Optimal. Leaf size=113 \[ \frac {3 \sinh ^{-1}(a x)}{8 a^5}+\frac {i x^4 \sqrt {a^2 x^2+1}}{5 a}+\frac {x^3 \sqrt {a^2 x^2+1}}{4 a^2}+\frac {(-45 a x+64 i) \sqrt {a^2 x^2+1}}{120 a^5}-\frac {4 i x^2 \sqrt {a^2 x^2+1}}{15 a^3} \]
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Rubi [A] time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5060, 833, 780, 215} \[ \frac {i x^4 \sqrt {a^2 x^2+1}}{5 a}+\frac {x^3 \sqrt {a^2 x^2+1}}{4 a^2}-\frac {4 i x^2 \sqrt {a^2 x^2+1}}{15 a^3}+\frac {(-45 a x+64 i) \sqrt {a^2 x^2+1}}{120 a^5}+\frac {3 \sinh ^{-1}(a x)}{8 a^5} \]
Antiderivative was successfully verified.
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Rule 215
Rule 780
Rule 833
Rule 5060
Rubi steps
\begin {align*} \int e^{i \tan ^{-1}(a x)} x^4 \, dx &=\int \frac {x^4 (1+i a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i x^4 \sqrt {1+a^2 x^2}}{5 a}+\frac {\int \frac {x^3 \left (-4 i a+5 a^2 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{5 a^2}\\ &=\frac {x^3 \sqrt {1+a^2 x^2}}{4 a^2}+\frac {i x^4 \sqrt {1+a^2 x^2}}{5 a}+\frac {\int \frac {x^2 \left (-15 a^2-16 i a^3 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{20 a^4}\\ &=-\frac {4 i x^2 \sqrt {1+a^2 x^2}}{15 a^3}+\frac {x^3 \sqrt {1+a^2 x^2}}{4 a^2}+\frac {i x^4 \sqrt {1+a^2 x^2}}{5 a}+\frac {\int \frac {x \left (32 i a^3-45 a^4 x\right )}{\sqrt {1+a^2 x^2}} \, dx}{60 a^6}\\ &=-\frac {4 i x^2 \sqrt {1+a^2 x^2}}{15 a^3}+\frac {x^3 \sqrt {1+a^2 x^2}}{4 a^2}+\frac {i x^4 \sqrt {1+a^2 x^2}}{5 a}+\frac {(64 i-45 a x) \sqrt {1+a^2 x^2}}{120 a^5}+\frac {3 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{8 a^4}\\ &=-\frac {4 i x^2 \sqrt {1+a^2 x^2}}{15 a^3}+\frac {x^3 \sqrt {1+a^2 x^2}}{4 a^2}+\frac {i x^4 \sqrt {1+a^2 x^2}}{5 a}+\frac {(64 i-45 a x) \sqrt {1+a^2 x^2}}{120 a^5}+\frac {3 \sinh ^{-1}(a x)}{8 a^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 64, normalized size = 0.57 \[ \frac {45 \sinh ^{-1}(a x)+\sqrt {a^2 x^2+1} \left (24 i a^4 x^4+30 a^3 x^3-32 i a^2 x^2-45 a x+64 i\right )}{120 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 67, normalized size = 0.59 \[ \frac {{\left (24 i \, a^{4} x^{4} + 30 \, a^{3} x^{3} - 32 i \, a^{2} x^{2} - 45 \, a x + 64 i\right )} \sqrt {a^{2} x^{2} + 1} - 45 \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{120 \, a^{5}} \]
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 = 128, normalized size = 1.13 \[ \frac {i x^{4} \sqrt {a^{2} x^{2}+1}}{5 a}-\frac {4 i x^{2} \sqrt {a^{2} x^{2}+1}}{15 a^{3}}+\frac {8 i \sqrt {a^{2} x^{2}+1}}{15 a^{5}}+\frac {x^{3} \sqrt {a^{2} x^{2}+1}}{4 a^{2}}-\frac {3 x \sqrt {a^{2} x^{2}+1}}{8 a^{4}}+\frac {3 \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 a^{4} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 100, normalized size = 0.88 \[ \frac {i \, \sqrt {a^{2} x^{2} + 1} x^{4}}{5 \, a} + \frac {\sqrt {a^{2} x^{2} + 1} x^{3}}{4 \, a^{2}} - \frac {4 i \, \sqrt {a^{2} x^{2} + 1} x^{2}}{15 \, a^{3}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1} x}{8 \, a^{4}} + \frac {3 \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{5}} + \frac {8 i \, \sqrt {a^{2} x^{2} + 1}}{15 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 98, normalized size = 0.87 \[ \frac {\sqrt {a^2\,x^2+1}\,\left (\frac {x^3\,{\left (a^2\right )}^{3/2}}{4\,a^4}-\frac {3\,x\,\sqrt {a^2}}{8\,a^4}+\frac {a\,8{}\mathrm {i}}{15\,{\left (a^2\right )}^{5/2}}-\frac {a^3\,x^2\,4{}\mathrm {i}}{15\,{\left (a^2\right )}^{5/2}}+\frac {a^5\,x^4\,1{}\mathrm {i}}{5\,{\left (a^2\right )}^{5/2}}\right )}{\sqrt {a^2}}+\frac {3\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{8\,a^4\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.02, size = 138, normalized size = 1.22 \[ i a \left (\begin {cases} \frac {x^{4} \sqrt {a^{2} x^{2} + 1}}{5 a^{2}} - \frac {4 x^{2} \sqrt {a^{2} x^{2} + 1}}{15 a^{4}} + \frac {8 \sqrt {a^{2} x^{2} + 1}}{15 a^{6}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {x^{5}}{4 \sqrt {a^{2} x^{2} + 1}} - \frac {x^{3}}{8 a^{2} \sqrt {a^{2} x^{2} + 1}} - \frac {3 x}{8 a^{4} \sqrt {a^{2} x^{2} + 1}} + \frac {3 \operatorname {asinh}{\left (a x \right )}}{8 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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