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

Optimal. Leaf size=206 \[ \frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{b d^2 (c x-1)^{9/2} (c x+1)^{9/2}}{81 c^5}-\frac{10 b d^2 (c x-1)^{7/2} (c x+1)^{7/2}}{441 c^5}-\frac{b d^2 (c x-1)^{5/2} (c x+1)^{5/2}}{525 c^5}+\frac{4 b d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{945 c^5}-\frac{8 b d^2 \sqrt{c x-1} \sqrt{c x+1}}{315 c^5} \]

[Out]

(-8*b*d^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(315*c^5) + (4*b*d^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2))/(945*c^5) - (b*
d^2*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2))/(525*c^5) - (10*b*d^2*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))/(441*c^5) - (b*d
^2*(-1 + c*x)^(9/2)*(1 + c*x)^(9/2))/(81*c^5) + (d^2*x^5*(a + b*ArcCosh[c*x]))/5 - (2*c^2*d^2*x^7*(a + b*ArcCo
sh[c*x]))/7 + (c^4*d^2*x^9*(a + b*ArcCosh[c*x]))/9

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Rubi [A]  time = 0.293386, antiderivative size = 264, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {270, 5731, 12, 520, 1251, 897, 1153} \[ \frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac{b d^2 \left (1-c^2 x^2\right )^5}{81 c^5 \sqrt{c x-1} \sqrt{c x+1}}-\frac{10 b d^2 \left (1-c^2 x^2\right )^4}{441 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b d^2 \left (1-c^2 x^2\right )^3}{525 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{4 b d^2 \left (1-c^2 x^2\right )^2}{945 c^5 \sqrt{c x-1} \sqrt{c x+1}}+\frac{8 b d^2 \left (1-c^2 x^2\right )}{315 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b*d^2*(1 - c^2*x^2))/(315*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (4*b*d^2*(1 - c^2*x^2)^2)/(945*c^5*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3)/(525*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (10*b*d^2*(1 - c^2*x^2)
^4)/(441*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^5)/(81*c^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
d^2*x^5*(a + b*ArcCosh[c*x]))/5 - (2*c^2*d^2*x^7*(a + b*ArcCosh[c*x]))/7 + (c^4*d^2*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 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(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)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], 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] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^4 \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{1}{315} \left (b c d^2\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{-1+c^2 x^2}} \, dx}{315 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b c d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt{-1+c^2 x}} \, dx,x,x^2\right )}{630 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}+\frac{x^2}{c^2}\right )^2 \left (8-20 x^2+35 x^4\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{315 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )-\frac{\left (b d^2 \sqrt{-1+c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{c^4}-\frac{4 x^2}{c^4}+\frac{3 x^4}{c^4}+\frac{50 x^6}{c^4}+\frac{35 x^8}{c^4}\right ) \, dx,x,\sqrt{-1+c^2 x^2}\right )}{315 c \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{8 b d^2 \left (1-c^2 x^2\right )}{315 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{4 b d^2 \left (1-c^2 x^2\right )^2}{945 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^3}{525 c^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{10 b d^2 \left (1-c^2 x^2\right )^4}{441 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 \left (1-c^2 x^2\right )^5}{81 c^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{1}{5} d^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \cosh ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.19838, size = 124, normalized size = 0.6 \[ \frac{d^2 \left (315 a c^5 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )-b \sqrt{c x-1} \sqrt{c x+1} \left (1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right )+315 b c^5 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right ) \cosh ^{-1}(c x)\right )}{99225 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) - b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(2104 + 1052*c^2*x^2 + 789
*c^4*x^4 - 2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcCosh[c*x]))/(99225*c
^5)

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Maple [A]  time = 0.013, size = 128, normalized size = 0.6 \begin{align*}{\frac{1}{{c}^{5}} \left ({d}^{2}a \left ({\frac{{c}^{9}{x}^{9}}{9}}-{\frac{2\,{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{5}{x}^{5}}{5}} \right ) +{d}^{2}b \left ({\frac{{\rm arccosh} \left (cx\right ){c}^{9}{x}^{9}}{9}}-{\frac{2\,{\rm arccosh} \left (cx\right ){c}^{7}{x}^{7}}{7}}+{\frac{{\rm arccosh} \left (cx\right ){c}^{5}{x}^{5}}{5}}-{\frac{1225\,{c}^{8}{x}^{8}-2650\,{c}^{6}{x}^{6}+789\,{c}^{4}{x}^{4}+1052\,{c}^{2}{x}^{2}+2104}{99225}\sqrt{cx-1}\sqrt{cx+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^5*(d^2*a*(1/9*c^9*x^9-2/7*c^7*x^7+1/5*c^5*x^5)+d^2*b*(1/9*arccosh(c*x)*c^9*x^9-2/7*arccosh(c*x)*c^7*x^7+1/
5*arccosh(c*x)*c^5*x^5-1/99225*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(1225*c^8*x^8-2650*c^6*x^6+789*c^4*x^4+1052*c^2*x^2
+2104)))

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Maxima [A]  time = 1.18024, size = 431, normalized size = 2.09 \begin{align*} \frac{1}{9} \, a c^{4} d^{2} x^{9} - \frac{2}{7} \, a c^{2} d^{2} 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^{4} d^{2} + \frac{1}{5} \, a d^{2} x^{5} - \frac{2}{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^{2} d^{2} + \frac{1}{75} \,{\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 d^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*x^7 + 1/2835*(315*x^9*arccosh(c*x) - (35*sqrt(c^2*x^2 - 1)*x^8/c^2 + 40*sqrt
(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^4*d^2 + 1/5*a*d^2*x^5 - 2/245*(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^2*d^2 + 1/75*(15*x^5*arccosh(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*d^2

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Fricas [A]  time = 1.86919, size = 387, normalized size = 1.88 \begin{align*} \frac{11025 \, a c^{9} d^{2} x^{9} - 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \,{\left (35 \, b c^{9} d^{2} x^{9} - 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (1225 \, b c^{8} d^{2} x^{8} - 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} + 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt{c^{2} x^{2} - 1}}{99225 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(11025*a*c^9*d^2*x^9 - 28350*a*c^7*d^2*x^7 + 19845*a*c^5*d^2*x^5 + 315*(35*b*c^9*d^2*x^9 - 90*b*c^7*d^
2*x^7 + 63*b*c^5*d^2*x^5)*log(c*x + sqrt(c^2*x^2 - 1)) - (1225*b*c^8*d^2*x^8 - 2650*b*c^6*d^2*x^6 + 789*b*c^4*
d^2*x^4 + 1052*b*c^2*d^2*x^2 + 2104*b*d^2)*sqrt(c^2*x^2 - 1))/c^5

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Sympy [A]  time = 26.2538, size = 236, normalized size = 1.15 \begin{align*} \begin{cases} \frac{a c^{4} d^{2} x^{9}}{9} - \frac{2 a c^{2} d^{2} x^{7}}{7} + \frac{a d^{2} x^{5}}{5} + \frac{b c^{4} d^{2} x^{9} \operatorname{acosh}{\left (c x \right )}}{9} - \frac{b c^{3} d^{2} x^{8} \sqrt{c^{2} x^{2} - 1}}{81} - \frac{2 b c^{2} d^{2} x^{7} \operatorname{acosh}{\left (c x \right )}}{7} + \frac{106 b c d^{2} x^{6} \sqrt{c^{2} x^{2} - 1}}{3969} + \frac{b d^{2} x^{5} \operatorname{acosh}{\left (c x \right )}}{5} - \frac{263 b d^{2} x^{4} \sqrt{c^{2} x^{2} - 1}}{33075 c} - \frac{1052 b d^{2} x^{2} \sqrt{c^{2} x^{2} - 1}}{99225 c^{3}} - \frac{2104 b d^{2} \sqrt{c^{2} x^{2} - 1}}{99225 c^{5}} & \text{for}\: c \neq 0 \\\frac{d^{2} x^{5} \left (a + \frac{i \pi b}{2}\right )}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**4*d**2*x**9/9 - 2*a*c**2*d**2*x**7/7 + a*d**2*x**5/5 + b*c**4*d**2*x**9*acosh(c*x)/9 - b*c**3*
d**2*x**8*sqrt(c**2*x**2 - 1)/81 - 2*b*c**2*d**2*x**7*acosh(c*x)/7 + 106*b*c*d**2*x**6*sqrt(c**2*x**2 - 1)/396
9 + b*d**2*x**5*acosh(c*x)/5 - 263*b*d**2*x**4*sqrt(c**2*x**2 - 1)/(33075*c) - 1052*b*d**2*x**2*sqrt(c**2*x**2
- 1)/(99225*c**3) - 2104*b*d**2*sqrt(c**2*x**2 - 1)/(99225*c**5), Ne(c, 0)), (d**2*x**5*(a + I*pi*b/2)/5, Tru
e))

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Giac [A]  time = 1.50041, size = 402, normalized size = 1.95 \begin{align*} \frac{1}{9} \, a c^{4} d^{2} x^{9} - \frac{2}{7} \, a c^{2} d^{2} 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^{4} d^{2} + \frac{1}{5} \, a d^{2} x^{5} - \frac{2}{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^{2} d^{2} + \frac{1}{75} \,{\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 d^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*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^4*d^2 + 1/5*a*d^2*x^5 - 2/245*(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^2*d^2 + 1/75*(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*d^2