^2*x^5)/160 - (21*a*c^2*d^2
*x^7)/80 + (a*c^4*d^2*x^9)/10) - (3*a*d^(5/2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))]
)/(256*c^5) + (b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*(36
*ArcCosh[c*x]^2 + Cosh[6*ArcCosh[c*x]] + 18*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 18*ArcCosh[c*x]*Sinh[4*ArcCosh
[c*x]] - 6*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]])))/(2304*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*d^2*Sqrt[
-(d*(-1 + c*x)*(1 + c*x))]*(1440*ArcCosh[c*x]^2 - 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cos
h[6*ArcCosh[c*x]] + 9*Cosh[8*ArcCosh[c*x]] + 1152*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 576*ArcCosh[c*x]*Sinh[4*
ArcCosh[c*x]] - 384*ArcCosh[c*x]*Sinh[6*ArcCosh[c*x]] - 72*ArcCosh[c*x]*Sinh[8*ArcCosh[c*x]]))/(36864*c^5*Sqrt
[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (b*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(50400*ArcCosh[c*x]^2 - 25200*Cosh[
2*ArcCosh[c*x]] + 3600*Cosh[4*ArcCosh[c*x]] + 2600*Cosh[6*ArcCosh[c*x]] + 675*Cosh[8*ArcCosh[c*x]] + 72*Cosh[1
0*ArcCosh[c*x]] + 50400*ArcCosh[c*x]*Sinh[2*ArcCosh[c*x]] - 14400*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]] - 15600*Ar
cCosh[c*x]*Sinh[6*ArcCosh[c*x]] - 5400*ArcCosh[c*x]*Sinh[8*ArcCosh[c*x]] - 720*ArcCosh[c*x]*Sinh[10*ArcCosh[c*
x]]))/(3686400*c^5*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a c^{4} d^{2} x^{8} - 2 \, a c^{2} d^{2} x^{6} + a d^{2} x^{4} + {\left (b c^{4} d^{2} x^{8} - 2 \, b c^{2} d^{2} x^{6} + b d^{2} x^{4}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^8 - 2*a*c^2*d^2*x^6 + a*d^2*x^4 + (b*c^4*d^2*x^8 - 2*b*c^2*d^2*x^6 + b*d^2*x^4)*arccosh(
c*x))*sqrt(-c^2*d*x^2 + d), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)*x^4, x)

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maple [A]  time = 0.81, size = 690, normalized size = 1.52 \[ -\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{10 c^{2} d}-\frac {3 a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{80 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{160 c^{4}}+\frac {a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{128 c^{4}}+\frac {3 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{256 c^{4}}+\frac {3 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{256 c^{4} \sqrt {c^{2} d}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{5} x^{10}}{100 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {21 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{3} x^{8}}{640 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {31 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c \,x^{6}}{960 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} x^{4}}{512 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} x^{2}}{512 \sqrt {c x +1}\, c^{3} \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{6} \mathrm {arccosh}\left (c x \right ) x^{11}}{10 \left (c x +1\right ) \left (c x -1\right )}-\frac {29 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{4} \mathrm {arccosh}\left (c x \right ) x^{9}}{80 \left (c x +1\right ) \left (c x -1\right )}+\frac {73 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} c^{2} \mathrm {arccosh}\left (c x \right ) x^{7}}{160 \left (c x +1\right ) \left (c x -1\right )}-\frac {129 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \mathrm {arccosh}\left (c x \right ) x^{5}}{640 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \mathrm {arccosh}\left (c x \right ) x^{3}}{256 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \mathrm {arccosh}\left (c x \right ) x}{256 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {101 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2}}{1228800 \sqrt {c x +1}\, c^{5} \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{512 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x)

[Out]

-1/10*a*x^3*(-c^2*d*x^2+d)^(7/2)/c^2/d-3/80*a/c^4*x*(-c^2*d*x^2+d)^(7/2)/d+1/160*a/c^4*x*(-c^2*d*x^2+d)^(5/2)+
1/128*a/c^4*d*x*(-c^2*d*x^2+d)^(3/2)+3/256*a/c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+3/256*a/c^4*d^3/(c^2*d)^(1/2)*arct
an((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/100*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^5/(c*x-1)^(1/2)*x^
10+21/640*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^8-31/960*b*(-d*(c^2*x^2-1))^(1/2)*d^2
/(c*x+1)^(1/2)*c/(c*x-1)^(1/2)*x^6+1/512*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^4+3/512*
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)*x^2+1/10*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^6
/(c*x-1)*arccosh(c*x)*x^11-29/80*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^9+73/160*b*(-
d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^7-129/640*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/(c*
x-1)*arccosh(c*x)*x^5-1/256*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x^3+3/256*b*(-d*(c^2
*x^2-1))^(1/2)*d^2/(c*x+1)/c^4/(c*x-1)*arccosh(c*x)*x-101/1228800*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x+1)^(1/2)/c
^5/(c*x-1)^(1/2)-3/512*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/c^5*arccosh(c*x)^2*d^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{1280} \, {\left (\frac {128 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x^{3}}{c^{2} d} - \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{4}} + \frac {48 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} x}{c^{4} d} - \frac {10 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x}{c^{4}} - \frac {15 \, \sqrt {-c^{2} d x^{2} + d} d^{2} x}{c^{4}} - \frac {15 \, d^{\frac {5}{2}} \arcsin \left (c x\right )}{c^{5}}\right )} a + b \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/1280*(128*(-c^2*d*x^2 + d)^(7/2)*x^3/(c^2*d) - 8*(-c^2*d*x^2 + d)^(5/2)*x/c^4 + 48*(-c^2*d*x^2 + d)^(7/2)*x
/(c^4*d) - 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^4 - 15*sqrt(-c^2*d*x^2 + d)*d^2*x/c^4 - 15*d^(5/2)*arcsin(c*x)/c^5)
*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*x^4*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2),x)

[Out]

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2), 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(x**4*(-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x)),x)

[Out]

Timed out

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