Optimal. Leaf size=71 \[ -i \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4625, 3717, 2190, 2531, 2282, 6589} \[ -i \sin ^{-1}(a x) \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )-\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{x} \, dx &=\operatorname {Subst}\left (\int x^2 \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a x)^3-2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-2 \operatorname {Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-i \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+i \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-i \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-i \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 71, normalized size = 1.00 \[ i \sin ^{-1}(a x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{-2 i \sin ^{-1}(a x)}\right )+\frac {1}{3} i \sin ^{-1}(a x)^3+\sin ^{-1}(a x)^2 \log \left (1-e^{-2 i \sin ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (a x\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 169, normalized size = 2.38 \[ -\frac {i \arcsin \left (a x \right )^{3}}{3}+\arcsin \left (a x \right )^{2} \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+2 \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right )+\arcsin \left (a x \right )^{2} \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right )-2 i \arcsin \left (a x \right ) \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right )+2 \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________