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

Optimal. Leaf size=54 \[ \frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{b c}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{b c} \]

[Out]

(CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(b*c) - (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(b*c)

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Rubi [A]  time = 0.0619756, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4624, 3303, 3299, 3302} \[ \frac{\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{b c}-\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}(c x)}{b}\right )}{b c} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCos[c*x])^(-1),x]

[Out]

(CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/(b*c) - (Cos[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(b*c)

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

Mathematica [A]  time = 0.0619107, size = 46, normalized size = 0.85 \[ -\frac{\cos \left (\frac{a}{b}\right ) \text{Si}\left (\frac{a}{b}+\cos ^{-1}(c x)\right )-\sin \left (\frac{a}{b}\right ) \text{CosIntegral}\left (\frac{a}{b}+\cos ^{-1}(c x)\right )}{b c} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCos[c*x])^(-1),x]

[Out]

-((-(CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/(b*c))

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Maple [A]  time = 0.049, size = 49, normalized size = 0.9 \begin{align*}{\frac{1}{c} \left ( -{\frac{1}{b}{\it Si} \left ( \arccos \left ( cx \right ) +{\frac{a}{b}} \right ) \cos \left ({\frac{a}{b}} \right ) }+{\frac{1}{b}{\it Ci} \left ( \arccos \left ( cx \right ) +{\frac{a}{b}} \right ) \sin \left ({\frac{a}{b}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \arccos \left (c x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(b*arccos(c*x) + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b \arccos \left (c x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(b*arccos(c*x) + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \operatorname{acos}{\left (c x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*acos(c*x)),x)

[Out]

Integral(1/(a + b*acos(c*x)), x)

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Giac [A]  time = 1.14686, size = 68, normalized size = 1.26 \begin{align*} \frac{\operatorname{Ci}\left (\frac{a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac{a}{b}\right )}{b c} - \frac{\cos \left (\frac{a}{b}\right ) \operatorname{Si}\left (\frac{a}{b} + \arccos \left (c x\right )\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos_integral(a/b + arccos(c*x))*sin(a/b)/(b*c) - cos(a/b)*sin_integral(a/b + arccos(c*x))/(b*c)