\(\int \frac {1}{\arccos (a+b x)} \, dx\) [34]

Optimal result

Integrand size = 8, antiderivative size = 12 \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\text {Si}(\arccos (a+b x))}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4888, 4720, 3380} \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\text {Si}(\arccos (a+b x))}{b} \]

[In]

Int[ArcCos[a + b*x]^(-1),x]

[Out]

-(SinIntegral[ArcCos[a + b*x]]/b)

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]

Rule 4888

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\arccos (x)} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arccos (a+b x)\right )}{b} \\ & = -\frac {\text {Si}(\arccos (a+b x))}{b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\text {Si}(\arccos (a+b x))}{b} \]

[In]

Integrate[ArcCos[a + b*x]^(-1),x]

[Out]

-(SinIntegral[ArcCos[a + b*x]]/b)

Maple [A] (verified)

Time = 0.58 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.08

method	result	size
derivativedivides	\(-\frac {\operatorname {Si}\left (\arccos \left (b x +a \right )\right )}{b}\)	\(13\)
default	\(-\frac {\operatorname {Si}\left (\arccos \left (b x +a \right )\right )}{b}\)	\(13\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(1/arccos(b*x + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(1/acos(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

integrate(1/arccos(b*x + a), x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\arccos (a+b x)} \, dx=-\frac {\operatorname {Si}\left (\arccos \left (b x + a\right )\right )}{b} \]

[In]

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

[Out]

-sin_integral(arccos(b*x + a))/b

Mupad [F(-1)]

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

[In]

int(1/acos(a + b*x),x)

[Out]

int(1/acos(a + b*x), x)