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

Optimal. Leaf size=292 \[ -\frac{2 d^2 \left (1-a^2 x^2\right )^{5/2} \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{525 a^9}+\frac{4 d \left (1-a^2 x^2\right )^{3/2} \left (189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right )}{945 a^9}-\frac{\sqrt{1-a^2 x^2} \left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right )}{315 a^9}+\frac{4 d^3 \left (1-a^2 x^2\right )^{7/2} \left (9 a^2 c+7 d\right )}{441 a^9}-\frac{d^4 \left (1-a^2 x^2\right )^{9/2}}{81 a^9}+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+c^4 x \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x) \]

[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)*Sqrt[1 - a^2*x^2])/(315*a^9) + (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)^(3/2))/(945*a^9) - (2*d^2*(63*a^4*c^2 +
90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^(5/2))/(525*a^9) + (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^(7/2))/(441*a^9) -
(d^4*(1 - a^2*x^2)^(9/2))/(81*a^9) + c^4*x*ArcCos[a*x] + (4*c^3*d*x^3*ArcCos[a*x])/3 + (6*c^2*d^2*x^5*ArcCos[a
*x])/5 + (4*c*d^3*x^7*ArcCos[a*x])/7 + (d^4*x^9*ArcCos[a*x])/9

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Rubi [A]  time = 0.326099, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {194, 4666, 12, 1799, 1850} \[ -\frac{2 d^2 \left (1-a^2 x^2\right )^{5/2} \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{525 a^9}+\frac{4 d \left (1-a^2 x^2\right )^{3/2} \left (189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right )}{945 a^9}-\frac{\sqrt{1-a^2 x^2} \left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right )}{315 a^9}+\frac{4 d^3 \left (1-a^2 x^2\right )^{7/2} \left (9 a^2 c+7 d\right )}{441 a^9}-\frac{d^4 \left (1-a^2 x^2\right )^{9/2}}{81 a^9}+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+c^4 x \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^4*ArcCos[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)*Sqrt[1 - a^2*x^2])/(315*a^9) + (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)^(3/2))/(945*a^9) - (2*d^2*(63*a^4*c^2 +
90*a^2*c*d + 35*d^2)*(1 - a^2*x^2)^(5/2))/(525*a^9) + (4*d^3*(9*a^2*c + 7*d)*(1 - a^2*x^2)^(7/2))/(441*a^9) -
(d^4*(1 - a^2*x^2)^(9/2))/(81*a^9) + c^4*x*ArcCos[a*x] + (4*c^3*d*x^3*ArcCos[a*x])/3 + (6*c^2*d^2*x^5*ArcCos[a
*x])/5 + (4*c*d^3*x^7*ArcCos[a*x])/7 + (d^4*x^9*ArcCos[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 4666

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcCos[c*x], u, x] + Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{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 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 \cos ^{-1}(a x) \, dx &=c^4 x \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-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^2 x^2}} \, dx\\ &=c^4 x \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-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^2 x^2}} \, dx\\ &=c^4 x \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x)+\frac{1}{630} a \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 )\\ &=c^4 x \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x)+\frac{1}{630} a \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 )\\ &=-\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 ) \sqrt{1-a^2 x^2}}{315 a^9}+\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 )^{3/2}}{945 a^9}-\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 )^{5/2}}{525 a^9}+\frac{4 d^3 \left (9 a^2 c+7 d\right ) \left (1-a^2 x^2\right )^{7/2}}{441 a^9}-\frac{d^4 \left (1-a^2 x^2\right )^{9/2}}{81 a^9}+c^4 x \cos ^{-1}(a x)+\frac{4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac{6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac{4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac{1}{9} d^4 x^9 \cos ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.184607, size = 212, normalized size = 0.73 \[ \frac{1}{315} x \cos ^{-1}(a x) \left (378 c^2 d^2 x^4+420 c^3 d x^2+315 c^4+180 c d^3 x^6+35 d^4 x^8\right )-\frac{\sqrt{1-a^2 x^2} \left (a^8 \left (23814 c^2 d^2 x^4+44100 c^3 d x^2+99225 c^4+8100 c d^3 x^6+1225 d^4 x^8\right )+8 a^6 d \left (3969 c^2 d x^2+11025 c^3+1215 c d^2 x^4+175 d^3 x^6\right )+48 a^4 d^2 \left (1323 c^2+270 c d x^2+35 d^2 x^4\right )+320 a^2 d^3 \left (81 c+7 d x^2\right )+4480 d^4\right )}{99225 a^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]*(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 + 2
3814*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)*ArcCos[a*x])/315

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Maple [A]  time = 0.004, size = 393, normalized size = 1.4 \begin{align*}{\frac{1}{a} \left ({\frac{a\arccos \left ( ax \right ){d}^{4}{x}^{9}}{9}}+{\frac{4\,a\arccos \left ( ax \right ) c{d}^{3}{x}^{7}}{7}}+{\frac{6\,a\arccos \left ( ax \right ){c}^{2}{d}^{2}{x}^{5}}{5}}+{\frac{4\,a\arccos \left ( ax \right ){c}^{3}d{x}^{3}}{3}}+\arccos \left ( ax \right ){c}^{4}ax+{\frac{1}{315\,{a}^{8}} \left ( 35\,{d}^{4} \left ( -1/9\,{a}^{8}{x}^{8}\sqrt{-{a}^{2}{x}^{2}+1}-{\frac{8\,{a}^{6}{x}^{6}\sqrt{-{a}^{2}{x}^{2}+1}}{63}}-{\frac{16\,{a}^{4}{x}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}{105}}-{\frac{64\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{315}}-{\frac{128\,\sqrt{-{a}^{2}{x}^{2}+1}}{315}} \right ) +180\,{a}^{2}c{d}^{3} \left ( -1/7\,{a}^{6}{x}^{6}\sqrt{-{a}^{2}{x}^{2}+1}-{\frac{6\,{a}^{4}{x}^{4}\sqrt{-{a}^{2}{x}^{2}+1}}{35}}-{\frac{8\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{35}}-{\frac{16\,\sqrt{-{a}^{2}{x}^{2}+1}}{35}} \right ) +378\,{a}^{4}{c}^{2}{d}^{2} \left ( -1/5\,{a}^{4}{x}^{4}\sqrt{-{a}^{2}{x}^{2}+1}-{\frac{4\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}}{15}}-{\frac{8\,\sqrt{-{a}^{2}{x}^{2}+1}}{15}} \right ) +420\,{a}^{6}{c}^{3}d \left ( -1/3\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2/3\,\sqrt{-{a}^{2}{x}^{2}+1} \right ) -315\,{a}^{8}{c}^{4}\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(1/9*a*arccos(a*x)*d^4*x^9+4/7*a*arccos(a*x)*c*d^3*x^7+6/5*a*arccos(a*x)*c^2*d^2*x^5+4/3*a*arccos(a*x)*c^3
*d*x^3+arccos(a*x)*c^4*a*x+1/315/a^8*(35*d^4*(-1/9*a^8*x^8*(-a^2*x^2+1)^(1/2)-8/63*a^6*x^6*(-a^2*x^2+1)^(1/2)-
16/105*a^4*x^4*(-a^2*x^2+1)^(1/2)-64/315*a^2*x^2*(-a^2*x^2+1)^(1/2)-128/315*(-a^2*x^2+1)^(1/2))+180*a^2*c*d^3*
(-1/7*a^6*x^6*(-a^2*x^2+1)^(1/2)-6/35*a^4*x^4*(-a^2*x^2+1)^(1/2)-8/35*a^2*x^2*(-a^2*x^2+1)^(1/2)-16/35*(-a^2*x
^2+1)^(1/2))+378*a^4*c^2*d^2*(-1/5*a^4*x^4*(-a^2*x^2+1)^(1/2)-4/15*a^2*x^2*(-a^2*x^2+1)^(1/2)-8/15*(-a^2*x^2+1
)^(1/2))+420*a^6*c^3*d*(-1/3*a^2*x^2*(-a^2*x^2+1)^(1/2)-2/3*(-a^2*x^2+1)^(1/2))-315*a^8*c^4*(-a^2*x^2+1)^(1/2)
))

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Maxima [A]  time = 1.49616, size = 540, normalized size = 1.85 \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 )} \arccos \left (a x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^4*arccos(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 + 16
80*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*sqrt(-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*sqrt(-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)*arccos(a*x)

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Fricas [A]  time = 2.49507, size = 568, normalized size = 1.95 \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 )} \arccos \left (a x\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*arccos(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)*ar
ccos(a*x) - (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 + 4480*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 = 23.4418, size = 502, normalized size = 1.72 \begin{align*} \begin{cases} c^{4} x \operatorname{acos}{\left (a x \right )} + \frac{4 c^{3} d x^{3} \operatorname{acos}{\left (a x \right )}}{3} + \frac{6 c^{2} d^{2} x^{5} \operatorname{acos}{\left (a x \right )}}{5} + \frac{4 c d^{3} x^{7} \operatorname{acos}{\left (a x \right )}}{7} + \frac{d^{4} x^{9} \operatorname{acos}{\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{\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*acos(a*x),x)

[Out]

Piecewise((c**4*x*acos(a*x) + 4*c**3*d*x**3*acos(a*x)/3 + 6*c**2*d**2*x**5*acos(a*x)/5 + 4*c*d**3*x**7*acos(a*
x)/7 + d**4*x**9*acos(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)), (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 = 1.19606, size = 551, normalized size = 1.89 \begin{align*} \frac{1}{9} \, d^{4} x^{9} \arccos \left (a x\right ) + \frac{4}{7} \, c d^{3} x^{7} \arccos \left (a x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} d^{4} x^{8}}{81 \, a} + \frac{6}{5} \, c^{2} d^{2} x^{5} \arccos \left (a x\right ) - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} c d^{3} x^{6}}{49 \, a} + \frac{4}{3} \, c^{3} d x^{3} \arccos \left (a x\right ) - \frac{6 \, \sqrt{-a^{2} x^{2} + 1} c^{2} d^{2} x^{4}}{25 \, a} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} d^{4} x^{6}}{567 \, a^{3}} + c^{4} x \arccos \left (a x\right ) - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} c^{3} d x^{2}}{9 \, a} - \frac{24 \, \sqrt{-a^{2} x^{2} + 1} c d^{3} x^{4}}{245 \, a^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{4}}{a} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} c^{2} d^{2} x^{2}}{25 \, a^{3}} - \frac{16 \, \sqrt{-a^{2} x^{2} + 1} d^{4} x^{4}}{945 \, a^{5}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} c^{3} d}{9 \, a^{3}} - \frac{32 \, \sqrt{-a^{2} x^{2} + 1} c d^{3} x^{2}}{245 \, a^{5}} - \frac{16 \, \sqrt{-a^{2} x^{2} + 1} c^{2} d^{2}}{25 \, a^{5}} - \frac{64 \, \sqrt{-a^{2} x^{2} + 1} d^{4} x^{2}}{2835 \, a^{7}} - \frac{64 \, \sqrt{-a^{2} x^{2} + 1} c d^{3}}{245 \, a^{7}} - \frac{128 \, \sqrt{-a^{2} x^{2} + 1} d^{4}}{2835 \, a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*d^4*x^9*arccos(a*x) + 4/7*c*d^3*x^7*arccos(a*x) - 1/81*sqrt(-a^2*x^2 + 1)*d^4*x^8/a + 6/5*c^2*d^2*x^5*arcc
os(a*x) - 4/49*sqrt(-a^2*x^2 + 1)*c*d^3*x^6/a + 4/3*c^3*d*x^3*arccos(a*x) - 6/25*sqrt(-a^2*x^2 + 1)*c^2*d^2*x^
4/a - 8/567*sqrt(-a^2*x^2 + 1)*d^4*x^6/a^3 + c^4*x*arccos(a*x) - 4/9*sqrt(-a^2*x^2 + 1)*c^3*d*x^2/a - 24/245*s
qrt(-a^2*x^2 + 1)*c*d^3*x^4/a^3 - sqrt(-a^2*x^2 + 1)*c^4/a - 8/25*sqrt(-a^2*x^2 + 1)*c^2*d^2*x^2/a^3 - 16/945*
sqrt(-a^2*x^2 + 1)*d^4*x^4/a^5 - 8/9*sqrt(-a^2*x^2 + 1)*c^3*d/a^3 - 32/245*sqrt(-a^2*x^2 + 1)*c*d^3*x^2/a^5 -
16/25*sqrt(-a^2*x^2 + 1)*c^2*d^2/a^5 - 64/2835*sqrt(-a^2*x^2 + 1)*d^4*x^2/a^7 - 64/245*sqrt(-a^2*x^2 + 1)*c*d^
3/a^7 - 128/2835*sqrt(-a^2*x^2 + 1)*d^4/a^9