\(\int \frac {1}{\arccos (a x)} \, dx\) [48]

Optimal result

Integrand size = 6, antiderivative size = 10 \[ \int \frac {1}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4720, 3380} \[ \int \frac {1}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))}{a} \]

[In]

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

[Out]

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

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 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*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arccos (a x)\right )}{a} \\ & = -\frac {\text {Si}(\arccos (a x))}{a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a x)} \, dx=-\frac {\text {Si}(\arccos (a x))}{a} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(-\frac {\operatorname {Si}\left (\arccos \left (a x \right )\right )}{a}\)	\(11\)
default	\(-\frac {\operatorname {Si}\left (\arccos \left (a x \right )\right )}{a}\)	\(11\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {1}{\arccos (a x)} \, dx=\int { \frac {1}{\arccos \left (a x\right )} \,d x } \]

[In]

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

[Out]

integral(1/arccos(a*x), x)

Sympy [F]

\[ \int \frac {1}{\arccos (a x)} \, dx=\int \frac {1}{\operatorname {acos}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/acos(a*x), x)

Maxima [F]

\[ \int \frac {1}{\arccos (a x)} \, dx=\int { \frac {1}{\arccos \left (a x\right )} \,d x } \]

[In]

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

[Out]

integrate(1/arccos(a*x), x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a x)} \, dx=-\frac {\operatorname {Si}\left (\arccos \left (a x\right )\right )}{a} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\arccos (a x)} \, dx=\int \frac {1}{\mathrm {acos}\left (a\,x\right )} \,d x \]

[In]

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

[Out]

int(1/acos(a*x), x)