Optimal. Leaf size=64 \[ \frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{2} i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5219, 4625, 3717, 2190, 2279, 2391} \[ \frac {1}{2} i b \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 5219
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 53, normalized size = 0.83 \[ a \log (x)+\frac {1}{2} i b \left (\csc ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\right )-b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arccsc}\left (c x\right ) + a}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 140, normalized size = 2.19 \[ a \ln \left (c x \right )+\frac {i b \mathrm {arccsc}\left (c x \right )^{2}}{2}-b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (c^{2} \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} \log \relax (x)}{c^{4} x^{3} - c^{2} x}\,{d x} + \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) \log \relax (x)\right )} b + a \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.76, size = 65, normalized size = 1.02 \[ \frac {b\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2}+a\,\ln \relax (x)-b\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________