3.169 \(\int \frac{x}{(a+b \cos ^{-1}(c x))^3} \, dx\)

Optimal. Leaf size=130 \[ -\frac{\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{b^3 c^2}+\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{b^3 c^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2} \]

[Out]

(x*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - 1/(2*b^2*c^2*(a + b*ArcCos[c*x])) + x^2/(b^2*(a + b*ArcC
os[c*x])) - (CosIntegral[(2*(a + b*ArcCos[c*x]))/b]*Sin[(2*a)/b])/(b^3*c^2) + (Cos[(2*a)/b]*SinIntegral[(2*(a
+ b*ArcCos[c*x]))/b])/(b^3*c^2)

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Rubi [A]  time = 0.311036, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {4634, 4720, 4636, 4406, 12, 3303, 3299, 3302, 4642} \[ -\frac{\sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (\frac{2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^3 c^2}+\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^3 c^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*ArcCos[c*x])^3,x]

[Out]

(x*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - 1/(2*b^2*c^2*(a + b*ArcCos[c*x])) + x^2/(b^2*(a + b*ArcC
os[c*x])) - (CosIntegral[(2*a)/b + 2*ArcCos[c*x]]*Sin[(2*a)/b])/(b^3*c^2) + (Cos[(2*a)/b]*SinIntegral[(2*a)/b
+ 2*ArcCos[c*x]])/(b^3*c^2)

Rule 4634

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcCo
s[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCos[c*x])^(n + 1
))/Sqrt[1 - c^2*x^2], x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCos[c*x])^(n + 1))/Sqrt[1 - c^2*
x^2], x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4720

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp
[((f*x)^m*(a + b*ArcCos[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] + Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)
^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && GtQ[d, 0]

Rule 4636

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Cos[x]^m*Sin[x], x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 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 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[c*x])
^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rubi steps

\begin{align*} \int \frac{x}{\left (a+b \cos ^{-1}(c x)\right )^3} \, dx &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{\int \frac{1}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx}{2 b c}+\frac{c \int \frac{x^2}{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx}{b}\\ &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}-\frac{2 \int \frac{x}{a+b \cos ^{-1}(c x)} \, dx}{b^2}\\ &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 (a+b x)} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\sin (2 x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{\cos \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\sin \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}-\frac{\sin \left (\frac{2 a}{b}\right ) \operatorname{Subst}\left (\int \frac{\cos \left (\frac{2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac{x \sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac{1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}-\frac{\text{Ci}\left (\frac{2 a}{b}+2 \cos ^{-1}(c x)\right ) \sin \left (\frac{2 a}{b}\right )}{b^3 c^2}+\frac{\cos \left (\frac{2 a}{b}\right ) \text{Si}\left (\frac{2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^3 c^2}\\ \end{align*}

Mathematica [A]  time = 0.278308, size = 107, normalized size = 0.82 \[ \frac{\frac{b^2 c x \sqrt{1-c^2 x^2}}{\left (a+b \cos ^{-1}(c x)\right )^2}+\frac{b \left (2 c^2 x^2-1\right )}{a+b \cos ^{-1}(c x)}-2 \sin \left (\frac{2 a}{b}\right ) \text{CosIntegral}\left (2 \left (\frac{a}{b}+\cos ^{-1}(c x)\right )\right )+2 \cos \left (\frac{2 a}{b}\right ) \text{Si}\left (2 \left (\frac{a}{b}+\cos ^{-1}(c x)\right )\right )}{2 b^3 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*ArcCos[c*x])^3,x]

[Out]

((b^2*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x])^2 + (b*(-1 + 2*c^2*x^2))/(a + b*ArcCos[c*x]) - 2*CosIntegral[
2*(a/b + ArcCos[c*x])]*Sin[(2*a)/b] + 2*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcCos[c*x])])/(2*b^3*c^2)

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Maple [A]  time = 0.053, size = 157, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{\sin \left ( 2\,\arccos \left ( cx \right ) \right ) }{4\, \left ( a+b\arccos \left ( cx \right ) \right ) ^{2}b}}+{\frac{1}{ \left ( 2\,a+2\,b\arccos \left ( cx \right ) \right ){b}^{3}} \left ( 2\,\arccos \left ( cx \right ){\it Si} \left ( 2\,\arccos \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) b-2\,\arccos \left ( cx \right ){\it Ci} \left ( 2\,\arccos \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) b+2\,{\it Si} \left ( 2\,\arccos \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \cos \left ( 2\,{\frac{a}{b}} \right ) a-2\,{\it Ci} \left ( 2\,\arccos \left ( cx \right ) +2\,{\frac{a}{b}} \right ) \sin \left ( 2\,{\frac{a}{b}} \right ) a+\cos \left ( 2\,\arccos \left ( cx \right ) \right ) b \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a+b*arccos(c*x))^3,x)

[Out]

1/c^2*(1/4*sin(2*arccos(c*x))/(a+b*arccos(c*x))^2/b+1/2*(2*arccos(c*x)*Si(2*arccos(c*x)+2*a/b)*cos(2*a/b)*b-2*
arccos(c*x)*Ci(2*arccos(c*x)+2*a/b)*sin(2*a/b)*b+2*Si(2*arccos(c*x)+2*a/b)*cos(2*a/b)*a-2*Ci(2*arccos(c*x)+2*a
/b)*sin(2*a/b)*a+cos(2*arccos(c*x))*b)/(a+b*arccos(c*x))/b^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, a c^{2} x^{2} + \sqrt{c x + 1} \sqrt{-c x + 1} b c x +{\left (2 \, b c^{2} x^{2} - b\right )} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) - a - \frac{4 \,{\left (b^{4} c^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )^{2} + 2 \, a b^{3} c^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a^{2} b^{2} c^{2}\right )} \int \frac{x}{b \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a}\,{d x}}{b^{2}}}{2 \,{\left (b^{4} c^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right )^{2} + 2 \, a b^{3} c^{2} \arctan \left (\sqrt{c x + 1} \sqrt{-c x + 1}, c x\right ) + a^{2} b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*arccos(c*x))^3,x, algorithm="maxima")

[Out]

1/2*(2*a*c^2*x^2 + sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x + (2*b*c^2*x^2 - b)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1)
, c*x) - 4*(b^4*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x
+ 1), c*x) + a^2*b^2*c^2)*integrate(x/(b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b^2), x) - a)/(b^4*
c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a^
2*b^2*c^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x}{b^{3} \arccos \left (c x\right )^{3} + 3 \, a b^{2} \arccos \left (c x\right )^{2} + 3 \, a^{2} b \arccos \left (c x\right ) + a^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/(b^3*arccos(c*x)^3 + 3*a*b^2*arccos(c*x)^2 + 3*a^2*b*arccos(c*x) + a^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\left (a + b \operatorname{acos}{\left (c x \right )}\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b*acos(c*x))**3,x)

[Out]

Integral(x/(a + b*acos(c*x))**3, x)

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Giac [B]  time = 1.33554, size = 1161, normalized size = 8.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*c^2*x^2*arccos(c*x)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*b^2*arccos(c*x)^2*
cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b
^3*c^2) + 2*b^2*arccos(c*x)^2*cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*
c^2*arccos(c*x) + a^2*b^3*c^2) + a*b*c^2*x^2/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) -
4*a*b*arccos(c*x)*cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*
arccos(c*x) + a^2*b^3*c^2) + 4*a*b*arccos(c*x)*cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(
c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*a^2*cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/
(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - b^2*arccos(c*x)^2*sin_integral(2*a/b + 2*arc
cos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 2*a^2*cos(a/b)^2*sin_integral(2*a/
b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 1/2*sqrt(-c^2*x^2 + 1)*b^
2*c*x/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*a*b*arccos(c*x)*sin_integral(2*a/b +
2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 1/2*b^2*arccos(c*x)/(b^5*c^2
*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - a^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arc
cos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 1/2*a*b/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x)
+ a^2*b^3*c^2)