Optimal. Leaf size=60 \[ -\frac {2 i (1+i a x)^2}{a \sqrt {a^2 x^2+1}}-\frac {3 i \sqrt {a^2 x^2+1}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5059, 853, 669, 641, 215} \[ -\frac {2 i (1+i a x)^2}{a \sqrt {a^2 x^2+1}}-\frac {3 i \sqrt {a^2 x^2+1}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 641
Rule 669
Rule 853
Rule 5059
Rubi steps
\begin {align*} \int e^{3 i \tan ^{-1}(a x)} \, dx &=\int \frac {(1+i a x)^2}{(1-i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \frac {(1+i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 i (1+i a x)^2}{a \sqrt {1+a^2 x^2}}-3 \int \frac {1+i a x}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 i (1+i a x)^2}{a \sqrt {1+a^2 x^2}}-\frac {3 i \sqrt {1+a^2 x^2}}{a}-3 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {2 i (1+i a x)^2}{a \sqrt {1+a^2 x^2}}-\frac {3 i \sqrt {1+a^2 x^2}}{a}-\frac {3 \sinh ^{-1}(a x)}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 42, normalized size = 0.70 \[ -\frac {3 \sinh ^{-1}(a x)}{a}+\frac {\sqrt {a^2 x^2+1} \left (\frac {4}{a x+i}-i\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 60, normalized size = 1.00 \[ \frac {4 \, a x + {\left (3 \, a x + 3 i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (-i \, a x + 5\right )} + 4 i}{a^{2} x + i \, a} \]
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 = 0.11, size = 81, normalized size = 1.35 \[ \frac {4 x}{\sqrt {a^{2} x^{2}+1}}-\frac {i a \,x^{2}}{\sqrt {a^{2} x^{2}+1}}-\frac {5 i}{a \sqrt {a^{2} x^{2}+1}}-\frac {3 \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.31, size = 57, normalized size = 0.95 \[ -\frac {i \, a x^{2}}{\sqrt {a^{2} x^{2} + 1}} + \frac {4 \, x}{\sqrt {a^{2} x^{2} + 1}} - \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a} - \frac {5 i}{\sqrt {a^{2} x^{2} + 1} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.42, size = 72, normalized size = 1.20 \[ -\frac {\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{a}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {4\,\sqrt {a^2\,x^2+1}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \left (\int \frac {i}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________