3.2.60 \(\int \frac {1}{a+b \text {ArcCos}(c x)} \, dx\) [160]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4720, 3384, 3380, 3383} \begin {gather*} \frac {\sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcCos}(c x)}{b}\right )}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}(c x)}{b}\right )}{b c} \end {gather*}

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 3380

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 3383

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 3384

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 4720

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

Rubi steps

\begin {align*} \int \frac {1}{a+b \cos ^{-1}(c x)} \, dx &=\frac {\text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.05, size = 46, normalized size = 0.85 \begin {gather*} -\frac {-\text {CosIntegral}\left (\frac {a}{b}+\text {ArcCos}(c x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcCos}(c x)\right )}{b c} \end {gather*}

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 = 0.08, size = 49, normalized size = 0.91

method	result	size
derivativedivides	\(\frac {-\frac {\sinIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b}+\frac {\cosineIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}}{c}\)	\(49\)
default	\(\frac {-\frac {\sinIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b}+\frac {\cosineIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}}{c}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{a + b \operatorname {acos}{\left (c x \right )}}\, dx \end {gather*}

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 = 0.41, size = 50, normalized size = 0.93 \begin {gather*} \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 {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b*acos(c*x)),x)

[Out]

int(1/(a + b*acos(c*x)), x)

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