Optimal. Leaf size=117 \[ \frac {14 a^2 \sqrt {a^2 x^2+1}}{3 x}-\frac {3 i a \sqrt {a^2 x^2+1}}{2 x^2}-\frac {\sqrt {a^2 x^2+1}}{3 x^3}+\frac {4 a^3 \sqrt {a^2 x^2+1}}{a x+i}+\frac {11}{2} i a^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.62, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5060, 6742, 271, 264, 266, 51, 63, 208, 651} \[ \frac {4 a^3 \sqrt {a^2 x^2+1}}{a x+i}+\frac {14 a^2 \sqrt {a^2 x^2+1}}{3 x}-\frac {3 i a \sqrt {a^2 x^2+1}}{2 x^2}-\frac {\sqrt {a^2 x^2+1}}{3 x^3}+\frac {11}{2} i a^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 264
Rule 266
Rule 271
Rule 651
Rule 5060
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{3 i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+i a x)^2}{x^4 (1-i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^4 \sqrt {1+a^2 x^2}}+\frac {3 i a}{x^3 \sqrt {1+a^2 x^2}}-\frac {4 a^2}{x^2 \sqrt {1+a^2 x^2}}-\frac {4 i a^3}{x \sqrt {1+a^2 x^2}}+\frac {4 i a^4}{(i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx\\ &=(3 i a) \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx-\left (4 a^2\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx-\left (4 i a^3\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx+\left (4 i a^4\right ) \int \frac {1}{(i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^4 \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}+\frac {4 a^2 \sqrt {1+a^2 x^2}}{x}+\frac {4 a^3 \sqrt {1+a^2 x^2}}{i+a x}+\frac {1}{2} (3 i a) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 a^2\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx-\left (2 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1+a^2 x^2}}{i+a x}-(4 i a) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )-\frac {1}{4} \left (3 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1+a^2 x^2}}{i+a x}+4 i a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )-\frac {1}{2} (3 i a) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{3 x^3}-\frac {3 i a \sqrt {1+a^2 x^2}}{2 x^2}+\frac {14 a^2 \sqrt {1+a^2 x^2}}{3 x}+\frac {4 a^3 \sqrt {1+a^2 x^2}}{i+a x}+\frac {11}{2} i a^3 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 89, normalized size = 0.76 \[ \frac {1}{6} \left (-33 i a^3 \log (x)+33 i a^3 \log \left (\sqrt {a^2 x^2+1}+1\right )+\frac {\sqrt {a^2 x^2+1} \left (52 a^3 x^3+19 i a^2 x^2+7 a x-2 i\right )}{x^3 (a x+i)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 139, normalized size = 1.19 \[ \frac {52 \, a^{4} x^{4} + 52 i \, a^{3} x^{3} - 33 \, {\left (-i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 33 \, {\left (i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + {\left (52 \, a^{3} x^{3} + 19 i \, a^{2} x^{2} + 7 \, a x - 2 i\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, a x^{4} + 6 i \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 141, normalized size = 1.21 \[ -\frac {1}{3 x^{3} \sqrt {a^{2} x^{2}+1}}-\frac {13 a^{2} \left (-\frac {1}{x \sqrt {a^{2} x^{2}+1}}-\frac {2 a^{2} x}{\sqrt {a^{2} x^{2}+1}}\right )}{3}-i a^{3} \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )+3 i a \left (-\frac {1}{2 x^{2} \sqrt {a^{2} x^{2}+1}}-\frac {3 a^{2} \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 100, normalized size = 0.85 \[ \frac {26 \, a^{4} x}{3 \, \sqrt {a^{2} x^{2} + 1}} + \frac {11}{2} i \, a^{3} \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) - \frac {11 i \, a^{3}}{2 \, \sqrt {a^{2} x^{2} + 1}} + \frac {13 \, a^{2}}{3 \, \sqrt {a^{2} x^{2} + 1} x} - \frac {3 i \, a}{2 \, \sqrt {a^{2} x^{2} + 1} x^{2}} - \frac {1}{3 \, \sqrt {a^{2} x^{2} + 1} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.41, size = 116, normalized size = 0.99 \[ \frac {11\,a^3\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )}{2}-\frac {\sqrt {a^2\,x^2+1}}{3\,x^3}-\frac {a\,\sqrt {a^2\,x^2+1}\,3{}\mathrm {i}}{2\,x^2}+\frac {14\,a^2\,\sqrt {a^2\,x^2+1}}{3\,x}+\frac {4\,a^4\,\sqrt {a^2\,x^2+1}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \left (\int \frac {i}{a^{2} x^{6} \sqrt {a^{2} x^{2} + 1} + x^{4} \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{6} \sqrt {a^{2} x^{2} + 1} + x^{4} \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{6} \sqrt {a^{2} x^{2} + 1} + x^{4} \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{6} \sqrt {a^{2} x^{2} + 1} + x^{4} \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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