Optimal. Leaf size=51 \[ -\frac {1}{2} i \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{2} i \sin ^{-1}(a x)^2+\sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4625, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{2} i \sin ^{-1}(a x)^2+\sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)}{x} \, dx &=\operatorname {Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{2} i \sin ^{-1}(a x)^2-2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{2} i \sin ^{-1}(a x)^2+\sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{2} i \sin ^{-1}(a x)^2+\sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-\frac {1}{2} i \sin ^{-1}(a x)^2+\sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{2} i \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 46, normalized size = 0.90 \[ \sin ^{-1}(a x) \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{2} i \left (\sin ^{-1}(a x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.40, size = 111, normalized size = 2.18 \[ -\frac {i \arcsin \left (a x \right )^{2}}{2}+\arcsin \left (a x \right ) \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right ) \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-i \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )-i \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.08, size = 41, normalized size = 0.80 \[ \ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (a\,x\right )-\frac {\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-\frac {{\mathrm {asin}\left (a\,x\right )}^2\,1{}\mathrm {i}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________