3.71 \(\int x^4 (d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=360 \[ \frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

1/8*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))-3/128*d*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^4-1/64*d*x
^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/16*d*x^5*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)+3/256*b*d*x^
2*(-c^2*d*x^2+d)^(1/2)/c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/256*b*d*x^4*(-c^2*d*x^2+d)^(1/2)/c/(c*x-1)^(1/2)/(c*x
+1)^(1/2)-1/32*b*c*d*x^6*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/64*b*c^3*d*x^8*(-c^2*d*x^2+d)^(1/2
)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-3/256*d*(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 = 1.06, antiderivative size = 372, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5798, 5745, 5743, 5759, 5676, 30, 14} \[ \frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(3*b*d*x^2*Sqrt[d - c^2*d*x^2])/(256*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*x^4*Sqrt[d - c^2*d*x^2])/(256*c*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d*x^6*Sqrt[d - c^2*d*x^2])/(32*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x
^8*Sqrt[d - c^2*d*x^2])/(64*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(
128*c^4) - (d*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(64*c^2) + (d*x^5*Sqrt[d - c^2*d*x^2]*(a + b*ArcCo
sh[c*x]))/16 + (d*x^5*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/8 - (3*d*Sqrt[d - c^2*d*x^
2]*(a + b*ArcCosh[c*x])^2)/(256*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 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 )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {\left (d \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}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (3 d \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}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (d \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}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (3 d \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}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (3 d \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}{128 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 4.82, size = 337, normalized size = 0.94 \[ \frac {d \left (-1728 a \sqrt {d} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-576 a c x \left (16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right ) \sqrt {d-c^2 d x^2}+\frac {32 b \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {b \sqrt {d-c^2 d x^2} \left (1440 \cosh ^{-1}(c x)^2-576 \cosh \left (2 \cosh ^{-1}(c x)\right )+144 \cosh \left (4 \cosh ^{-1}(c x)\right )+64 \cosh \left (6 \cosh ^{-1}(c x)\right )+9 \cosh \left (8 \cosh ^{-1}(c x)\right )-24 \cosh ^{-1}(c x) \left (-48 \sinh \left (2 \cosh ^{-1}(c x)\right )+24 \sinh \left (4 \cosh ^{-1}(c x)\right )+16 \sinh \left (6 \cosh ^{-1}(c x)\right )+3 \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right )}{73728 c^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]),x]

[Out]

(d*(-576*a*c*x*Sqrt[d - c^2*d*x^2]*(3 + 2*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6) - 1728*a*Sqrt[d]*ArcTan[(c*x*Sqrt
[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + (32*b*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh
[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCosh[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4
*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (b*Sqrt[d - c^2*d*x^2]*(1440
*ArcCosh[c*x]^2 - 576*Cosh[2*ArcCosh[c*x]] + 144*Cosh[4*ArcCosh[c*x]] + 64*Cosh[6*ArcCosh[c*x]] + 9*Cosh[8*Arc
Cosh[c*x]] - 24*ArcCosh[c*x]*(-48*Sinh[2*ArcCosh[c*x]] + 24*Sinh[4*ArcCosh[c*x]] + 16*Sinh[6*ArcCosh[c*x]] + 3
*Sinh[8*ArcCosh[c*x]])))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(73728*c^5)

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fricas [F]  time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d x^{6} - a d x^{4} + {\left (b c^{2} d x^{6} - b d x^{4}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(-(a*c^2*d*x^6 - a*d*x^4 + (b*c^2*d*x^6 - b*d*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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*d*x^2 + d)^(3/2)*(b*arccosh(c*x) + a)*x^4, x)

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maple [A]  time = 0.62, size = 561, normalized size = 1.56 \[ -\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{4} \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{8}}{64 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \,x^{6}}{32 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4}}{256 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2}}{256 \sqrt {c x +1}\, c^{3} \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} \mathrm {arccosh}\left (c x \right ) x^{9}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{7}}{16 \left (c x +1\right ) \left (c x -1\right )}-\frac {13 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x^{5}}{64 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x^{3}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {15 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d}{8192 \sqrt {c x +1}\, c^{5} \sqrt {c x -1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x)

[Out]

-1/8*a*x^3*(-c^2*d*x^2+d)^(5/2)/c^2/d-1/16*a/c^4*x*(-c^2*d*x^2+d)^(5/2)/d+1/64*a/c^4*x*(-c^2*d*x^2+d)^(3/2)+3/
128*a/c^4*d*x*(-c^2*d*x^2+d)^(1/2)+3/128*a/c^4*d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+
1/64*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)*c^3/(c*x-1)^(1/2)*x^8-1/32*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1
/2)*c/(c*x-1)^(1/2)*x^6+1/256*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/c/(c*x-1)^(1/2)*x^4+3/256*b*(-d*(c^2*x^
2-1))^(1/2)*d/(c*x+1)^(1/2)/c^3/(c*x-1)^(1/2)*x^2-3/256*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-1/8*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)*c^4/(c*x-1)*arccosh(c*x)*x^9+5/16*b*(-d*(c^2*x^2-1)
)^(1/2)*d/(c*x+1)*c^2/(c*x-1)*arccosh(c*x)*x^7-13/64*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x
^5-1/128*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/c^2/(c*x-1)*arccosh(c*x)*x^3+3/128*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x
+1)/c^4/(c*x-1)*arccosh(c*x)*x-15/8192*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/c^5/(c*x-1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{128} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}{c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{4}} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{4} d} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{4}} - \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{5}}\right )} a + b \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x)),x, algorithm="maxima")

[Out]

-1/128*(16*(-c^2*d*x^2 + d)^(5/2)*x^3/(c^2*d) - 2*(-c^2*d*x^2 + d)^(3/2)*x/c^4 + 8*(-c^2*d*x^2 + d)^(5/2)*x/(c
^4*d) - 3*sqrt(-c^2*d*x^2 + d)*d*x/c^4 - 3*d^(3/2)*arcsin(c*x)/c^5)*a + b*integrate((-c^2*d*x^2 + d)^(3/2)*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 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]

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)^(3/2),x)

[Out]

int(x^4*(a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x)),x)

[Out]

Integral(x**4*(-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x)), x)

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