\(\int \frac {a+b \arccos (c x)}{x} \, dx\) [144]

Optimal result

Integrand size = 12, antiderivative size = 63 \[ \int \frac {a+b \arccos (c x)}{x} \, dx=-\frac {i (a+b \arccos (c x))^2}{2 b}+(a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4722, 3800, 2221, 2317, 2438} \[ \int \frac {a+b \arccos (c x)}{x} \, dx=-\frac {i (a+b \arccos (c x))^2}{2 b}+\log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]

[In]

Int[(a + b*ArcCos[c*x])/x,x]

[Out]

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 3800

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

Rule 4722

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arccos (c x)}{x} \, dx=-\frac {1}{2} i b \arccos (c x)^2+b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]

[In]

Integrate[(a + b*ArcCos[c*x])/x,x]

[Out]

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

Maple [A] (verified)

Time = 0.95 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19

method	result	size
parts	\(a \ln \left (x \right )+b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(75\)
derivativedivides	\(a \ln \left (c x \right )+b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(77\)
default	\(a \ln \left (c x \right )+b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\)	\(77\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*arccos(c*x) + a)/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*acos(c*x))/x, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((b*arccos(c*x) + a)/x, x)

Mupad [F(-1)]

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

[In]

int((a + b*acos(c*x))/x,x)

[Out]

int((a + b*acos(c*x))/x, x)