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

Optimal. Leaf size=370 \[ \frac{2 d^2 \left (1-a^2 x^2\right )^3 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{525 a^9 \sqrt{a x-1} \sqrt{a x+1}}-\frac{4 d \left (1-a^2 x^2\right )^2 \left (189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right )}{945 a^9 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\left (1-a^2 x^2\right ) \left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right )}{315 a^9 \sqrt{a x-1} \sqrt{a x+1}}-\frac{4 d^3 \left (1-a^2 x^2\right )^4 \left (9 a^2 c+7 d\right )}{441 a^9 \sqrt{a x-1} \sqrt{a x+1}}+\frac{d^4 \left (1-a^2 x^2\right )^5}{81 a^9 \sqrt{a x-1} \sqrt{a x+1}}+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+c^4 x \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x) \]

[Out]

((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*(1 - a^2*x^2))/(315*a^9*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]) - (4*d*(105*a^6*c^3 + 189*a^4*c^2*d + 135*a^2*c*d^2 + 35*d^3)*(1 - a^2*x^2)^2)/(945*a^9*Sqrt
[-1 + a*x]*Sqrt[1 + a*x]) + (2*d^2*(63*a^4*c^2 + 90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^3)/(525*a^9*Sqrt[-1 + a*x]
*Sqrt[1 + a*x]) - (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^4)/(441*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (d^4*(1 - a
^2*x^2)^5)/(81*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^4*x*ArcCosh[a*x] + (4*c^3*d*x^3*ArcCosh[a*x])/3 + (6*c^2*
d^2*x^5*ArcCosh[a*x])/5 + (4*c*d^3*x^7*ArcCosh[a*x])/7 + (d^4*x^9*ArcCosh[a*x])/9

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Rubi [A]  time = 0.459296, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {194, 5705, 12, 1610, 1799, 1850} \[ \frac{2 d^2 \left (1-a^2 x^2\right )^3 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{525 a^9 \sqrt{a x-1} \sqrt{a x+1}}-\frac{4 d \left (1-a^2 x^2\right )^2 \left (189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right )}{945 a^9 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\left (1-a^2 x^2\right ) \left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right )}{315 a^9 \sqrt{a x-1} \sqrt{a x+1}}-\frac{4 d^3 \left (1-a^2 x^2\right )^4 \left (9 a^2 c+7 d\right )}{441 a^9 \sqrt{a x-1} \sqrt{a x+1}}+\frac{d^4 \left (1-a^2 x^2\right )^5}{81 a^9 \sqrt{a x-1} \sqrt{a x+1}}+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+c^4 x \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4*ArcCosh[a*x],x]

[Out]

((315*a^8*c^4 + 420*a^6*c^3*d + 378*a^4*c^2*d^2 + 180*a^2*c*d^3 + 35*d^4)*(1 - a^2*x^2))/(315*a^9*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]) - (4*d*(105*a^6*c^3 + 189*a^4*c^2*d + 135*a^2*c*d^2 + 35*d^3)*(1 - a^2*x^2)^2)/(945*a^9*Sqrt
[-1 + a*x]*Sqrt[1 + a*x]) + (2*d^2*(63*a^4*c^2 + 90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^3)/(525*a^9*Sqrt[-1 + a*x]
*Sqrt[1 + a*x]) - (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^4)/(441*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (d^4*(1 - a
^2*x^2)^5)/(81*a^9*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + c^4*x*ArcCosh[a*x] + (4*c^3*d*x^3*ArcCosh[a*x])/3 + (6*c^2*
d^2*x^5*ArcCosh[a*x])/5 + (4*c*d^3*x^7*ArcCosh[a*x])/7 + (d^4*x^9*ArcCosh[a*x])/9

Rule 194

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

Rule 5705

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (c+d x^2\right )^4 \cosh ^{-1}(a x) \, dx &=c^4 x \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x)-a \int \frac{x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )}{315 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=c^4 x \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x)-\frac{1}{315} a \int \frac{x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=c^4 x \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )}{\sqrt{-1+a^2 x^2}} \, dx}{315 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=c^4 x \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{315 c^4+420 c^3 d x+378 c^2 d^2 x^2+180 c d^3 x^3+35 d^4 x^4}{\sqrt{-1+a^2 x}} \, dx,x,x^2\right )}{630 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=c^4 x \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x)-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4}{a^8 \sqrt{-1+a^2 x}}+\frac{4 d \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right ) \sqrt{-1+a^2 x}}{a^8}+\frac{6 d^2 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right ) \left (-1+a^2 x\right )^{3/2}}{a^8}+\frac{20 d^3 \left (9 a^2 c+7 d\right ) \left (-1+a^2 x\right )^{5/2}}{a^8}+\frac{35 d^4 \left (-1+a^2 x\right )^{7/2}}{a^8}\right ) \, dx,x,x^2\right )}{630 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \left (1-a^2 x^2\right )}{315 a^9 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{4 d \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right ) \left (1-a^2 x^2\right )^2}{945 a^9 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{2 d^2 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right ) \left (1-a^2 x^2\right )^3}{525 a^9 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{4 d^3 \left (9 a^2 c+7 d\right ) \left (1-a^2 x^2\right )^4}{441 a^9 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{d^4 \left (1-a^2 x^2\right )^5}{81 a^9 \sqrt{-1+a x} \sqrt{1+a x}}+c^4 x \cosh ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cosh ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cosh ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cosh ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cosh ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.281725, size = 216, normalized size = 0.58 \[ \frac{1}{315} x \cosh ^{-1}(a x) \left (378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4+180 c d^3 x^6+35 d^4 x^8\right )-\frac{\sqrt{a x-1} \sqrt{a x+1} \left (a^8 \left (23814 c^2 d^2 x^4+44100 c^3 d x^2+99225 c^4+8100 c d^3 x^6+1225 d^4 x^8\right )+8 a^6 d \left (3969 c^2 d x^2+11025 c^3+1215 c d^2 x^4+175 d^3 x^6\right )+48 a^4 d^2 \left (1323 c^2+270 c d x^2+35 d^2 x^4\right )+320 a^2 d^3 \left (81 c+7 d x^2\right )+4480 d^4\right )}{99225 a^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(4480*d^4 + 320*a^2*d^3*(81*c + 7*d*x^2) + 48*a^4*d^2*(1323*c^2 + 270*c*d*x^2 +
35*d^2*x^4) + 8*a^6*d*(11025*c^3 + 3969*c^2*d*x^2 + 1215*c*d^2*x^4 + 175*d^3*x^6) + a^8*(99225*c^4 + 44100*c^
3*d*x^2 + 23814*c^2*d^2*x^4 + 8100*c*d^3*x^6 + 1225*d^4*x^8)))/(99225*a^9) + (x*(315*c^4 + 420*c^3*d*x^2 + 378
*c^2*d^2*x^4 + 180*c*d^3*x^6 + 35*d^4*x^8)*ArcCosh[a*x])/315

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Maple [A]  time = 0.03, size = 255, normalized size = 0.7 \begin{align*}{\frac{1}{a} \left ({\frac{a{\rm arccosh} \left (ax\right ){d}^{4}{x}^{9}}{9}}+{\frac{4\,a{\rm arccosh} \left (ax\right )c{d}^{3}{x}^{7}}{7}}+{\frac{6\,a{\rm arccosh} \left (ax\right ){c}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{4\,a{\rm arccosh} \left (ax\right ){c}^{3}d{x}^{3}}{3}}+{\rm arccosh} \left (ax\right ){c}^{4}ax-{\frac{1225\,{a}^{8}{d}^{4}{x}^{8}+8100\,{a}^{8}c{d}^{3}{x}^{6}+23814\,{a}^{8}{c}^{2}{d}^{2}{x}^{4}+1400\,{a}^{6}{d}^{4}{x}^{6}+44100\,{a}^{8}{c}^{3}d{x}^{2}+9720\,{a}^{6}c{d}^{3}{x}^{4}+99225\,{a}^{8}{c}^{4}+31752\,{a}^{6}{c}^{2}{d}^{2}{x}^{2}+1680\,{a}^{4}{d}^{4}{x}^{4}+88200\,{a}^{6}{c}^{3}d+12960\,{a}^{4}c{d}^{3}{x}^{2}+63504\,{a}^{4}{c}^{2}{d}^{2}+2240\,{a}^{2}{d}^{4}{x}^{2}+25920\,{a}^{2}c{d}^{3}+4480\,{d}^{4}}{99225\,{a}^{8}}\sqrt{ax-1}\sqrt{ax+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(1/9*a*arccosh(a*x)*d^4*x^9+4/7*a*arccosh(a*x)*c*d^3*x^7+6/5*a*arccosh(a*x)*c^2*d^2*x^5+4/3*a*arccosh(a*x)
*c^3*d*x^3+arccosh(a*x)*c^4*a*x-1/99225/a^8*(a*x-1)^(1/2)*(a*x+1)^(1/2)*(1225*a^8*d^4*x^8+8100*a^8*c*d^3*x^6+2
3814*a^8*c^2*d^2*x^4+1400*a^6*d^4*x^6+44100*a^8*c^3*d*x^2+9720*a^6*c*d^3*x^4+99225*a^8*c^4+31752*a^6*c^2*d^2*x
^2+1680*a^4*d^4*x^4+88200*a^6*c^3*d+12960*a^4*c*d^3*x^2+63504*a^4*c^2*d^2+2240*a^2*d^4*x^2+25920*a^2*c*d^3+448
0*d^4))

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Maxima [A]  time = 1.13737, size = 520, normalized size = 1.41 \begin{align*} -\frac{1}{99225} \,{\left (\frac{1225 \, \sqrt{a^{2} x^{2} - 1} d^{4} x^{8}}{a^{2}} + \frac{8100 \, \sqrt{a^{2} x^{2} - 1} c d^{3} x^{6}}{a^{2}} + \frac{23814 \, \sqrt{a^{2} x^{2} - 1} c^{2} d^{2} x^{4}}{a^{2}} + \frac{1400 \, \sqrt{a^{2} x^{2} - 1} d^{4} x^{6}}{a^{4}} + \frac{44100 \, \sqrt{a^{2} x^{2} - 1} c^{3} d x^{2}}{a^{2}} + \frac{9720 \, \sqrt{a^{2} x^{2} - 1} c d^{3} x^{4}}{a^{4}} + \frac{99225 \, \sqrt{a^{2} x^{2} - 1} c^{4}}{a^{2}} + \frac{31752 \, \sqrt{a^{2} x^{2} - 1} c^{2} d^{2} x^{2}}{a^{4}} + \frac{1680 \, \sqrt{a^{2} x^{2} - 1} d^{4} x^{4}}{a^{6}} + \frac{88200 \, \sqrt{a^{2} x^{2} - 1} c^{3} d}{a^{4}} + \frac{12960 \, \sqrt{a^{2} x^{2} - 1} c d^{3} x^{2}}{a^{6}} + \frac{63504 \, \sqrt{a^{2} x^{2} - 1} c^{2} d^{2}}{a^{6}} + \frac{2240 \, \sqrt{a^{2} x^{2} - 1} d^{4} x^{2}}{a^{8}} + \frac{25920 \, \sqrt{a^{2} x^{2} - 1} c d^{3}}{a^{8}} + \frac{4480 \, \sqrt{a^{2} x^{2} - 1} d^{4}}{a^{10}}\right )} a + \frac{1}{315} \,{\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \operatorname{arcosh}\left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/99225*(1225*sqrt(a^2*x^2 - 1)*d^4*x^8/a^2 + 8100*sqrt(a^2*x^2 - 1)*c*d^3*x^6/a^2 + 23814*sqrt(a^2*x^2 - 1)*
c^2*d^2*x^4/a^2 + 1400*sqrt(a^2*x^2 - 1)*d^4*x^6/a^4 + 44100*sqrt(a^2*x^2 - 1)*c^3*d*x^2/a^2 + 9720*sqrt(a^2*x
^2 - 1)*c*d^3*x^4/a^4 + 99225*sqrt(a^2*x^2 - 1)*c^4/a^2 + 31752*sqrt(a^2*x^2 - 1)*c^2*d^2*x^2/a^4 + 1680*sqrt(
a^2*x^2 - 1)*d^4*x^4/a^6 + 88200*sqrt(a^2*x^2 - 1)*c^3*d/a^4 + 12960*sqrt(a^2*x^2 - 1)*c*d^3*x^2/a^6 + 63504*s
qrt(a^2*x^2 - 1)*c^2*d^2/a^6 + 2240*sqrt(a^2*x^2 - 1)*d^4*x^2/a^8 + 25920*sqrt(a^2*x^2 - 1)*c*d^3/a^8 + 4480*s
qrt(a^2*x^2 - 1)*d^4/a^10)*a + 1/315*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315*c^4*x
)*arccosh(a*x)

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Fricas [A]  time = 2.33466, size = 590, normalized size = 1.59 \begin{align*} \frac{315 \,{\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) -{\left (1225 \, a^{8} d^{4} x^{8} + 99225 \, a^{8} c^{4} + 88200 \, a^{6} c^{3} d + 63504 \, a^{4} c^{2} d^{2} + 100 \,{\left (81 \, a^{8} c d^{3} + 14 \, a^{6} d^{4}\right )} x^{6} + 25920 \, a^{2} c d^{3} + 6 \,{\left (3969 \, a^{8} c^{2} d^{2} + 1620 \, a^{6} c d^{3} + 280 \, a^{4} d^{4}\right )} x^{4} + 4480 \, d^{4} + 4 \,{\left (11025 \, a^{8} c^{3} d + 7938 \, a^{6} c^{2} d^{2} + 3240 \, a^{4} c d^{3} + 560 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt{a^{2} x^{2} - 1}}{99225 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(315*(35*a^9*d^4*x^9 + 180*a^9*c*d^3*x^7 + 378*a^9*c^2*d^2*x^5 + 420*a^9*c^3*d*x^3 + 315*a^9*c^4*x)*lo
g(a*x + sqrt(a^2*x^2 - 1)) - (1225*a^8*d^4*x^8 + 99225*a^8*c^4 + 88200*a^6*c^3*d + 63504*a^4*c^2*d^2 + 100*(81
*a^8*c*d^3 + 14*a^6*d^4)*x^6 + 25920*a^2*c*d^3 + 6*(3969*a^8*c^2*d^2 + 1620*a^6*c*d^3 + 280*a^4*d^4)*x^4 + 448
0*d^4 + 4*(11025*a^8*c^3*d + 7938*a^6*c^2*d^2 + 3240*a^4*c*d^3 + 560*a^2*d^4)*x^2)*sqrt(a^2*x^2 - 1))/a^9

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Sympy [A]  time = 25.8003, size = 503, normalized size = 1.36 \begin{align*} \begin{cases} c^{4} x \operatorname{acosh}{\left (a x \right )} + \frac{4 c^{3} d x^{3} \operatorname{acosh}{\left (a x \right )}}{3} + \frac{6 c^{2} d^{2} x^{5} \operatorname{acosh}{\left (a x \right )}}{5} + \frac{4 c d^{3} x^{7} \operatorname{acosh}{\left (a x \right )}}{7} + \frac{d^{4} x^{9} \operatorname{acosh}{\left (a x \right )}}{9} - \frac{c^{4} \sqrt{a^{2} x^{2} - 1}}{a} - \frac{4 c^{3} d x^{2} \sqrt{a^{2} x^{2} - 1}}{9 a} - \frac{6 c^{2} d^{2} x^{4} \sqrt{a^{2} x^{2} - 1}}{25 a} - \frac{4 c d^{3} x^{6} \sqrt{a^{2} x^{2} - 1}}{49 a} - \frac{d^{4} x^{8} \sqrt{a^{2} x^{2} - 1}}{81 a} - \frac{8 c^{3} d \sqrt{a^{2} x^{2} - 1}}{9 a^{3}} - \frac{8 c^{2} d^{2} x^{2} \sqrt{a^{2} x^{2} - 1}}{25 a^{3}} - \frac{24 c d^{3} x^{4} \sqrt{a^{2} x^{2} - 1}}{245 a^{3}} - \frac{8 d^{4} x^{6} \sqrt{a^{2} x^{2} - 1}}{567 a^{3}} - \frac{16 c^{2} d^{2} \sqrt{a^{2} x^{2} - 1}}{25 a^{5}} - \frac{32 c d^{3} x^{2} \sqrt{a^{2} x^{2} - 1}}{245 a^{5}} - \frac{16 d^{4} x^{4} \sqrt{a^{2} x^{2} - 1}}{945 a^{5}} - \frac{64 c d^{3} \sqrt{a^{2} x^{2} - 1}}{245 a^{7}} - \frac{64 d^{4} x^{2} \sqrt{a^{2} x^{2} - 1}}{2835 a^{7}} - \frac{128 d^{4} \sqrt{a^{2} x^{2} - 1}}{2835 a^{9}} & \text{for}\: a \neq 0 \\\frac{i \pi \left (c^{4} x + \frac{4 c^{3} d x^{3}}{3} + \frac{6 c^{2} d^{2} x^{5}}{5} + \frac{4 c d^{3} x^{7}}{7} + \frac{d^{4} x^{9}}{9}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((c**4*x*acosh(a*x) + 4*c**3*d*x**3*acosh(a*x)/3 + 6*c**2*d**2*x**5*acosh(a*x)/5 + 4*c*d**3*x**7*acos
h(a*x)/7 + d**4*x**9*acosh(a*x)/9 - c**4*sqrt(a**2*x**2 - 1)/a - 4*c**3*d*x**2*sqrt(a**2*x**2 - 1)/(9*a) - 6*c
**2*d**2*x**4*sqrt(a**2*x**2 - 1)/(25*a) - 4*c*d**3*x**6*sqrt(a**2*x**2 - 1)/(49*a) - d**4*x**8*sqrt(a**2*x**2
- 1)/(81*a) - 8*c**3*d*sqrt(a**2*x**2 - 1)/(9*a**3) - 8*c**2*d**2*x**2*sqrt(a**2*x**2 - 1)/(25*a**3) - 24*c*d
**3*x**4*sqrt(a**2*x**2 - 1)/(245*a**3) - 8*d**4*x**6*sqrt(a**2*x**2 - 1)/(567*a**3) - 16*c**2*d**2*sqrt(a**2*
x**2 - 1)/(25*a**5) - 32*c*d**3*x**2*sqrt(a**2*x**2 - 1)/(245*a**5) - 16*d**4*x**4*sqrt(a**2*x**2 - 1)/(945*a*
*5) - 64*c*d**3*sqrt(a**2*x**2 - 1)/(245*a**7) - 64*d**4*x**2*sqrt(a**2*x**2 - 1)/(2835*a**7) - 128*d**4*sqrt(
a**2*x**2 - 1)/(2835*a**9), Ne(a, 0)), (I*pi*(c**4*x + 4*c**3*d*x**3/3 + 6*c**2*d**2*x**5/5 + 4*c*d**3*x**7/7
+ d**4*x**9/9)/2, True))

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Giac [A]  time = 1.15932, size = 478, normalized size = 1.29 \begin{align*} \frac{1}{315} \,{\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{99225 \, \sqrt{a^{2} x^{2} - 1} a^{8} c^{4} + 44100 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} a^{6} c^{3} d + 132300 \, \sqrt{a^{2} x^{2} - 1} a^{6} c^{3} d + 23814 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} a^{4} c^{2} d^{2} + 79380 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} a^{4} c^{2} d^{2} + 8100 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{7}{2}} a^{2} c d^{3} + 119070 \, \sqrt{a^{2} x^{2} - 1} a^{4} c^{2} d^{2} + 34020 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} a^{2} c d^{3} + 1225 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{9}{2}} d^{4} + 56700 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} a^{2} c d^{3} + 6300 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{7}{2}} d^{4} + 56700 \, \sqrt{a^{2} x^{2} - 1} a^{2} c d^{3} + 13230 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} d^{4} + 14700 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} d^{4} + 11025 \, \sqrt{a^{2} x^{2} - 1} d^{4}}{99225 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*(35*d^4*x^9 + 180*c*d^3*x^7 + 378*c^2*d^2*x^5 + 420*c^3*d*x^3 + 315*c^4*x)*log(a*x + sqrt(a^2*x^2 - 1))
- 1/99225*(99225*sqrt(a^2*x^2 - 1)*a^8*c^4 + 44100*(a^2*x^2 - 1)^(3/2)*a^6*c^3*d + 132300*sqrt(a^2*x^2 - 1)*a^
6*c^3*d + 23814*(a^2*x^2 - 1)^(5/2)*a^4*c^2*d^2 + 79380*(a^2*x^2 - 1)^(3/2)*a^4*c^2*d^2 + 8100*(a^2*x^2 - 1)^(
7/2)*a^2*c*d^3 + 119070*sqrt(a^2*x^2 - 1)*a^4*c^2*d^2 + 34020*(a^2*x^2 - 1)^(5/2)*a^2*c*d^3 + 1225*(a^2*x^2 -
1)^(9/2)*d^4 + 56700*(a^2*x^2 - 1)^(3/2)*a^2*c*d^3 + 6300*(a^2*x^2 - 1)^(7/2)*d^4 + 56700*sqrt(a^2*x^2 - 1)*a^
2*c*d^3 + 13230*(a^2*x^2 - 1)^(5/2)*d^4 + 14700*(a^2*x^2 - 1)^(3/2)*d^4 + 11025*sqrt(a^2*x^2 - 1)*d^4)/a^9