Optimal. Leaf size=118 \[ \frac {14 a^2 \sqrt {a^2 x^2+1}}{3 x}+\frac {3 i a \sqrt {a^2 x^2+1}}{2 x^2}-\frac {\sqrt {a^2 x^2+1}}{3 x^3}-\frac {4 a^3 \sqrt {a^2 x^2+1}}{-a x+i}-\frac {11}{2} i a^3 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.60, antiderivative size = 118, 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\\ &=-\left ((3 i a) \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx\right )-\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 = 0.75 \[ \frac {1}{6} \left (33 i a^3 \log (x)-33 i a^3 \log \left (\sqrt {a^2 x^2+1}+1\right )+\frac {\sqrt {a^2 x^2+1} \left (52 a^3 x^3-19 i a^2 x^2+7 a x+2 i\right )}{x^3 (a x-i)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 139, normalized size = 1.18 \[ \frac {52 \, a^{4} x^{4} - 52 i \, a^{3} x^{3} - 33 \, {\left (i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) - 33 \, {\left (-i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + {\left (52 \, a^{3} x^{3} - 19 i \, a^{2} x^{2} + 7 \, a x + 2 i\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, a x^{4} - 6 i \, x^{3}} \]
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 [B] time = 0.18, size = 392, normalized size = 3.32 \[ -\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x^{3}}+\frac {16 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{3 x}-\frac {16 a^{4} x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}-8 a^{4} x \sqrt {a^{2} x^{2}+1}-\frac {8 a^{4} \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {2 i a \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{\left (x -\frac {i}{a}\right )^{2}}-\frac {11 i a^{3} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{2}+8 a^{4} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x +\frac {8 a^{4} \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}+\frac {11 i a^{3} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{6}+\frac {3 i a \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{2 x^{2}}+\frac {11 i a^{3} \sqrt {a^{2} x^{2}+1}}{2}-\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{\left (x -\frac {i}{a}\right )^{3}}-\frac {16 i a^{3} \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 117, normalized size = 0.99 \[ \frac {14\,a^2\,\sqrt {a^2\,x^2+1}}{3\,x}-\frac {\sqrt {a^2\,x^2+1}}{3\,x^3}+\frac {a\,\sqrt {a^2\,x^2+1}\,3{}\mathrm {i}}{2\,x^2}-\frac {11\,a^3\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )}{2}-\frac {4\,a^4\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{7} - 3 i a^{2} x^{6} - 3 a x^{5} + i x^{4}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{7} - 3 i a^{2} x^{6} - 3 a x^{5} + i x^{4}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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