Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c} \]
-Ci((a+b*arccos(c*x))/b)*cos(a/b)/b^2/c-Si((a+b*arccos(c*x))/b)*sin(a/b)/b ^2/c+(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))
Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}-\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right )-\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{b^2 c} \]
((b*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) - Cos[a/b]*CosIntegral[a/b + Ar cCos[c*x]] - Sin[a/b]*SinIntegral[a/b + ArcCos[c*x]])/(b^2*c)
Time = 0.54 (sec) , antiderivative size = 81, 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 = {5133, 5225, 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.
\(\displaystyle \int \frac {1}{(a+b \arccos (c x))^2} \, dx\) |
\(\Big \downarrow \) 5133 |
\(\displaystyle \frac {c \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))}dx}{b}+\frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\int \frac {\cos \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\sin \left (\frac {a}{b}\right ) \int -\frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\cos \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\sin \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))+\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \int \frac {\sin \left (\frac {a+b \arccos (c x)}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )+\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{b^2 c}\) |
Sqrt[1 - c^2*x^2]/(b*c*(a + b*ArcCos[c*x])) - (Cos[a/b]*CosIntegral[(a + b *ArcCos[c*x])/b] + Sin[a/b]*SinIntegral[(a + b*ArcCos[c*x])/b])/(b^2*c)
3.2.65.3.1 Defintions of rubi rules used
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[(-Sqrt[1 - c ^2*x^2])*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1 )) Int[x*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.65 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-c^{2} x^{2}+1}}{\left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}}{c}\) | \(74\) |
default | \(\frac {\frac {\sqrt {-c^{2} x^{2}+1}}{\left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )+\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )}{b^{2}}}{c}\) | \(74\) |
1/c*((-c^2*x^2+1)^(1/2)/(a+b*arccos(c*x))/b-(Si(arccos(c*x)+a/b)*sin(a/b)+ Ci(arccos(c*x)+a/b)*cos(a/b))/b^2)
\[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]
-((b^2*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c^2)*integrate (sqrt(c*x + 1)*sqrt(-c*x + 1)*x/(a*b*c^2*x^2 - a*b + (b^2*c^2*x^2 - b^2)*a rctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)), x) - sqrt(c*x + 1)*sqrt(-c*x + 1))/(b^2*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c)
Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (84) = 168\).
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=-\frac {b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} - \frac {b \arccos \left (c x\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} - \frac {a \cos \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} - \frac {a \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} + \frac {\sqrt {-c^{2} x^{2} + 1} b}{b^{3} c \arccos \left (c x\right ) + a b^{2} c} \]
-b*arccos(c*x)*cos(a/b)*cos_integral(a/b + arccos(c*x))/(b^3*c*arccos(c*x) + a*b^2*c) - b*arccos(c*x)*sin(a/b)*sin_integral(a/b + arccos(c*x))/(b^3* c*arccos(c*x) + a*b^2*c) - a*cos(a/b)*cos_integral(a/b + arccos(c*x))/(b^3 *c*arccos(c*x) + a*b^2*c) - a*sin(a/b)*sin_integral(a/b + arccos(c*x))/(b^ 3*c*arccos(c*x) + a*b^2*c) + sqrt(-c^2*x^2 + 1)*b/(b^3*c*arccos(c*x) + a*b ^2*c)
Timed out. \[ \int \frac {1}{(a+b \arccos (c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]