3.48 \(\int \frac {1}{\cos ^{-1}(a x)} \, dx\)

Optimal. Leaf size=10 \[ -\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{a} \]

[Out]

-Si(arccos(a*x))/a

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Rubi [A]  time = 0.02, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4624, 3299} \[ -\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^(-1),x]

[Out]

-(SinIntegral[ArcCos[a*x]]/a)

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 10, normalized size = 1.00 \[ -\frac {\text {Si}\left (\cos ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^(-1),x]

[Out]

-(SinIntegral[ArcCos[a*x]]/a)

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fricas [F]  time = 2.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\arccos \left (a x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/arccos(a*x), x)

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giac [A]  time = 0.98, size = 10, normalized size = 1.00 \[ -\frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sin_integral(arccos(a*x))/a

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maple [A]  time = 0.03, size = 11, normalized size = 1.10 \[ -\frac {\Si \left (\arccos \left (a x \right )\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccos(a*x),x)

[Out]

-Si(arccos(a*x))/a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\arccos \left (a x\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arccos(a*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\mathrm {acos}\left (a\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/acos(a*x),x)

[Out]

int(1/acos(a*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\operatorname {acos}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acos(a*x),x)

[Out]

Integral(1/acos(a*x), x)

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