Optimal. Leaf size=137 \[ -\frac {51 i \sinh ^{-1}(a x)}{8 a^4}-\frac {x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {i x^3 \sqrt {a^2 x^2+1}}{4 a}-\frac {9 i (-3 a x+2 i) \sqrt {a^2 x^2+1}}{8 a^4}+\frac {27 \sqrt {a^2 x^2+1}}{4 a^4}+\frac {(1+i a x)^3}{a^4 \sqrt {a^2 x^2+1}} \]
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Rubi [A] time = 0.62, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5060, 1633, 1593, 12, 852, 1635, 1815, 27, 743, 641, 215} \[ -\frac {i x^3 \sqrt {a^2 x^2+1}}{4 a}-\frac {x^2 \sqrt {a^2 x^2+1}}{a^2}-\frac {9 i (-3 a x+2 i) \sqrt {a^2 x^2+1}}{8 a^4}+\frac {27 \sqrt {a^2 x^2+1}}{4 a^4}+\frac {(1+i a x)^3}{a^4 \sqrt {a^2 x^2+1}}-\frac {51 i \sinh ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 215
Rule 641
Rule 743
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1815
Rule 5060
Rubi steps
\begin {align*} \int e^{3 i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac {x^3 (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^3}{a}-x^4\right )}{(1-i a x)^2} \, dx\right )\\ &=-\left ((i a) \int \frac {\left (\frac {i}{a}-x\right ) x^3 \sqrt {1+a^2 x^2}}{(1-i a x)^2} \, dx\right )\\ &=a^2 \int \frac {x^3 \left (1+a^2 x^2\right )^{3/2}}{a^2 (1-i a x)^3} \, dx\\ &=\int \frac {x^3 \left (1+a^2 x^2\right )^{3/2}}{(1-i a x)^3} \, dx\\ &=\int \frac {x^3 (1+i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\int \frac {(1+i a x)^2 \left (\frac {3 i}{a^3}-\frac {x}{a^2}-\frac {i x^2}{a}\right )}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {\int \frac {\frac {12 i}{a}-28 x-27 i a x^2+12 a^2 x^3}{\sqrt {1+a^2 x^2}} \, dx}{4 a^2}\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {\int \frac {36 i a-108 a^2 x-81 i a^3 x^2}{\sqrt {1+a^2 x^2}} \, dx}{12 a^4}\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {\int -\frac {9 i a (-2 i+3 a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{12 a^4}\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}+\frac {(3 i) \int \frac {(-2 i+3 a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{4 a^3}\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i-3 a x) \sqrt {1+a^2 x^2}}{8 a^4}+\frac {(3 i) \int \frac {-17 a^2-18 i a^3 x}{\sqrt {1+a^2 x^2}} \, dx}{8 a^5}\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}+\frac {27 \sqrt {1+a^2 x^2}}{4 a^4}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i-3 a x) \sqrt {1+a^2 x^2}}{8 a^4}-\frac {(51 i) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a^3}\\ &=\frac {(1+i a x)^3}{a^4 \sqrt {1+a^2 x^2}}+\frac {27 \sqrt {1+a^2 x^2}}{4 a^4}-\frac {x^2 \sqrt {1+a^2 x^2}}{a^2}-\frac {i x^3 \sqrt {1+a^2 x^2}}{4 a}-\frac {9 i (2 i-3 a x) \sqrt {1+a^2 x^2}}{8 a^4}-\frac {51 i \sinh ^{-1}(a x)}{8 a^4}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 80, normalized size = 0.58 \[ \sqrt {a^2 x^2+1} \left (\frac {4 i}{a^4 (a x+i)}+\frac {6}{a^4}+\frac {19 i x}{8 a^3}-\frac {x^2}{a^2}-\frac {i x^3}{4 a}\right )-\frac {51 i \sinh ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 88, normalized size = 0.64 \[ \frac {32 i \, a x - 51 \, {\left (-i \, a x + 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (-2 i \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 11 i \, a^{2} x^{2} + 29 \, a x + 80 i\right )} \sqrt {a^{2} x^{2} + 1} - 32}{8 \, a^{5} x + 8 i \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 143, normalized size = 1.04 \[ -\frac {i a \,x^{5}}{4 \sqrt {a^{2} x^{2}+1}}+\frac {17 i x^{3}}{8 a \sqrt {a^{2} x^{2}+1}}+\frac {51 i x}{8 a^{3} \sqrt {a^{2} x^{2}+1}}-\frac {51 i \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 a^{3} \sqrt {a^{2}}}-\frac {x^{4}}{\sqrt {a^{2} x^{2}+1}}+\frac {5 x^{2}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {10}{a^{4} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 114, normalized size = 0.83 \[ -\frac {i \, a x^{5}}{4 \, \sqrt {a^{2} x^{2} + 1}} - \frac {x^{4}}{\sqrt {a^{2} x^{2} + 1}} + \frac {17 i \, x^{3}}{8 \, \sqrt {a^{2} x^{2} + 1} a} + \frac {5 \, x^{2}}{\sqrt {a^{2} x^{2} + 1} a^{2}} + \frac {51 i \, x}{8 \, \sqrt {a^{2} x^{2} + 1} a^{3}} - \frac {51 i \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{4}} + \frac {10}{\sqrt {a^{2} x^{2} + 1} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.46, size = 137, normalized size = 1.00 \[ \frac {\sqrt {a^2\,x^2+1}\,\left (\frac {4}{{\left (a^2\right )}^{3/2}}+\frac {2\,\sqrt {a^2}}{a^4}-\frac {x^2\,\sqrt {a^2}}{a^2}-\frac {x^3\,{\left (a^2\right )}^{3/2}\,1{}\mathrm {i}}{4\,a^3}+\frac {x\,\sqrt {a^2}\,19{}\mathrm {i}}{8\,a^3}\right )}{\sqrt {a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,51{}\mathrm {i}}{8\,a^3\,\sqrt {a^2}}+\frac {\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{a^3\,\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \left (\int \frac {i x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{4}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{6}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{5}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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