Optimal. Leaf size=102 \[ \frac {11 \sinh ^{-1}(a x)}{2 a^3}+\frac {i (1+i a x)^3}{a^3 \sqrt {a^2 x^2+1}}+\frac {i (3+i a x)^2 \sqrt {a^2 x^2+1}}{3 a^3}+\frac {(-3 a x+28 i) \sqrt {a^2 x^2+1}}{6 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5060, 1633, 1593, 12, 852, 1635, 1654, 780, 215} \[ \frac {i (1+i a x)^3}{a^3 \sqrt {a^2 x^2+1}}+\frac {i (3+i a x)^2 \sqrt {a^2 x^2+1}}{3 a^3}+\frac {(-3 a x+28 i) \sqrt {a^2 x^2+1}}{6 a^3}+\frac {11 \sinh ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 215
Rule 780
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1654
Rule 5060
Rubi steps
\begin {align*} \int e^{3 i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1+i a x)^2}{(1-i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=-\left ((i a) \int \frac {\sqrt {1+a^2 x^2} \left (\frac {i x^2}{a}-x^3\right )}{(1-i a x)^2} \, dx\right )\\ &=-\left ((i a) \int \frac {\left (\frac {i}{a}-x\right ) x^2 \sqrt {1+a^2 x^2}}{(1-i a x)^2} \, dx\right )\\ &=a^2 \int \frac {x^2 \left (1+a^2 x^2\right )^{3/2}}{a^2 (1-i a x)^3} \, dx\\ &=\int \frac {x^2 \left (1+a^2 x^2\right )^{3/2}}{(1-i a x)^3} \, dx\\ &=\int \frac {x^2 (1+i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {i (1+i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\int \frac {\left (-\frac {3}{a^2}-\frac {i x}{a}\right ) (1+i a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i (1+i a x)^3}{a^3 \sqrt {1+a^2 x^2}}+\frac {i (3+i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}+\frac {1}{3} \int \frac {\left (-\frac {3}{a^2}-\frac {i x}{a}\right ) (-5-3 i a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i (1+i a x)^3}{a^3 \sqrt {1+a^2 x^2}}+\frac {(28 i-3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {i (3+i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}+\frac {11 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}\\ &=\frac {i (1+i a x)^3}{a^3 \sqrt {1+a^2 x^2}}+\frac {(28 i-3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {i (3+i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}+\frac {11 \sinh ^{-1}(a x)}{2 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 63, normalized size = 0.62 \[ \frac {33 \sinh ^{-1}(a x)+\frac {\sqrt {a^2 x^2+1} \left (-2 i a^3 x^3-7 a^2 x^2+19 i a x-52\right )}{a x+i}}{6 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 81, normalized size = 0.79 \[ -\frac {24 \, a x + {\left (33 \, a x + 33 i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (-2 i \, a^{3} x^{3} - 7 \, a^{2} x^{2} + 19 i \, a x - 52\right )} \sqrt {a^{2} x^{2} + 1} + 24 i}{6 \, a^{4} x + 6 i \, a^{3}} \]
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 [A] time = 0.16, size = 123, normalized size = 1.21 \[ -\frac {i a \,x^{4}}{3 \sqrt {a^{2} x^{2}+1}}+\frac {13 i x^{2}}{3 a \sqrt {a^{2} x^{2}+1}}+\frac {26 i}{3 a^{3} \sqrt {a^{2} x^{2}+1}}-\frac {3 x^{3}}{2 \sqrt {a^{2} x^{2}+1}}-\frac {11 x}{2 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {11 \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 95, normalized size = 0.93 \[ -\frac {i \, a x^{4}}{3 \, \sqrt {a^{2} x^{2} + 1}} - \frac {3 \, x^{3}}{2 \, \sqrt {a^{2} x^{2} + 1}} + \frac {13 i \, x^{2}}{3 \, \sqrt {a^{2} x^{2} + 1} a} - \frac {11 \, x}{2 \, \sqrt {a^{2} x^{2} + 1} a^{2}} + \frac {11 \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{3}} + \frac {26 i}{3 \, \sqrt {a^{2} x^{2} + 1} a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 114, normalized size = 1.12 \[ \frac {11\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,x\,\sqrt {a^2}}{2\,a^2}-\frac {a\,14{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}+\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}-\frac {4\,\sqrt {a^2\,x^2+1}}{a^2\,\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \left (\int \frac {i x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{5}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{4}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________