Optimal. Leaf size=63 \[ \frac {i a \sqrt {a^2 x^2+1}}{x}-\frac {\sqrt {a^2 x^2+1}}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5060, 835, 807, 266, 63, 208} \[ \frac {i a \sqrt {a^2 x^2+1}}{x}-\frac {\sqrt {a^2 x^2+1}}{2 x^2}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rule 807
Rule 835
Rule 5060
Rubi steps
\begin {align*} \int \frac {e^{-i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac {1-i a x}{x^3 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{2 x^2}-\frac {1}{2} \int \frac {2 i a+a^2 x}{x^2 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {i a \sqrt {1+a^2 x^2}}{x}-\frac {1}{2} a^2 \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {i a \sqrt {1+a^2 x^2}}{x}-\frac {1}{4} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {i a \sqrt {1+a^2 x^2}}{x}-\frac {1}{2} \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 {\sqrt {1+a^2 x^2}}{2 x^2}+\frac {i a \sqrt {1+a^2 x^2}}{x}+\frac {1}{2} a^2 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 57, normalized size = 0.90 \[ \frac {1}{2} \left (\frac {(-1+2 i a x) \sqrt {a^2 x^2+1}}{x^2}+a^2 \log \left (\sqrt {a^2 x^2+1}+1\right )+a^2 (-\log (x))\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 83, normalized size = 1.32 \[ \frac {a^{2} x^{2} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - a^{2} x^{2} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 2 i \, a^{2} x^{2} + \sqrt {a^{2} x^{2} + 1} {\left (2 i \, a x - 1\right )}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.17, size = 219, normalized size = 3.48 \[ \frac {a^{2} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}-\frac {a^{2} \sqrt {a^{2} x^{2}+1}}{2}+\frac {i a \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{x}-i a^{3} x \sqrt {a^{2} x^{2}+1}-\frac {i a^{3} \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+a^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a^{3} \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}-\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a^{2} x^{2} + 1}}{{\left (i \, a x + 1\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.04, size = 52, normalized size = 0.83 \[ \frac {a^2\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )}{2}-\frac {\sqrt {a^2\,x^2+1}}{2\,x^2}+\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x^{4} - i x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________