Optimal. Leaf size=90 \[ \frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(16-9 i a x) \sqrt {1+a^2 x^2}}{24 a^4}-\frac {3 i \sinh ^{-1}(a x)}{8 a^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5168, 847, 794,
221} \begin {gather*} -\frac {3 i \sinh ^{-1}(a x)}{8 a^4}+\frac {x^2 \sqrt {a^2 x^2+1}}{3 a^2}-\frac {i x^3 \sqrt {a^2 x^2+1}}{4 a}-\frac {(16-9 i a x) \sqrt {a^2 x^2+1}}{24 a^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 794
Rule 847
Rule 5168
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (1-i a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}+\frac {\int \frac {x^2 \left (3 i a+4 a^2 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}+\frac {\int \frac {x \left (-8 a^2+9 i a^3 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{12 a^4}\\ &=\frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(16-9 i a x) \sqrt {1+a^2 x^2}}{24 a^4}-\frac {(3 i) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a^3}\\ &=\frac {x^2 \sqrt {1+a^2 x^2}}{3 a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {(16-9 i a x) \sqrt {1+a^2 x^2}}{24 a^4}-\frac {3 i \sinh ^{-1}(a x)}{8 a^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 56, normalized size = 0.62 \begin {gather*} \frac {\sqrt {1+a^2 x^2} \left (-16+9 i a x+8 a^2 x^2-6 i a^3 x^3\right )-9 i \sinh ^{-1}(a x)}{24 a^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 240 vs. \(2 (73 ) = 146\).
time = 0.10, size = 241, normalized size = 2.68
method | result | size |
risch | \(-\frac {i \left (6 a^{3} x^{3}+8 i a^{2} x^{2}-9 a x -16 i\right ) \sqrt {a^{2} x^{2}+1}}{24 a^{4}}-\frac {3 i \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 a^{3} \sqrt {a^{2}}}\) | \(77\) |
default | \(-\frac {i \left (\frac {x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{2}}-\frac {\frac {x \sqrt {a^{2} x^{2}+1}}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}}{4 a^{2}}\right )}{a}+\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a^{4}}+\frac {i \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}\right )}{a^{3}}-\frac {\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}}{a^{4}}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 76, normalized size = 0.84 \begin {gather*} -\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{3}} + \frac {5 i \, \sqrt {a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{4}} - \frac {3 i \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{4}} - \frac {\sqrt {a^{2} x^{2} + 1}}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.39, size = 59, normalized size = 0.66 \begin {gather*} \frac {{\left (-6 i \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 i \, a x - 16\right )} \sqrt {a^{2} x^{2} + 1} + 9 i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{24 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i \int \frac {x^{3} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.06, size = 85, normalized size = 0.94 \begin {gather*} -\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,3{}\mathrm {i}}{8\,a^3\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {2}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^2\,x^2}{3\,{\left (a^2\right )}^{3/2}}+\frac {x^3\,{\left (a^2\right )}^{3/2}\,1{}\mathrm {i}}{4\,a^3}-\frac {x\,\sqrt {a^2}\,3{}\mathrm {i}}{8\,a^3}\right )}{\sqrt {a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________