Integrand size = 12, antiderivative size = 63 \[ \int \frac {a+b \arccos (c x)}{x} \, dx=-\frac {i (a+b \arccos (c x))^2}{2 b}+(a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]
-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)
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arccos (c x)}{x} \, dx=-\frac {1}{2} i b \arccos (c x)^2+b \arccos (c x) \log \left (1+e^{2 i \arccos (c x)}\right )+a \log (x)-\frac {1}{2} i b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right ) \]
(-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])]
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5137, 3042, 4202, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a+b \arccos (c x)}{x} \, dx\) |
\(\Big \downarrow \) 5137 |
\(\displaystyle -\int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c x}d\arccos (c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int (a+b \arccos (c x)) \tan (\arccos (c x))d\arccos (c x)\) |
\(\Big \downarrow \) 4202 |
\(\displaystyle 2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1+e^{2 i \arccos (c x)}}d\arccos (c x)-\frac {i (a+b \arccos (c x))^2}{2 b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle 2 i \left (\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1+e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle 2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\) |
((-1/2*I)*(a + b*ArcCos[c*x])^2)/b + (2*I)*((-1/2*I)*(a + b*ArcCos[c*x])*L og[1 + E^((2*I)*ArcCos[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcCos[c*x])])/4)
3.2.44.3.1 Defintions of rubi rules used
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] - Si mp[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]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[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]
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[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] && IGt Q[m, 0]
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 ]
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 | \(a \ln \left (c x \right )+b \left (-\frac {i \arccos \left (c x \right )^{2}}{2}+\arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )\) | \(77\) |
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))
\[ \int \frac {a+b \arccos (c x)}{x} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \arccos (c x)}{x} \, dx=\int \frac {a + b \operatorname {acos}{\left (c x \right )}}{x}\, dx \]
\[ \int \frac {a+b \arccos (c x)}{x} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \arccos (c x)}{x} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{x} \,d x } \]
Timed out. \[ \int \frac {a+b \arccos (c x)}{x} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x} \,d x \]