\(\int \frac {a+b \arccos (c x)}{x (d-c^2 d x^2)} \, dx\) [5]

Optimal result

Integrand size = 25, antiderivative size = 71 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\frac {2 (a+b \arccos (c x)) \text {arctanh}\left (e^{2 i \arccos (c x)}\right )}{d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4770, 4504, 4268, 2317, 2438} \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=\frac {2 \text {arctanh}\left (e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))}{d}-\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{2 d} \]

[In]

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

[Out]

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

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]

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(143\) vs. \(2(71)=142\).

Time = 0.32 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.01 \[ \int \frac {a+b \arccos (c x)}{x \left (d-c^2 d x^2\right )} \, dx=-\frac {2 b \arccos (c x) \log \left (1-e^{i \arccos (c x)}\right )+2 b \arccos (c x) \log \left (1+e^{i \arccos (c x)}\right )-2 b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )-2 a \log (x)+a \log \left (1-c^2 x^2\right )-2 i b \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )+i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.76

method	result	size
parts	\(-\frac {a \left (-\ln \left (x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(196\)
derivativedivides	\(-\frac {a \left (-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(198\)
default	\(-\frac {a \left (-\ln \left (c x \right )+\frac {\ln \left (c x -1\right )}{2}+\frac {\ln \left (c x +1\right )}{2}\right )}{d}-\frac {b \left (\arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )-\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}+\arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {polylog}\left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d}\)	\(198\)

[In]

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

[Out]

-a/d*(-ln(x)+1/2*ln(c*x-1)+1/2*ln(c*x+1))-b/d*(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]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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