Integrand size = 10, antiderivative size = 53 \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c}+\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c} \] Output:
cos(a/b)*Ci((a+b*arccos(c*x))/b)/b/c+sin(a/b)*Si((a+b*arccos(c*x))/b)/b/c
Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.87 \[ \int \frac {1}{a+b \arccos (c x)} \, dx=-\frac {-\operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{b c} \] Input:
Integrate[(a + b*ArcCos[c*x])^(-1),x]
Output:
-((-(CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b]) + Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]])/(b*c))
Time = 0.39 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5135, 25, 3042, 3784, 25, 3042, 3780, 3783}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{a+b \arccos (c x)} \, dx\) |
\(\Big \downarrow \) 5135 |
\(\displaystyle -\frac {\int -\frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle -\frac {-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\cos \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle -\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b c}\) |
Input:
Int[(a + b*ArcCos[c*x])^(-1),x]
Output:
-((-(CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b]) + Cos[a/b]*SinIntegral[( a + b*ArcCos[c*x])/b])/(b*c))
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[-(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]
Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92
method | result | size |
derivativedivides | \(\frac {-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b}+\frac {\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}}{c}\) | \(49\) |
default | \(\frac {-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b}+\frac {\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}}{c}\) | \(49\) |
Input:
int(1/(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
1/c*(-Si(arccos(c*x)+a/b)*cos(a/b)/b+Ci(arccos(c*x)+a/b)*sin(a/b)/b)
\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int { \frac {1}{b \arccos \left (c x\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
integral(1/(b*arccos(c*x) + a), x)
\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int \frac {1}{a + b \operatorname {acos}{\left (c x \right )}}\, dx \] Input:
integrate(1/(a+b*acos(c*x)),x)
Output:
Integral(1/(a + b*acos(c*x)), x)
\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int { \frac {1}{b \arccos \left (c x\right ) + a} \,d x } \] Input:
integrate(1/(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
integrate(1/(b*arccos(c*x) + a), x)
Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.94 \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\frac {\operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c} - \frac {\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b c} \] Input:
integrate(1/(a+b*arccos(c*x)),x, algorithm="giac")
Output:
cos_integral(a/b + arccos(c*x))*sin(a/b)/(b*c) - cos(a/b)*sin_integral(a/b + arccos(c*x))/(b*c)
Timed out. \[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int \frac {1}{a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \] Input:
int(1/(a + b*acos(c*x)),x)
Output:
int(1/(a + b*acos(c*x)), x)
\[ \int \frac {1}{a+b \arccos (c x)} \, dx=\int \frac {1}{\mathit {acos} \left (c x \right ) b +a}d x \] Input:
int(1/(a+b*acos(c*x)),x)
Output:
int(1/(acos(c*x)*b + a),x)