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 [A]
time = 0.07, size = 117, normalized size = 1.30
method | result | size |
risch | \(\frac {i \left (6 a^{3} x^{3}-8 i a^{2} x^{2}-9 a x +16 i\right ) \sqrt {a^{2} x^{2}+1}}{24 a^{4}}+\frac {3 i \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 a^{3} \sqrt {a^{2}}}\) | \(77\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right ) \sqrt {a^{2} x^{2}+1}}{6}}{2 a^{4} \sqrt {\pi }}+\frac {i \left (-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {5}{2}} \left (-10 a^{2} x^{2}+15\right ) \sqrt {a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (a^{2}\right )^{\frac {5}{2}} \arcsinh \left (a x \right )}{4 a^{5}}\right )}{2 a^{3} \sqrt {\pi }\, \sqrt {a^{2}}}\) | \(109\) |
default | \(i a \left (\frac {x^{3} \sqrt {a^{2} x^{2}+1}}{4 a^{2}}-\frac {3 \left (\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}\right )+\frac {x^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {a^{2} x^{2}+1}}{3 a^{4}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 81, normalized size = 0.90 \begin {gather*} \frac {i \, \sqrt {a^{2} x^{2} + 1} x^{3}}{4 \, a} + \frac {\sqrt {a^{2} x^{2} + 1} x^{2}}{3 \, a^{2}} - \frac {3 i \, \sqrt {a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac {3 i \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{4}} - \frac {2 \, \sqrt {a^{2} x^{2} + 1}}{3 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.24, 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 [A]
time = 3.55, size = 119, normalized size = 1.32 \begin {gather*} \frac {i a x^{5}}{4 \sqrt {a^{2} x^{2} + 1}} + \begin {cases} \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases} - \frac {i x^{3}}{8 a \sqrt {a^{2} x^{2} + 1}} - \frac {3 i x}{8 a^{3} \sqrt {a^{2} x^{2} + 1}} + \frac {3 i \operatorname {asinh}{\left (a x \right )}}{8 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 70, normalized size = 0.78 \begin {gather*} -\frac {1}{24} \, \sqrt {a^{2} x^{2} + 1} {\left ({\left (2 \, x {\left (-\frac {3 i \, x}{a} - \frac {4}{a^{2}}\right )} + \frac {9 i}{a^{3}}\right )} x + \frac {16}{a^{4}}\right )} - \frac {3 i \, \log \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} + 1}\right )}{8 \, a^{3} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.43, 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]
________________________________________________________________________________________