\(\int \frac {x}{(a+b \arccos (c x))^3} \, dx\) [169]

Optimal result

Integrand size = 12, antiderivative size = 130 \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^3 c^2} \]

[Out]

-1/2/b^2/c^2/(a+b*arccos(c*x))+x^2/b^2/(a+b*arccos(c*x))+cos(2*a/b)*Si(2*(a+b*arccos(c*x))/b)/b^3/c^2-Ci(2*(a+
b*arccos(c*x))/b)*sin(2*a/b)/b^3/c^2+1/2*x*(-c^2*x^2+1)^(1/2)/b/c/(a+b*arccos(c*x))^2

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4730, 4808, 4732, 4491, 12, 3384, 3380, 3383, 4738} \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=-\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^3 c^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}+\frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2} \]

[In]

Int[x/(a + b*ArcCos[c*x])^3,x]

[Out]

(x*Sqrt[1 - c^2*x^2])/(2*b*c*(a + b*ArcCos[c*x])^2) - 1/(2*b^2*c^2*(a + b*ArcCos[c*x])) + x^2/(b^2*(a + b*ArcC
os[c*x])) - (CosIntegral[(2*(a + b*ArcCos[c*x]))/b]*Sin[(2*a)/b])/(b^3*c^2) + (Cos[(2*a)/b]*SinIntegral[(2*(a
+ b*ArcCos[c*x]))/b])/(b^3*c^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4730

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[c*((m + 1)/(b*(n + 1))), Int[x^(m + 1)*((a + b*ArcCos[c*x])^(n +
1)/Sqrt[1 - c^2*x^2]), x], x] + Dist[m/(b*c*(n + 1)), Int[x^(m - 1)*((a + b*ArcCos[c*x])^(n + 1)/Sqrt[1 - c^2
*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 4732

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(b*c^(m + 1))^(-1), Subst[Int[x^n*C
os[-a/b + x/b]^m*Sin[-a/b + x/b], x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4738

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(-(b*c*(n + 1))^(-1)
)*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && E
qQ[c^2*d + e, 0] && NeQ[n, -1]

Rule 4808

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(-(f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcCos[c*x])^(n + 1), x] + Dist[f*(m/(
b*c*(n + 1)))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]], Int[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n + 1), x], x] /
; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {\int \frac {1}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2} \, dx}{2 b c}+\frac {c \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2} \, dx}{b} \\ & = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {2 \int \frac {x}{a+b \arccos (c x)} \, dx}{b^2} \\ & = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {2 \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^3 c^2} \\ & = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {2 \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x} \, dx,x,a+b \arccos (c x)\right )}{b^3 c^2} \\ & = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{b^3 c^2} \\ & = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}+\frac {\cos \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^3 c^2}-\frac {\sin \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^3 c^2} \\ & = \frac {x \sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}-\frac {1}{2 b^2 c^2 (a+b \arccos (c x))}+\frac {x^2}{b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {2 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arccos (c x))}{b}\right )}{b^3 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\frac {\frac {b^2 c x \sqrt {1-c^2 x^2}}{(a+b \arccos (c x))^2}+\frac {b \left (-1+2 c^2 x^2\right )}{a+b \arccos (c x)}-2 \operatorname {CosIntegral}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )+2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\arccos (c x)\right )\right )}{2 b^3 c^2} \]

[In]

Integrate[x/(a + b*ArcCos[c*x])^3,x]

[Out]

((b^2*c*x*Sqrt[1 - c^2*x^2])/(a + b*ArcCos[c*x])^2 + (b*(-1 + 2*c^2*x^2))/(a + b*ArcCos[c*x]) - 2*CosIntegral[
2*(a/b + ArcCos[c*x])]*Sin[(2*a)/b] + 2*Cos[(2*a)/b]*SinIntegral[2*(a/b + ArcCos[c*x])])/(2*b^3*c^2)

Maple [A] (verified)

Time = 0.44 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{4 \left (a +b \arccos \left (c x \right )\right )^{2} b}-\frac {2 \arccos \left (c x \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) b -2 \arccos \left (c x \right ) \operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) a -2 \,\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\cos \left (2 \arccos \left (c x \right )\right ) b}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{2}}\)	\(158\)
default	\(\frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{4 \left (a +b \arccos \left (c x \right )\right )^{2} b}-\frac {2 \arccos \left (c x \right ) \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) b -2 \arccos \left (c x \right ) \operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +2 \sin \left (\frac {2 a}{b}\right ) \operatorname {Ci}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) a -2 \,\operatorname {Si}\left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -\cos \left (2 \arccos \left (c x \right )\right ) b}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{2}}\)	\(158\)

[In]

int(x/(a+b*arccos(c*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/c^2*(1/4*sin(2*arccos(c*x))/(a+b*arccos(c*x))^2/b-1/2*(2*arccos(c*x)*sin(2*a/b)*Ci(2*arccos(c*x)+2*a/b)*b-2*
arccos(c*x)*Si(2*arccos(c*x)+2*a/b)*cos(2*a/b)*b+2*sin(2*a/b)*Ci(2*arccos(c*x)+2*a/b)*a-2*Si(2*arccos(c*x)+2*a
/b)*cos(2*a/b)*a-cos(2*arccos(c*x))*b)/(a+b*arccos(c*x))/b^3)

Fricas [F]

\[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(x/(a+b*arccos(c*x))^3,x, algorithm="fricas")

[Out]

integral(x/(b^3*arccos(c*x)^3 + 3*a*b^2*arccos(c*x)^2 + 3*a^2*b*arccos(c*x) + a^3), x)

Sympy [F]

\[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\int \frac {x}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}\, dx \]

[In]

integrate(x/(a+b*acos(c*x))**3,x)

[Out]

Integral(x/(a + b*acos(c*x))**3, x)

Maxima [F]

\[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\int { \frac {x}{{\left (b \arccos \left (c x\right ) + a\right )}^{3}} \,d x } \]

[In]

integrate(x/(a+b*arccos(c*x))^3,x, algorithm="maxima")

[Out]

1/2*(2*a*c^2*x^2 + sqrt(c*x + 1)*sqrt(-c*x + 1)*b*c*x + (2*b*c^2*x^2 - b)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1)
, c*x) - 4*(b^4*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x
+ 1), c*x) + a^2*b^2*c^2)*integrate(x/(b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a*b^2), x) - a)/(b^4*
c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b^3*c^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) + a^
2*b^2*c^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (124) = 248\).

Time = 0.33 (sec) , antiderivative size = 860, normalized size of antiderivative = 6.62 \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(x/(a+b*arccos(c*x))^3,x, algorithm="giac")

[Out]

b^2*c^2*x^2*arccos(c*x)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*b^2*arccos(c*x)^2*
cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b
^3*c^2) + 2*b^2*arccos(c*x)^2*cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*
c^2*arccos(c*x) + a^2*b^3*c^2) + a*b*c^2*x^2/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) -
4*a*b*arccos(c*x)*cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*
arccos(c*x) + a^2*b^3*c^2) + 4*a*b*arccos(c*x)*cos(a/b)^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arccos(
c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*a^2*cos(a/b)*cos_integral(2*a/b + 2*arccos(c*x))*sin(a/b)/
(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - b^2*arccos(c*x)^2*sin_integral(2*a/b + 2*arc
cos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 2*a^2*cos(a/b)^2*sin_integral(2*a/
b + 2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) + 1/2*sqrt(-c^2*x^2 + 1)*b^
2*c*x/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 2*a*b*arccos(c*x)*sin_integral(2*a/b +
2*arccos(c*x))/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 1/2*b^2*arccos(c*x)/(b^5*c^2
*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - a^2*sin_integral(2*a/b + 2*arccos(c*x))/(b^5*c^2*arc
cos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x) + a^2*b^3*c^2) - 1/2*a*b/(b^5*c^2*arccos(c*x)^2 + 2*a*b^4*c^2*arccos(c*x)
+ a^2*b^3*c^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{(a+b \arccos (c x))^3} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3} \,d x \]

[In]

int(x/(a + b*acos(c*x))^3,x)

[Out]

int(x/(a + b*acos(c*x))^3, x)