Optimal. Leaf size=119 \[ -6 i a^2 \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-2 i a^2 \sin ^{-1}(a x)^3+6 a^2 \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^4}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.216142, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4627, 4681, 4625, 3717, 2190, 2531, 2282, 6589} \[ -6 i a^2 \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+3 a^2 \text{PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-2 i a^2 \sin ^{-1}(a x)^3+6 a^2 \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\frac{\sin ^{-1}(a x)^4}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4627
Rule 4681
Rule 4625
Rule 3717
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(a x)^4}{x^3} \, dx &=-\frac{\sin ^{-1}(a x)^4}{2 x^2}+(2 a) \int \frac{\sin ^{-1}(a x)^3}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}+\left (6 a^2\right ) \int \frac{\sin ^{-1}(a x)^2}{x} \, dx\\ &=-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}+\left (6 a^2\right ) \operatorname{Subst}\left (\int x^2 \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 i a^2 \sin ^{-1}(a x)^3-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}-\left (12 i a^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 i a^2 \sin ^{-1}(a x)^3-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}+6 a^2 \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-\left (12 a^2\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 i a^2 \sin ^{-1}(a x)^3-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}+6 a^2 \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-6 i a^2 \sin ^{-1}(a x) \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\left (6 i a^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-2 i a^2 \sin ^{-1}(a x)^3-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}+6 a^2 \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-6 i a^2 \sin ^{-1}(a x) \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-2 i a^2 \sin ^{-1}(a x)^3-\frac{2 a \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-\frac{\sin ^{-1}(a x)^4}{2 x^2}+6 a^2 \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-6 i a^2 \sin ^{-1}(a x) \text{Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+3 a^2 \text{Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end{align*}
Mathematica [A] time = 0.282873, size = 124, normalized size = 1.04 \[ -\frac{\sin ^{-1}(a x)^4}{2 x^2}+\frac{1}{4} a^2 \left (24 i \sin ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \sin ^{-1}(a x)}\right )+12 \text{PolyLog}\left (3,e^{-2 i \sin ^{-1}(a x)}\right )-\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{a x}+8 i \sin ^{-1}(a x)^3+24 \sin ^{-1}(a x)^2 \log \left (1-e^{-2 i \sin ^{-1}(a x)}\right )-i \pi ^3\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.091, size = 227, normalized size = 1.9 \begin{align*} -2\,i{a}^{2} \left ( \arcsin \left ( ax \right ) \right ) ^{3}-2\,{\frac{a \left ( \arcsin \left ( ax \right ) \right ) ^{3}\sqrt{-{a}^{2}{x}^{2}+1}}{x}}-{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{2\,{x}^{2}}}+6\,{a}^{2} \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1+iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) -12\,i{a}^{2}\arcsin \left ( ax \right ){\it polylog} \left ( 2,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +12\,{a}^{2}{\it polylog} \left ( 3,-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) +6\,{a}^{2} \left ( \arcsin \left ( ax \right ) \right ) ^{2}\ln \left ( 1-iax-\sqrt{-{a}^{2}{x}^{2}+1} \right ) -12\,i{a}^{2}\arcsin \left ( ax \right ){\it polylog} \left ( 2,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) +12\,{a}^{2}{\it polylog} \left ( 3,iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{4} + \frac{1}{2} \,{\left (\sqrt{a x + 1} \sqrt{-a x + 1} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} + 8 \, x \int \frac{7 \, \sqrt{a x + 1} \sqrt{-a x + 1} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{3} - 3 \,{\left (a^{3} x^{3} - a x\right )} \arctan \left (a x, \sqrt{a x + 1} \sqrt{-a x + 1}\right )^{2}}{8 \,{\left (a^{2} x^{4} - x^{2}\right )}}\,{d x}\right )} a x}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\arcsin \left (a x\right )^{4}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{asin}^{4}{\left (a x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{4}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]