\(\int \frac {a+b \arccos (c x)}{x (d-c^2 d x^2)^2} \, dx\) [13]

Optimal result

Integrand size = 25, antiderivative size = 122 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4794, 4770, 4504, 4268, 2317, 2438, 197} \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {2 \text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{d^2}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

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

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

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 4504

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rule 4770

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

Rule 4794

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/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Dist[b*c*(n/(2*f*(p + 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]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\text {Subst}(\int (a+b x) \csc (x) \sec (x) \, dx,x,\arccos (c x))}{d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 \text {Subst}(\int (a+b x) \csc (2 x) \, dx,x,\arccos (c x))}{d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{d^2}-\frac {b \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\arccos (c x)\right )}{d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}-\frac {(i b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {(i b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \arccos (c x)}\right )}{2 d^2} \\ & = \frac {b c x}{2 d^2 \sqrt {1-c^2 x^2}}+\frac {a+b \arccos (c x)}{2 d^2 \left (1-c^2 x^2\right )}+\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d^2}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d^2}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.91 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {\frac {b \sqrt {1-c^2 x^2}}{1-c x}-\frac {b \sqrt {1-c^2 x^2}}{1+c x}-\frac {2 a}{-1+c^2 x^2}+\frac {b \arccos (c x)}{1-c x}+\frac {b \arccos (c x)}{1+c x}-4 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )-4 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )+4 b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )+4 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )+4 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{4 d^2} \]

[In]

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

[Out]

((b*Sqrt[1 - c^2*x^2])/(1 - c*x) - (b*Sqrt[1 - c^2*x^2])/(1 + c*x) - (2*a)/(-1 + c^2*x^2) + (b*ArcCos[c*x])/(1
- c*x) + (b*ArcCos[c*x])/(1 + c*x) - 4*b*ArcCos[c*x]*Log[1 - E^(I*ArcCos[c*x])] - 4*b*ArcCos[c*x]*Log[1 + E^(
I*ArcCos[c*x])] + 4*b*ArcCos[c*x]*Log[1 + E^((2*I)*ArcCos[c*x])] + 4*a*Log[x] - 2*a*Log[1 - c^2*x^2] + (4*I)*b
*PolyLog[2, -E^(I*ArcCos[c*x])] + (4*I)*b*PolyLog[2, E^(I*ArcCos[c*x])] - (2*I)*b*PolyLog[2, -E^((2*I)*ArcCos[
c*x])])/(4*d^2)

Maple [A] (verified)

Time = 2.22 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.09

method	result	size
parts	\(\frac {a \left (\ln \left (x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}-\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(255\)
derivativedivides	\(\frac {a \left (\ln \left (c x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}-\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(257\)
default	\(\frac {a \left (\ln \left (c x \right )-\frac {1}{4 \left (c x -1\right )}-\frac {\ln \left (c x -1\right )}{2}+\frac {1}{4 c x +4}-\frac {\ln \left (c x +1\right )}{2}\right )}{d^{2}}+\frac {b \left (-\frac {i c^{2} x^{2}+c x \sqrt {-c^{2} x^{2}+1}+\arccos \left (c x \right )-i}{2 \left (c^{2} x^{2}-1\right )}-\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}\)	\(257\)

[In]

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

[Out]

a/d^2*(ln(x)-1/4/(c*x-1)-1/2*ln(c*x-1)+1/4/(c*x+1)-1/2*ln(c*x+1))+b/d^2*(-1/2*(I*c^2*x^2+c*x*(-c^2*x^2+1)^(1/2
)+arccos(c*x)-I)/(c^2*x^2-1)-arccos(c*x)*ln(1-c*x-I*(-c^2*x^2+1)^(1/2))+I*polylog(2,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))^2)-1/2*I*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)-arccos(c*x)*ln(1
+c*x+I*(-c^2*x^2+1)^(1/2))+I*polylog(2,-c*x-I*(-c^2*x^2+1)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

integral((b*arccos(c*x) + a)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)

Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{4} x^{5} - 2 c^{2} x^{3} + x}\, dx}{d^{2}} \]

[In]

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

[Out]

(Integral(a/(c**4*x**5 - 2*c**2*x**3 + x), x) + Integral(b*acos(c*x)/(c**4*x**5 - 2*c**2*x**3 + x), x))/d**2

Maxima [F]

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

[In]

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

[Out]

-1/2*a*(1/(c^2*d^2*x^2 - d^2) + log(c*x + 1)/d^2 + log(c*x - 1)/d^2 - 2*log(x)/d^2) + b*integrate(arctan2(sqrt
(c*x + 1)*sqrt(-c*x + 1), c*x)/(c^4*d^2*x^5 - 2*c^2*d^2*x^3 + d^2*x), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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