Optimal. Leaf size=56 \[ -\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sin ^{-1}(a x)}{3 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 56, 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 = {4627, 266, 51, 63, 208} \[ -\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-\frac {\sin ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 208
Rule 266
Rule 4627
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)}{x^4} \, dx &=-\frac {\sin ^{-1}(a x)}{3 x^3}+\frac {1}{3} a \int \frac {1}{x^3 \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {\sin ^{-1}(a x)}{3 x^3}+\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {\sin ^{-1}(a x)}{3 x^3}+\frac {1}{12} a^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {\sin ^{-1}(a x)}{3 x^3}-\frac {1}{6} a \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )\\ &=-\frac {a \sqrt {1-a^2 x^2}}{6 x^2}-\frac {\sin ^{-1}(a x)}{3 x^3}-\frac {1}{6} a^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 53, normalized size = 0.95 \[ -\frac {a x \sqrt {1-a^2 x^2}+a^3 x^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+2 \sin ^{-1}(a x)}{6 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 73, normalized size = 1.30 \[ -\frac {a^{3} x^{3} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a^{3} x^{3} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right ) + 2 \, \sqrt {-a^{2} x^{2} + 1} a x + 4 \, \arcsin \left (a x\right )}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 77, normalized size = 1.38 \[ -\frac {a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - a^{4} \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) + \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2}}{x^{2}}}{12 \, a} - \frac {\arcsin \left (a x\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 53, normalized size = 0.95 \[ a^{3} \left (-\frac {\arcsin \left (a x \right )}{3 a^{3} x^{3}}-\frac {\sqrt {-a^{2} x^{2}+1}}{6 a^{2} x^{2}}-\frac {\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 60, normalized size = 1.07 \[ -\frac {1}{6} \, {\left (a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-a^{2} x^{2} + 1}}{x^{2}}\right )} a - \frac {\arcsin \left (a x\right )}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {asin}\left (a\,x\right )}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.33, size = 109, normalized size = 1.95 \[ \frac {a \left (\begin {cases} - \frac {a^{2} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} - \frac {a \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{2 x} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{2} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a}{2 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i}{2 a x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{3} - \frac {\operatorname {asin}{\left (a x \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________