\(\int \frac {a+b \arccos (c x)}{x^2 (d-c^2 d x^2)} \, dx\) [6]

Optimal result

Integrand size = 25, antiderivative size = 107 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {a+b \arccos (c x)}{d x}+\frac {2 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d}-\frac {i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d}+\frac {i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d} \]

[Out]

(-a-b*arccos(c*x))/d/x+2*c*(a+b*arccos(c*x))*arctanh(c*x+I*(-c^2*x^2+1)^(1/2))/d+b*c*arctanh((-c^2*x^2+1)^(1/2
))/d-I*b*c*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2))/d+I*b*c*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4790, 4750, 4268, 2317, 2438, 272, 65, 214} \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=\frac {2 c \text {arctanh}\left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))}{d}-\frac {a+b \arccos (c x)}{d x}-\frac {i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d}+\frac {i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d} \]

[In]

Int[(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)),x]

[Out]

-((a + b*ArcCos[c*x])/(d*x)) + (2*c*(a + b*ArcCos[c*x])*ArcTanh[E^(I*ArcCos[c*x])])/d + (b*c*ArcTanh[Sqrt[1 -
c^2*x^2]])/d - (I*b*c*PolyLog[2, -E^(I*ArcCos[c*x])])/d + (I*b*c*PolyLog[2, E^(I*ArcCos[c*x])])/d

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4750

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

Rule 4790

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(m
+ 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d + e*x
^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arccos (c x)}{d x}+c^2 \int \frac {a+b \arccos (c x)}{d-c^2 d x^2} \, dx-\frac {(b c) \int \frac {1}{x \sqrt {1-c^2 x^2}} \, dx}{d} \\ & = -\frac {a+b \arccos (c x)}{d x}-\frac {c \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arccos (c x))}{d}-\frac {(b c) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2 d} \\ & = -\frac {a+b \arccos (c x)}{d x}+\frac {2 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d}+\frac {b \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c d}+\frac {(b c) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{d}-\frac {(b c) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arccos (c x)\right )}{d} \\ & = -\frac {a+b \arccos (c x)}{d x}+\frac {2 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d}-\frac {(i b c) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{d}+\frac {(i b c) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arccos (c x)}\right )}{d} \\ & = -\frac {a+b \arccos (c x)}{d x}+\frac {2 c (a+b \arccos (c x)) \text {arctanh}\left (e^{i \arccos (c x)}\right )}{d}+\frac {b c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d}-\frac {i b c \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{d}+\frac {i b c \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.48 \[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=-\frac {2 a+2 b \arccos (c x)+2 b c x \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-2 b c x \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+2 b c x \log (x)+a c x \log (1-c x)-a c x \log (1+c x)-2 b c x \log \left (1+\sqrt {1-c^2 x^2}\right )+2 i b c x \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-2 i b c x \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{2 d x} \]

[In]

Integrate[(a + b*ArcCos[c*x])/(x^2*(d - c^2*d*x^2)),x]

[Out]

-1/2*(2*a + 2*b*ArcCos[c*x] + 2*b*c*x*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 2*b*c*x*ArcCos[c*x]*Log[1 + E^(
I*ArcCos[c*x])] + 2*b*c*x*Log[x] + a*c*x*Log[1 - c*x] - a*c*x*Log[1 + c*x] - 2*b*c*x*Log[1 + Sqrt[1 - c^2*x^2]
] + (2*I)*b*c*x*PolyLog[2, -E^(I*ArcCos[c*x])] - (2*I)*b*c*x*PolyLog[2, E^(I*ArcCos[c*x])])/(d*x)

Maple [A] (verified)

Time = 3.49 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.36

method	result	size
parts	\(-\frac {a \left (\frac {c \ln \left (c x -1\right )}{2}-\frac {c \ln \left (c x +1\right )}{2}+\frac {1}{x}\right )}{d}-\frac {b c \left (\frac {\arccos \left (c x \right )}{c x}+i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(146\)
derivativedivides	\(c \left (-\frac {a \left (\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {\arccos \left (c x \right )}{c x}+i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(149\)
default	\(c \left (-\frac {a \left (\frac {1}{c x}+\frac {\ln \left (c x -1\right )}{2}-\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\frac {\arccos \left (c x \right )}{c x}+i \operatorname {dilog}\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {dilog}\left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+2 i \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\right )\)	\(149\)

[In]

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

[Out]

-a/d*(1/2*c*ln(c*x-1)-1/2*c*ln(c*x+1)+1/x)-b/d*c*(1/c/x*arccos(c*x)+I*dilog(c*x+I*(-c^2*x^2+1)^(1/2))+I*dilog(
1+c*x+I*(-c^2*x^2+1)^(1/2))+2*I*arctan(c*x+I*(-c^2*x^2+1)^(1/2))-arccos(c*x)*ln(1+c*x+I*(-c^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral(-(b*arccos(c*x) + a)/(c^2*d*x^4 - d*x^2), x)

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx=- \frac {\int \frac {a}{c^{2} x^{4} - x^{2}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{4} - x^{2}}\, dx}{d} \]

[In]

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

[Out]

-(Integral(a/(c**2*x**4 - x**2), x) + Integral(b*acos(c*x)/(c**2*x**4 - x**2), x))/d

Maxima [F]

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

[In]

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

[Out]

1/2*a*(c*log(c*x + 1)/d - c*log(c*x - 1)/d - 2/(d*x)) - 1/2*(2*d*x*integrate(1/2*(c^2*x*log(c*x + 1) - c^2*x*l
og(-c*x + 1) - 2*c)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^2*d*x^3 - d*x), x) - (c*x*log(c*x + 1) - c*x*log(-c*x + 1)
- 2)*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*b/(d*x)

Giac [F]

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

[In]

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

[Out]

integrate(-(b*arccos(c*x) + a)/((c^2*d*x^2 - d)*x^2), x)

Mupad [F(-1)]

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

[In]

int((a + b*acos(c*x))/(x^2*(d - c^2*d*x^2)),x)

[Out]

int((a + b*acos(c*x))/(x^2*(d - c^2*d*x^2)), x)