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

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

[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 + (x^5*(d - c^2*d*x^2)^(3/2)*(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])

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Rubi [A]  time = 1.06425, 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 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]

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 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 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 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 30

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

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]]

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*}

Mathematica [A]  time = 4.51278, size = 337, normalized size = 0.94 \[ \frac{d \left (-576 a c x \left (16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right ) \sqrt{d-c^2 d x^2}-1728 a \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+\frac{32 b \sqrt{d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{b \sqrt{d-c^2 d x^2} \left (1440 \cosh ^{-1}(c x)^2-576 \cosh \left (2 \cosh ^{-1}(c x)\right )+144 \cosh \left (4 \cosh ^{-1}(c x)\right )+64 \cosh \left (6 \cosh ^{-1}(c x)\right )+9 \cosh \left (8 \cosh ^{-1}(c x)\right )-24 \cosh ^{-1}(c x) \left (-48 \sinh \left (2 \cosh ^{-1}(c x)\right )+24 \sinh \left (4 \cosh ^{-1}(c x)\right )+16 \sinh \left (6 \cosh ^{-1}(c x)\right )+3 \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right )}{73728 c^5} \]

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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Maple [A]  time = 0.353, size = 561, normalized size = 1.6 \begin{align*} -{\frac{{x}^{3}a}{8\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}-{\frac{ax}{16\,d{c}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{ax}{64\,{c}^{4}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{128\,{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{3\,a{d}^{2}}{128\,{c}^{4}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{bd{c}^{4}{\rm arccosh} \left (cx\right ){x}^{9}}{ \left ( 8\,cx+8 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{5\,b{c}^{2}d{\rm arccosh} \left (cx\right ){x}^{7}}{ \left ( 16\,cx+16 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{13\,bd{\rm arccosh} \left (cx\right ){x}^{5}}{ \left ( 64\,cx+64 \right ) \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{bd{\rm arccosh} \left (cx\right ){x}^{3}}{ \left ( 128\,cx+128 \right ){c}^{2} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{3\,bd{\rm arccosh} \left (cx\right )x}{ \left ( 128\,cx+128 \right ){c}^{4} \left ( cx-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{15\,bd}{8192\,{c}^{5}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{3\,db \left ({\rm arccosh} \left (cx\right ) \right ) ^{2}}{256\,{c}^{5}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bd{c}^{3}{x}^{8}}{64}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}-{\frac{bdc{x}^{6}}{32}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{bd{x}^{4}}{256\,c}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}}+{\frac{3\,bd{x}^{2}}{256\,{c}^{3}}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{cx-1}}}{\frac{1}{\sqrt{cx+1}}}} \end{align*}

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))-
1/8*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)*arcc
osh(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)-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+1/64*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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)