Integrand size = 12, antiderivative size = 91 \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2} \]
-Ci(2*(a+b*arccos(c*x))/b)*cos(2*a/b)/b^2/c^2-Si(2*(a+b*arccos(c*x))/b)*si n(2*a/b)/b^2/c^2+x*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))
Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\frac {\frac {b c x \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}-\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right )-\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{b^2 c^2} \]
((b*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x]) - Cos[(2*a)/b]*CosIntegral[ 2*(a/b + ArcCos[c*x])] - Sin[(2*a)/b]*SinIntegral[2*(a/b + ArcCos[c*x])])/ (b^2*c^2)
Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5143, 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.
\(\displaystyle \int \frac {x}{(a+b \arccos (c x))^2} \, dx\) |
\(\Big \downarrow \) 5143 |
\(\displaystyle \frac {\int -\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arccos (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^2}\) |
\(\Big \downarrow \) 3784 |
\(\displaystyle \frac {\sin \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {-\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arccos (c x))}{b}+\frac {\pi }{2}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))-\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle \frac {-\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right )-\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}\) |
(x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCos[c*x])) + (-(Cos[(2*a)/b]*CosInteg ral[(2*(a + b*ArcCos[c*x]))/b]) - Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcCo s[c*x]))/b])/(b^2*c^2)
3.2.64.3.1 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_)^(m_.), x_Symbol] :> Simp[( -x^m)*Sqrt[1 - c^2*x^2]*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - S imp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Cos[- a/b + x/b]^(m - 1)*(m - (m + 1)*Cos[-a/b + x/b]^2), x], x], x, a + b*ArcCos [c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
Time = 0.53 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{2 \left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )+\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{2}}}{c^{2}}\) | \(78\) |
default | \(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{2 \left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )+\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{2}}}{c^{2}}\) | \(78\) |
1/c^2*(1/2*sin(2*arccos(c*x))/(a+b*arccos(c*x))/b-(Ci(2*arccos(c*x)+2*a/b) *cos(2*a/b)+Si(2*arccos(c*x)+2*a/b)*sin(2*a/b))/b^2)
\[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int \frac {x}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]
\[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]
(sqrt(c*x + 1)*sqrt(-c*x + 1)*x - (b^2*c*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b*c)*integrate((2*c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(-c*x + 1)/ (a*b*c^3*x^2 - a*b*c + (b^2*c^3*x^2 - b^2*c)*arctan2(sqrt(c*x + 1)*sqrt(-c *x + 1), c*x)), x))/(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. 323 vs. \(2 (89) = 178\).
Time = 0.32 (sec) , antiderivative size = 323, normalized size of antiderivative = 3.55 \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=-\frac {2 \, b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, a \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, a \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b c x}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {b \arccos \left (c x\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {a \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} \]
-2*b*arccos(c*x)*cos(a/b)^2*cos_integral(2*a/b + 2*arccos(c*x))/(b^3*c^2*a rccos(c*x) + a*b^2*c^2) - 2*b*arccos(c*x)*cos(a/b)*sin(a/b)*sin_integral(2 *a/b + 2*arccos(c*x))/(b^3*c^2*arccos(c*x) + a*b^2*c^2) - 2*a*cos(a/b)^2*c os_integral(2*a/b + 2*arccos(c*x))/(b^3*c^2*arccos(c*x) + a*b^2*c^2) - 2*a *cos(a/b)*sin(a/b)*sin_integral(2*a/b + 2*arccos(c*x))/(b^3*c^2*arccos(c*x ) + a*b^2*c^2) + sqrt(-c^2*x^2 + 1)*b*c*x/(b^3*c^2*arccos(c*x) + a*b^2*c^2 ) + b*arccos(c*x)*cos_integral(2*a/b + 2*arccos(c*x))/(b^3*c^2*arccos(c*x) + a*b^2*c^2) + a*cos_integral(2*a/b + 2*arccos(c*x))/(b^3*c^2*arccos(c*x) + a*b^2*c^2)
Timed out. \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]