\(\int \frac {x}{(a+b \arccos (c x))^2} \, dx\) [164]

Optimal result

Integrand size = 12, antiderivative size = 91 \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2} \]

[Out]

-Ci(2*(a+b*arccos(c*x))/b)*cos(2*a/b)/b^2/c^2-Si(2*(a+b*arccos(c*x))/b)*sin(2*a/b)/b^2/c^2+x*(-c^2*x^2+1)^(1/2
)/b/c/(a+b*arccos(c*x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4728, 3384, 3380, 3383} \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=-\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))} \]

[In]

Int[x/(a + b*ArcCos[c*x])^2,x]

[Out]

(x*Sqrt[1 - c^2*x^2])/(b*c*(a + b*ArcCos[c*x])) - (Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcCos[c*x]))/b])/(b^2*c
^2) - (Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcCos[c*x]))/b])/(b^2*c^2)

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 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[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]

Rule 4728

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(-x^m)*Sqrt[1 - c^2*x^2]*((a + b*Arc
Cos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^(n + 1), C
os[-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]

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^2 c^2} \\ & = \frac {x \sqrt {1-c^2 x^2}}{b c (a+b \arccos (c x))}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^2 c^2} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\frac {\frac {b c x \sqrt {1-c^2 x^2}}{a+b \arccos (c x)}-\cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right )-\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{b^2 c^2} \]

[In]

Integrate[x/(a + b*ArcCos[c*x])^2,x]

[Out]

((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)

Maple [A] (verified)

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	\(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{2 \left (a +b \arccos \left (c x \right )\right ) b}-\frac {\operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )+\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{2}}}{c^{2}}\)	\(78\)

[In]

int(x/(a+b*arccos(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [F]

\[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x/(a+b*arccos(c*x))^2,x, algorithm="fricas")

[Out]

integral(x/(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^2), x)

Sympy [F]

\[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int \frac {x}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]

[In]

integrate(x/(a+b*acos(c*x))**2,x)

[Out]

Integral(x/(a + b*acos(c*x))**2, x)

Maxima [F]

\[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x/(a+b*arccos(c*x))^2,x, algorithm="maxima")

[Out]

(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)

Giac [B] (verification not implemented)

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 \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=-\frac {2 \, b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, a \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, a \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b c x}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {b \arccos \left (c x\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {a \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} \]

[In]

integrate(x/(a+b*arccos(c*x))^2,x, algorithm="giac")

[Out]

-2*b*arccos(c*x)*cos(a/b)^2*cos_integral(2*a/b + 2*arccos(c*x))/(b^3*c^2*arccos(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
*cos_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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(a+b \arccos (c x))^2} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int(x/(a + b*acos(c*x))^2,x)

[Out]

int(x/(a + b*acos(c*x))^2, x)