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

Optimal. Leaf size=111 \[ -\frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{2 b^3 c}+\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{2 b^3 c}+\frac{x}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{\sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.165866, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4622, 4720, 4624, 3303, 3299, 3302} \[ -\frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{2 b^3 c}+\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{2 b^3 c}+\frac{x}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac{\sqrt{1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])^(-3),x]

[Out]

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

Rule 4622

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

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 4624

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

Rubi steps

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

Mathematica [A]  time = 0.233061, size = 89, normalized size = 0.8 \[ \frac{\frac{b \left (a c x+b \sqrt{1-c^2 x^2}+b c x \cos ^{-1}(c x)\right )}{\left (a+b \cos ^{-1}(c x)\right )^2}-\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\cos ^{-1}(c x)\right )+\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\cos ^{-1}(c x)\right )}{2 b^3 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])^(-3),x]

[Out]

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

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Maple [A]  time = 0.058, size = 139, normalized size = 1.3 \begin{align*}{\frac{1}{c} \left ({\frac{1}{2\, \left ( a+b\arccos \left ( cx \right ) \right ) ^{2}b}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{1}{ \left ( 2\,a+2\,b\arccos \left ( cx \right ) \right ){b}^{3}} \left ( \arccos \left ( cx \right ){\it Si} \left ( \arccos \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) b-\arccos \left ( cx \right ){\it Ci} \left ( \arccos \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) b+{\it Si} \left ( \arccos \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) a-{\it Ci} \left ( \arccos \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) a+xbc \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*arccos(c*x))^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^3,x, algorithm="maxima")

[Out]

1/2*(b*c*x*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*c*x + sqrt(c*x + 1)*sqrt(-c*x + 1)*b - 2*(b^4*c*arct
an2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a^2*b^2*c)*i
ntegrate(1/2/(b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b^2), x))/(b^4*c*arctan2(sqrt(c*x + 1)*sqrt(-
c*x + 1), c*x)^2 + 2*a*b^3*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a^2*b^2*c)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{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(1/(a+b*arccos(c*x))^3,x, algorithm="fricas")

[Out]

integral(1/(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{1}{\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(1/(a+b*acos(c*x))**3,x)

[Out]

Integral((a + b*acos(c*x))**(-3), x)

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Giac [B]  time = 1.17704, size = 649, normalized size = 5.85 \begin{align*} -\frac{b^{2} \arccos \left (c x\right )^{2} \operatorname{Ci}\left (\frac{a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac{b^{2} \arccos \left (c x\right )^{2} \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arccos \left (c x\right )\right )}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac{b^{2} c x \arccos \left (c x\right )}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac{a b \arccos \left (c x\right ) \operatorname{Ci}\left (\frac{a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c} + \frac{a b \arccos \left (c x\right ) \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arccos \left (c x\right )\right )}{b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c} + \frac{a b c x}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac{a^{2} \operatorname{Ci}\left (\frac{a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac{a^{2} \cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arccos \left (c x\right )\right )}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac{\sqrt{-c^{2} x^{2} + 1} b^{2}}{2 \,{\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*arccos(c*x))^3,x, algorithm="giac")

[Out]

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