3.25 \(\int (c+d x^2)^3 \cos ^{-1}(a x) \, dx\)

Optimal. Leaf size=205 \[ \frac{d \left (1-a^2 x^2\right )^{3/2} \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7}-\frac{\sqrt{1-a^2 x^2} \left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right )}{35 a^7}-\frac{3 d^2 \left (1-a^2 x^2\right )^{5/2} \left (7 a^2 c+5 d\right )}{175 a^7}+\frac{d^3 \left (1-a^2 x^2\right )^{7/2}}{49 a^7}+c^2 d x^3 \cos ^{-1}(a x)+c^3 x \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x) \]

[Out]

-((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Sqrt[1 - a^2*x^2])/(35*a^7) + (d*(35*a^4*c^2 + 42*a^2*c*d
+ 15*d^2)*(1 - a^2*x^2)^(3/2))/(105*a^7) - (3*d^2*(7*a^2*c + 5*d)*(1 - a^2*x^2)^(5/2))/(175*a^7) + (d^3*(1 -
a^2*x^2)^(7/2))/(49*a^7) + c^3*x*ArcCos[a*x] + c^2*d*x^3*ArcCos[a*x] + (3*c*d^2*x^5*ArcCos[a*x])/5 + (d^3*x^7*
ArcCos[a*x])/7

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Rubi [A]  time = 0.243708, antiderivative size = 205, 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} \[ \frac{d \left (1-a^2 x^2\right )^{3/2} \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7}-\frac{\sqrt{1-a^2 x^2} \left (35 a^4 c^2 d+35 a^6 c^3+21 a^2 c d^2+5 d^3\right )}{35 a^7}-\frac{3 d^2 \left (1-a^2 x^2\right )^{5/2} \left (7 a^2 c+5 d\right )}{175 a^7}+\frac{d^3 \left (1-a^2 x^2\right )^{7/2}}{49 a^7}+c^2 d x^3 \cos ^{-1}(a x)+c^3 x \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3*ArcCos[a*x],x]

[Out]

-((35*a^6*c^3 + 35*a^4*c^2*d + 21*a^2*c*d^2 + 5*d^3)*Sqrt[1 - a^2*x^2])/(35*a^7) + (d*(35*a^4*c^2 + 42*a^2*c*d
+ 15*d^2)*(1 - a^2*x^2)^(3/2))/(105*a^7) - (3*d^2*(7*a^2*c + 5*d)*(1 - a^2*x^2)^(5/2))/(175*a^7) + (d^3*(1 -
a^2*x^2)^(7/2))/(49*a^7) + c^3*x*ArcCos[a*x] + c^2*d*x^3*ArcCos[a*x] + (3*c*d^2*x^5*ArcCos[a*x])/5 + (d^3*x^7*
ArcCos[a*x])/7

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 )^3 \cos ^{-1}(a x) \, dx &=c^3 x \cos ^{-1}(a x)+c^2 d x^3 \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)+a \int \frac{x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{35 \sqrt{1-a^2 x^2}} \, dx\\ &=c^3 x \cos ^{-1}(a x)+c^2 d x^3 \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)+\frac{1}{35} a \int \frac{x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt{1-a^2 x^2}} \, dx\\ &=c^3 x \cos ^{-1}(a x)+c^2 d x^3 \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)+\frac{1}{70} a \operatorname{Subst}\left (\int \frac{35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=c^3 x \cos ^{-1}(a x)+c^2 d x^3 \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)+\frac{1}{70} a \operatorname{Subst}\left (\int \left (\frac{35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3}{a^6 \sqrt{1-a^2 x}}-\frac{d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \sqrt{1-a^2 x}}{a^6}+\frac{3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x\right )^{3/2}}{a^6}-\frac{5 d^3 \left (1-a^2 x\right )^{5/2}}{a^6}\right ) \, dx,x,x^2\right )\\ &=-\frac{\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \sqrt{1-a^2 x^2}}{35 a^7}+\frac{d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^{3/2}}{105 a^7}-\frac{3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^{5/2}}{175 a^7}+\frac{d^3 \left (1-a^2 x^2\right )^{7/2}}{49 a^7}+c^3 x \cos ^{-1}(a x)+c^2 d x^3 \cos ^{-1}(a x)+\frac{3}{5} c d^2 x^5 \cos ^{-1}(a x)+\frac{1}{7} d^3 x^7 \cos ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.160836, size = 149, normalized size = 0.73 \[ \cos ^{-1}(a x) \left (c^2 d x^3+c^3 x+\frac{3}{5} c d^2 x^5+\frac{d^3 x^7}{7}\right )-\frac{\sqrt{1-a^2 x^2} \left (a^6 \left (1225 c^2 d x^2+3675 c^3+441 c d^2 x^4+75 d^3 x^6\right )+2 a^4 d \left (1225 c^2+294 c d x^2+45 d^2 x^4\right )+24 a^2 d^2 \left (49 c+5 d x^2\right )+240 d^3\right )}{3675 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)^3*ArcCos[a*x],x]

[Out]

-(Sqrt[1 - a^2*x^2]*(240*d^3 + 24*a^2*d^2*(49*c + 5*d*x^2) + 2*a^4*d*(1225*c^2 + 294*c*d*x^2 + 45*d^2*x^4) + a
^6*(3675*c^3 + 1225*c^2*d*x^2 + 441*c*d^2*x^4 + 75*d^3*x^6)))/(3675*a^7) + (c^3*x + c^2*d*x^3 + (3*c*d^2*x^5)/
5 + (d^3*x^7)/7)*ArcCos[a*x]

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Maple [A]  time = 0.007, size = 270, normalized size = 1.3 \begin{align*}{\frac{1}{a} \left ({\frac{a\arccos \left ( ax \right ){d}^{3}{x}^{7}}{7}}+{\frac{3\,a\arccos \left ( ax \right ) c{d}^{2}{x}^{5}}{5}}+a\arccos \left ( ax \right ){c}^{2}d{x}^{3}+\arccos \left ( ax \right ){c}^{3}ax+{\frac{1}{35\,{a}^{6}} \left ( 5\,{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 ) +21\,{a}^{2}c{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 ) +35\,{a}^{4}{c}^{2}d \left ( -1/3\,{a}^{2}{x}^{2}\sqrt{-{a}^{2}{x}^{2}+1}-2/3\,\sqrt{-{a}^{2}{x}^{2}+1} \right ) -35\,{a}^{6}{c}^{3}\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)^3*arccos(a*x),x)

[Out]

1/a*(1/7*a*arccos(a*x)*d^3*x^7+3/5*a*arccos(a*x)*c*d^2*x^5+a*arccos(a*x)*c^2*d*x^3+arccos(a*x)*c^3*a*x+1/35/a^
6*(5*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)-16/3
5*(-a^2*x^2+1)^(1/2))+21*a^2*c*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))+35*a^4*c^2*d*(-1/3*a^2*x^2*(-a^2*x^2+1)^(1/2)-2/3*(-a^2*x^2+1)^(1/2))-35*a^6*c^3*(-a^2*x^2+1)^(
1/2)))

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Maxima [A]  time = 1.46483, size = 360, normalized size = 1.76 \begin{align*} -\frac{1}{3675} \,{\left (\frac{75 \, \sqrt{-a^{2} x^{2} + 1} d^{3} x^{6}}{a^{2}} + \frac{441 \, \sqrt{-a^{2} x^{2} + 1} c d^{2} x^{4}}{a^{2}} + \frac{1225 \, \sqrt{-a^{2} x^{2} + 1} c^{2} d x^{2}}{a^{2}} + \frac{90 \, \sqrt{-a^{2} x^{2} + 1} d^{3} x^{4}}{a^{4}} + \frac{3675 \, \sqrt{-a^{2} x^{2} + 1} c^{3}}{a^{2}} + \frac{588 \, \sqrt{-a^{2} x^{2} + 1} c d^{2} x^{2}}{a^{4}} + \frac{2450 \, \sqrt{-a^{2} x^{2} + 1} c^{2} d}{a^{4}} + \frac{120 \, \sqrt{-a^{2} x^{2} + 1} d^{3} x^{2}}{a^{6}} + \frac{1176 \, \sqrt{-a^{2} x^{2} + 1} c d^{2}}{a^{6}} + \frac{240 \, \sqrt{-a^{2} x^{2} + 1} d^{3}}{a^{8}}\right )} a + \frac{1}{35} \,{\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} 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)^3*arccos(a*x),x, algorithm="maxima")

[Out]

-1/3675*(75*sqrt(-a^2*x^2 + 1)*d^3*x^6/a^2 + 441*sqrt(-a^2*x^2 + 1)*c*d^2*x^4/a^2 + 1225*sqrt(-a^2*x^2 + 1)*c^
2*d*x^2/a^2 + 90*sqrt(-a^2*x^2 + 1)*d^3*x^4/a^4 + 3675*sqrt(-a^2*x^2 + 1)*c^3/a^2 + 588*sqrt(-a^2*x^2 + 1)*c*d
^2*x^2/a^4 + 2450*sqrt(-a^2*x^2 + 1)*c^2*d/a^4 + 120*sqrt(-a^2*x^2 + 1)*d^3*x^2/a^6 + 1176*sqrt(-a^2*x^2 + 1)*
c*d^2/a^6 + 240*sqrt(-a^2*x^2 + 1)*d^3/a^8)*a + 1/35*(5*d^3*x^7 + 21*c*d^2*x^5 + 35*c^2*d*x^3 + 35*c^3*x)*arcc
os(a*x)

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Fricas [A]  time = 2.4291, size = 385, normalized size = 1.88 \begin{align*} \frac{105 \,{\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \arccos \left (a x\right ) -{\left (75 \, a^{6} d^{3} x^{6} + 3675 \, a^{6} c^{3} + 2450 \, a^{4} c^{2} d + 1176 \, a^{2} c d^{2} + 9 \,{\left (49 \, a^{6} c d^{2} + 10 \, a^{4} d^{3}\right )} x^{4} + 240 \, d^{3} +{\left (1225 \, a^{6} c^{2} d + 588 \, a^{4} c d^{2} + 120 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt{-a^{2} x^{2} + 1}}{3675 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(105*(5*a^7*d^3*x^7 + 21*a^7*c*d^2*x^5 + 35*a^7*c^2*d*x^3 + 35*a^7*c^3*x)*arccos(a*x) - (75*a^6*d^3*x^6
+ 3675*a^6*c^3 + 2450*a^4*c^2*d + 1176*a^2*c*d^2 + 9*(49*a^6*c*d^2 + 10*a^4*d^3)*x^4 + 240*d^3 + (1225*a^6*c^
2*d + 588*a^4*c*d^2 + 120*a^2*d^3)*x^2)*sqrt(-a^2*x^2 + 1))/a^7

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Sympy [A]  time = 8.37067, size = 326, normalized size = 1.59 \begin{align*} \begin{cases} c^{3} x \operatorname{acos}{\left (a x \right )} + c^{2} d x^{3} \operatorname{acos}{\left (a x \right )} + \frac{3 c d^{2} x^{5} \operatorname{acos}{\left (a x \right )}}{5} + \frac{d^{3} x^{7} \operatorname{acos}{\left (a x \right )}}{7} - \frac{c^{3} \sqrt{- a^{2} x^{2} + 1}}{a} - \frac{c^{2} d x^{2} \sqrt{- a^{2} x^{2} + 1}}{3 a} - \frac{3 c d^{2} x^{4} \sqrt{- a^{2} x^{2} + 1}}{25 a} - \frac{d^{3} x^{6} \sqrt{- a^{2} x^{2} + 1}}{49 a} - \frac{2 c^{2} d \sqrt{- a^{2} x^{2} + 1}}{3 a^{3}} - \frac{4 c d^{2} x^{2} \sqrt{- a^{2} x^{2} + 1}}{25 a^{3}} - \frac{6 d^{3} x^{4} \sqrt{- a^{2} x^{2} + 1}}{245 a^{3}} - \frac{8 c d^{2} \sqrt{- a^{2} x^{2} + 1}}{25 a^{5}} - \frac{8 d^{3} x^{2} \sqrt{- a^{2} x^{2} + 1}}{245 a^{5}} - \frac{16 d^{3} \sqrt{- a^{2} x^{2} + 1}}{245 a^{7}} & \text{for}\: a \neq 0 \\\frac{\pi \left (c^{3} x + c^{2} d x^{3} + \frac{3 c d^{2} x^{5}}{5} + \frac{d^{3} x^{7}}{7}\right )}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)**3*acos(a*x),x)

[Out]

Piecewise((c**3*x*acos(a*x) + c**2*d*x**3*acos(a*x) + 3*c*d**2*x**5*acos(a*x)/5 + d**3*x**7*acos(a*x)/7 - c**3
*sqrt(-a**2*x**2 + 1)/a - c**2*d*x**2*sqrt(-a**2*x**2 + 1)/(3*a) - 3*c*d**2*x**4*sqrt(-a**2*x**2 + 1)/(25*a) -
d**3*x**6*sqrt(-a**2*x**2 + 1)/(49*a) - 2*c**2*d*sqrt(-a**2*x**2 + 1)/(3*a**3) - 4*c*d**2*x**2*sqrt(-a**2*x**
2 + 1)/(25*a**3) - 6*d**3*x**4*sqrt(-a**2*x**2 + 1)/(245*a**3) - 8*c*d**2*sqrt(-a**2*x**2 + 1)/(25*a**5) - 8*d
**3*x**2*sqrt(-a**2*x**2 + 1)/(245*a**5) - 16*d**3*sqrt(-a**2*x**2 + 1)/(245*a**7), Ne(a, 0)), (pi*(c**3*x + c
**2*d*x**3 + 3*c*d**2*x**5/5 + d**3*x**7/7)/2, True))

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Giac [A]  time = 1.18997, size = 365, normalized size = 1.78 \begin{align*} \frac{1}{7} \, d^{3} x^{7} \arccos \left (a x\right ) + \frac{3}{5} \, c d^{2} x^{5} \arccos \left (a x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} d^{3} x^{6}}{49 \, a} + c^{2} d x^{3} \arccos \left (a x\right ) - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} c d^{2} x^{4}}{25 \, a} + c^{3} x \arccos \left (a x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} c^{2} d x^{2}}{3 \, a} - \frac{6 \, \sqrt{-a^{2} x^{2} + 1} d^{3} x^{4}}{245 \, a^{3}} - \frac{\sqrt{-a^{2} x^{2} + 1} c^{3}}{a} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} c d^{2} x^{2}}{25 \, a^{3}} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} c^{2} d}{3 \, a^{3}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} d^{3} x^{2}}{245 \, a^{5}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} c d^{2}}{25 \, a^{5}} - \frac{16 \, \sqrt{-a^{2} x^{2} + 1} d^{3}}{245 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*d^3*x^7*arccos(a*x) + 3/5*c*d^2*x^5*arccos(a*x) - 1/49*sqrt(-a^2*x^2 + 1)*d^3*x^6/a + c^2*d*x^3*arccos(a*x
) - 3/25*sqrt(-a^2*x^2 + 1)*c*d^2*x^4/a + c^3*x*arccos(a*x) - 1/3*sqrt(-a^2*x^2 + 1)*c^2*d*x^2/a - 6/245*sqrt(
-a^2*x^2 + 1)*d^3*x^4/a^3 - sqrt(-a^2*x^2 + 1)*c^3/a - 4/25*sqrt(-a^2*x^2 + 1)*c*d^2*x^2/a^3 - 2/3*sqrt(-a^2*x
^2 + 1)*c^2*d/a^3 - 8/245*sqrt(-a^2*x^2 + 1)*d^3*x^2/a^5 - 8/25*sqrt(-a^2*x^2 + 1)*c*d^2/a^5 - 16/245*sqrt(-a^
2*x^2 + 1)*d^3/a^7