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

Optimal. Leaf size=227 \[ -\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b d^3 (c x-1)^{9/2} (c x+1)^{9/2}}{81 c^3}+\frac{b d^3 (c x-1)^{7/2} (c x+1)^{7/2}}{441 c^3}-\frac{2 b d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{525 c^3}+\frac{8 b d^3 (c x-1)^{3/2} (c x+1)^{3/2}}{945 c^3}-\frac{16 b d^3 \sqrt{c x-1} \sqrt{c x+1}}{315 c^3} \]

[Out]

(-16*b*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(315*c^3) + (8*b*d^3*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(945*c^3) - (2
*b*d^3*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(525*c^3) + (b*d^3*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(441*c^3) + (b*d
^3*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(81*c^3) + (d^3*x^3*(a + b*ArcCosh[c*x]))/3 - (3*c^2*d^3*x^5*(a + b*ArcCo
sh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcCosh[c*x]))/7 - (c^6*d^3*x^9*(a + b*ArcCosh[c*x]))/9

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Rubi [A]  time = 0.395913, antiderivative size = 285, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {270, 5731, 12, 1610, 1799, 1620} \[ -\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b d^3 \left (1-c^2 x^2\right )^5}{81 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^3 \left (1-c^2 x^2\right )^4}{441 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{2 b d^3 \left (1-c^2 x^2\right )^3}{525 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^2}{945 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{16 b d^3 \left (1-c^2 x^2\right )}{315 c^3 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b*d^3*(1 - c^2*x^2))/(315*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (8*b*d^3*(1 - c^2*x^2)^2)/(945*c^3*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (2*b*d^3*(1 - c^2*x^2)^3)/(525*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^3*(1 - c^2*x^2)
^4)/(441*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^3*(1 - c^2*x^2)^5)/(81*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
d^3*x^3*(a + b*ArcCosh[c*x]))/3 - (3*c^2*d^3*x^5*(a + b*ArcCosh[c*x]))/5 + (3*c^4*d^3*x^7*(a + b*ArcCosh[c*x])
)/7 - (c^6*d^3*x^9*(a + b*ArcCosh[c*x]))/9

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(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, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

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

Rule 1610

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

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^2 \left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^3 x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{315 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{315} \left (b c d^3\right ) \int \frac{x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^3 \left (105-189 c^2 x^2+135 c^4 x^4-35 c^6 x^6\right )}{\sqrt{-1+c^2 x^2}} \, dx}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x \left (105-189 c^2 x+135 c^4 x^2-35 c^6 x^3\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^3 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{16}{c^2 \sqrt{-1+c^2 x}}-\frac{8 \sqrt{-1+c^2 x}}{c^2}+\frac{6 \left (-1+c^2 x\right )^{3/2}}{c^2}-\frac{5 \left (-1+c^2 x\right )^{5/2}}{c^2}-\frac{35 \left (-1+c^2 x\right )^{7/2}}{c^2}\right ) \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{16 b d^3 \left (1-c^2 x^2\right )}{315 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{8 b d^3 \left (1-c^2 x^2\right )^2}{945 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b d^3 \left (1-c^2 x^2\right )^3}{525 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^3 \left (1-c^2 x^2\right )^4}{441 c^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^3 \left (1-c^2 x^2\right )^5}{81 c^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{3} d^3 x^3 \left (a+b \cosh ^{-1}(c x)\right )-\frac{3}{5} c^2 d^3 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{3}{7} c^4 d^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{9} c^6 d^3 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.268577, size = 139, normalized size = 0.61 \[ -\frac{d^3 \left (315 a c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right )+b \sqrt{c x-1} \sqrt{c x+1} \left (-1225 c^8 x^8+4675 c^6 x^6-6297 c^4 x^4+2629 c^2 x^2+5258\right )+315 b c^3 x^3 \left (35 c^6 x^6-135 c^4 x^4+189 c^2 x^2-105\right ) \cosh ^{-1}(c x)\right )}{99225 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(315*a*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 + 35*c^6*x^6) + b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5258 +
2629*c^2*x^2 - 6297*c^4*x^4 + 4675*c^6*x^6 - 1225*c^8*x^8) + 315*b*c^3*x^3*(-105 + 189*c^2*x^2 - 135*c^4*x^4 +
35*c^6*x^6)*ArcCosh[c*x]))/(99225*c^3)

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Maple [A]  time = 0.013, size = 150, normalized size = 0.7 \begin{align*}{\frac{1}{{c}^{3}} \left ( -{d}^{3}a \left ({\frac{{c}^{9}{x}^{9}}{9}}-{\frac{3\,{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{5}{x}^{5}}{5}}-{\frac{{c}^{3}{x}^{3}}{3}} \right ) -{d}^{3}b \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{9}{x}^{9}}{9}}-{\frac{3\,{\rm arccosh} \left (cx\right ){c}^{7}{x}^{7}}{7}}+{\frac{3\,{\rm arccosh} \left (cx\right ){c}^{5}{x}^{5}}{5}}-{\frac{{c}^{3}{x}^{3}{\rm arccosh} \left (cx\right )}{3}}-{\frac{1225\,{c}^{8}{x}^{8}-4675\,{c}^{6}{x}^{6}+6297\,{c}^{4}{x}^{4}-2629\,{c}^{2}{x}^{2}-5258}{99225}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(-d^3*a*(1/9*c^9*x^9-3/7*c^7*x^7+3/5*c^5*x^5-1/3*c^3*x^3)-d^3*b*(1/9*arccosh(c*x)*c^9*x^9-3/7*arccosh(c*
x)*c^7*x^7+3/5*arccosh(c*x)*c^5*x^5-1/3*c^3*x^3*arccosh(c*x)-1/99225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*x^8
-4675*c^6*x^6+6297*c^4*x^4-2629*c^2*x^2-5258)))

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Maxima [B]  time = 1.28282, size = 524, normalized size = 2.31 \begin{align*} -\frac{1}{9} \, a c^{6} d^{3} x^{9} + \frac{3}{7} \, a c^{4} d^{3} x^{7} - \frac{1}{2835} \,{\left (315 \, x^{9} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{35 \, \sqrt{c^{2} x^{2} - 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{c^{2} x^{2} - 1}}{c^{10}}\right )} c\right )} b c^{6} d^{3} - \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{3}{245} \,{\left (35 \, x^{7} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{5 \, \sqrt{c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b c^{4} d^{3} - \frac{1}{25} \,{\left (15 \, x^{5} \operatorname{arcosh}\left (c x\right ) -{\left (\frac{3 \, \sqrt{c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b c^{2} d^{3} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \operatorname{arcosh}\left (c x\right ) - c{\left (\frac{\sqrt{c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 - 1/2835*(315*x^9*arccosh(c*x) - (35*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqr
t(c^2*x^2 - 1)*x^6/c^4 + 48*sqrt(c^2*x^2 - 1)*x^4/c^6 + 64*sqrt(c^2*x^2 - 1)*x^2/c^8 + 128*sqrt(c^2*x^2 - 1)/c
^10)*c)*b*c^6*d^3 - 3/5*a*c^2*d^3*x^5 + 3/245*(35*x^7*arccosh(c*x) - (5*sqrt(c^2*x^2 - 1)*x^6/c^2 + 6*sqrt(c^2
*x^2 - 1)*x^4/c^4 + 8*sqrt(c^2*x^2 - 1)*x^2/c^6 + 16*sqrt(c^2*x^2 - 1)/c^8)*c)*b*c^4*d^3 - 1/25*(15*x^5*arccos
h(c*x) - (3*sqrt(c^2*x^2 - 1)*x^4/c^2 + 4*sqrt(c^2*x^2 - 1)*x^2/c^4 + 8*sqrt(c^2*x^2 - 1)/c^6)*c)*b*c^2*d^3 +
1/3*a*d^3*x^3 + 1/9*(3*x^3*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x^2/c^2 + 2*sqrt(c^2*x^2 - 1)/c^4))*b*d^3

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Fricas [A]  time = 1.81398, size = 450, normalized size = 1.98 \begin{align*} -\frac{11025 \, a c^{9} d^{3} x^{9} - 42525 \, a c^{7} d^{3} x^{7} + 59535 \, a c^{5} d^{3} x^{5} - 33075 \, a c^{3} d^{3} x^{3} + 315 \,{\left (35 \, b c^{9} d^{3} x^{9} - 135 \, b c^{7} d^{3} x^{7} + 189 \, b c^{5} d^{3} x^{5} - 105 \, b c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (1225 \, b c^{8} d^{3} x^{8} - 4675 \, b c^{6} d^{3} x^{6} + 6297 \, b c^{4} d^{3} x^{4} - 2629 \, b c^{2} d^{3} x^{2} - 5258 \, b d^{3}\right )} \sqrt{c^{2} x^{2} - 1}}{99225 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/99225*(11025*a*c^9*d^3*x^9 - 42525*a*c^7*d^3*x^7 + 59535*a*c^5*d^3*x^5 - 33075*a*c^3*d^3*x^3 + 315*(35*b*c^
9*d^3*x^9 - 135*b*c^7*d^3*x^7 + 189*b*c^5*d^3*x^5 - 105*b*c^3*d^3*x^3)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*
c^8*d^3*x^8 - 4675*b*c^6*d^3*x^6 + 6297*b*c^4*d^3*x^4 - 2629*b*c^2*d^3*x^2 - 5258*b*d^3)*sqrt(c^2*x^2 - 1))/c^
3

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Sympy [A]  time = 26.3989, size = 272, normalized size = 1.2 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{9}}{9} + \frac{3 a c^{4} d^{3} x^{7}}{7} - \frac{3 a c^{2} d^{3} x^{5}}{5} + \frac{a d^{3} x^{3}}{3} - \frac{b c^{6} d^{3} x^{9} \operatorname{acosh}{\left (c x \right )}}{9} + \frac{b c^{5} d^{3} x^{8} \sqrt{c^{2} x^{2} - 1}}{81} + \frac{3 b c^{4} d^{3} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} - \frac{187 b c^{3} d^{3} x^{6} \sqrt{c^{2} x^{2} - 1}}{3969} - \frac{3 b c^{2} d^{3} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} + \frac{2099 b c d^{3} x^{4} \sqrt{c^{2} x^{2} - 1}}{33075} + \frac{b d^{3} x^{3} \operatorname{acosh}{\left (c x \right )}}{3} - \frac{2629 b d^{3} x^{2} \sqrt{c^{2} x^{2} - 1}}{99225 c} - \frac{5258 b d^{3} \sqrt{c^{2} x^{2} - 1}}{99225 c^{3}} & \text{for}\: c \neq 0 \\\frac{d^{3} x^{3} \left (a + \frac{i \pi b}{2}\right )}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**6*d**3*x**9/9 + 3*a*c**4*d**3*x**7/7 - 3*a*c**2*d**3*x**5/5 + a*d**3*x**3/3 - b*c**6*d**3*x**
9*acosh(c*x)/9 + b*c**5*d**3*x**8*sqrt(c**2*x**2 - 1)/81 + 3*b*c**4*d**3*x**7*acosh(c*x)/7 - 187*b*c**3*d**3*x
**6*sqrt(c**2*x**2 - 1)/3969 - 3*b*c**2*d**3*x**5*acosh(c*x)/5 + 2099*b*c*d**3*x**4*sqrt(c**2*x**2 - 1)/33075
+ b*d**3*x**3*acosh(c*x)/3 - 2629*b*d**3*x**2*sqrt(c**2*x**2 - 1)/(99225*c) - 5258*b*d**3*sqrt(c**2*x**2 - 1)/
(99225*c**3), Ne(c, 0)), (d**3*x**3*(a + I*pi*b/2)/3, True))

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Giac [A]  time = 1.55787, size = 501, normalized size = 2.21 \begin{align*} -\frac{1}{9} \, a c^{6} d^{3} x^{9} + \frac{3}{7} \, a c^{4} d^{3} x^{7} - \frac{1}{2835} \,{\left (315 \, x^{9} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{9}{2}} + 180 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 378 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 420 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 315 \, \sqrt{c^{2} x^{2} - 1}}{c^{9}}\right )} b c^{6} d^{3} - \frac{3}{5} \, a c^{2} d^{3} x^{5} + \frac{3}{245} \,{\left (35 \, x^{7} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{5 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{7}{2}} + 21 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 35 \, \sqrt{c^{2} x^{2} - 1}}{c^{7}}\right )} b c^{4} d^{3} - \frac{1}{25} \,{\left (15 \, x^{5} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 10 \,{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{c^{2} x^{2} - 1}}{c^{5}}\right )} b c^{2} d^{3} + \frac{1}{3} \, a d^{3} x^{3} + \frac{1}{9} \,{\left (3 \, x^{3} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 3 \, \sqrt{c^{2} x^{2} - 1}}{c^{3}}\right )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a*c^6*d^3*x^9 + 3/7*a*c^4*d^3*x^7 - 1/2835*(315*x^9*log(c*x + sqrt(c^2*x^2 - 1)) - (35*(c^2*x^2 - 1)^(9/2
) + 180*(c^2*x^2 - 1)^(7/2) + 378*(c^2*x^2 - 1)^(5/2) + 420*(c^2*x^2 - 1)^(3/2) + 315*sqrt(c^2*x^2 - 1))/c^9)*
b*c^6*d^3 - 3/5*a*c^2*d^3*x^5 + 3/245*(35*x^7*log(c*x + sqrt(c^2*x^2 - 1)) - (5*(c^2*x^2 - 1)^(7/2) + 21*(c^2*
x^2 - 1)^(5/2) + 35*(c^2*x^2 - 1)^(3/2) + 35*sqrt(c^2*x^2 - 1))/c^7)*b*c^4*d^3 - 1/25*(15*x^5*log(c*x + sqrt(c
^2*x^2 - 1)) - (3*(c^2*x^2 - 1)^(5/2) + 10*(c^2*x^2 - 1)^(3/2) + 15*sqrt(c^2*x^2 - 1))/c^5)*b*c^2*d^3 + 1/3*a*
d^3*x^3 + 1/9*(3*x^3*log(c*x + sqrt(c^2*x^2 - 1)) - ((c^2*x^2 - 1)^(3/2) + 3*sqrt(c^2*x^2 - 1))/c^3)*b*d^3