Integrand size = 12, antiderivative size = 64 \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}-\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
1/2*I*(a+b*arccsc(c*x))^2/b-(a+b*arccsc(c*x))*ln(1-(I/c/x+(1-1/c^2/x^2)^(1 /2))^2)+1/2*I*b*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=-b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+a \log (x)+\frac {1}{2} i b \left (\csc ^{-1}(c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right ) \]
-(b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])]) + a*Log[x] + (I/2)*b*(ArcC sc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])])
Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5742, 5136, 3042, 25, 4200, 25, 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.
\(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx\) |
\(\Big \downarrow \) 5742 |
\(\displaystyle -\int x \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )d\frac {1}{x}\) |
\(\Big \downarrow \) 5136 |
\(\displaystyle -\int c \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )d\arcsin \left (\frac {1}{c x}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\int -\left (\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \tan \left (\arcsin \left (\frac {1}{c x}\right )+\frac {\pi }{2}\right )\right )d\arcsin \left (\frac {1}{c x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \tan \left (\arcsin \left (\frac {1}{c x}\right )+\frac {\pi }{2}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )d\arcsin \left (\frac {1}{c x}\right )\) |
\(\Big \downarrow \) 4200 |
\(\displaystyle \frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b}-2 i \int -\frac {e^{2 i \arcsin \left (\frac {1}{c x}\right )} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}}d\arcsin \left (\frac {1}{c x}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle 2 i \int \frac {e^{2 i \arcsin \left (\frac {1}{c x}\right )} \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}}d\arcsin \left (\frac {1}{c x}\right )+\frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )-\frac {1}{2} i b \int \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )d\arcsin \left (\frac {1}{c x}\right )\right )+\frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )-\frac {1}{4} b \int x \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )de^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )+\frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle 2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )\right )+\frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b}\) |
((I/2)*(a + b*ArcSin[1/(c*x)])^2)/b + (2*I)*((I/2)*(a + b*ArcSin[1/(c*x)]) *Log[1 - E^((2*I)*ArcSin[1/(c*x)])] + (b*PolyLog[2, E^((2*I)*ArcSin[1/(c*x )])])/4)
3.1.8.3.1 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_.) + Pi*(k_.) + (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*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] , x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[( a + b*x)^n*Cot[x], x], x, ArcSin[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b *ArcSin[x/c])/x, x], x, 1/x] /; FreeQ[{a, b, c}, x]
Time = 0.89 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.12
method | result | size |
parts | \(a \ln \left (x \right )+b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(136\) |
derivativedivides | \(a \ln \left (c x \right )+b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(138\) |
default | \(a \ln \left (c x \right )+b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(138\) |
a*ln(x)+b*(1/2*I*arccsc(c*x)^2-arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2)) +I*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))-arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^ 2)^(1/2))+I*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2)))
\[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{x} \,d x } \]
\[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x}\, dx \]
\[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{x} \,d x } \]
(c^2*integrate(sqrt(c*x + 1)*sqrt(c*x - 1)*log(x)/(c^4*x^3 - c^2*x), x) + arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(x))*b + a*log(x)
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Limit: Max order reached or unable to make series expan sion Error: Bad Argument Value
Time = 0.94 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\frac {b\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2}+a\,\ln \left (x\right )-b\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right ) \]