Integrand size = 12, antiderivative size = 64 \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}-\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5327, 4721, 3798, 2221, 2317, 2438} \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{2} i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
[In]
[Out]
Rule 2221
Rule 2317
Rule 2438
Rule 3798
Rule 4721
Rule 5327
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {a+b \arcsin \left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \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 )+b \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right ) \\ & = \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) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right ) \\ & = \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 ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=-b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+a \log (x)+\frac {1}{2} i b \left (\csc ^{-1}(c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right ) \]
[In]
[Out]
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 | \(a \ln \left (c x \right )+b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2}}{2}-\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\) | \(138\) |
[In]
[Out]
\[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{x} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Time = 0.94 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx=\frac {b\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2}+a\,\ln \left (x\right )-b\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right ) \]
[In]
[Out]