Optimal. Leaf size=139 \[ \frac {19 a^2 \sqrt {a^2 x^2+1}}{8 x^2}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}+\frac {i a \sqrt {a^2 x^2+1}}{x^3}+\frac {4 i a^4 \sqrt {a^2 x^2+1}}{-a x+i}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-\frac {6 i a^3 \sqrt {a^2 x^2+1}}{x} \]
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Rubi [A] time = 0.66, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5060, 6742, 266, 51, 63, 208, 271, 264, 651} \[ \frac {4 i a^4 \sqrt {a^2 x^2+1}}{-a x+i}-\frac {6 i a^3 \sqrt {a^2 x^2+1}}{x}+\frac {19 a^2 \sqrt {a^2 x^2+1}}{8 x^2}+\frac {i a \sqrt {a^2 x^2+1}}{x^3}-\frac {\sqrt {a^2 x^2+1}}{4 x^4}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right ) \]
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^5} \, dx &=\int \frac {(1-i a x)^2}{x^5 (1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \left (\frac {1}{x^5 \sqrt {1+a^2 x^2}}-\frac {3 i a}{x^4 \sqrt {1+a^2 x^2}}-\frac {4 a^2}{x^3 \sqrt {1+a^2 x^2}}+\frac {4 i a^3}{x^2 \sqrt {1+a^2 x^2}}+\frac {4 a^4}{x \sqrt {1+a^2 x^2}}-\frac {4 a^5}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx\\ &=-\left ((3 i a) \int \frac {1}{x^4 \sqrt {1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac {1}{x^3 \sqrt {1+a^2 x^2}} \, dx+\left (4 i a^3\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (4 a^4\right ) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx-\left (4 a^5\right ) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^5 \sqrt {1+a^2 x^2}} \, dx\\ &=\frac {i a \sqrt {1+a^2 x^2}}{x^3}-\frac {4 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+a^2 x}} \, dx,x,x^2\right )-\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )+\left (2 i a^3\right ) \int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx+\left (2 a^4\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}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {2 a^2 \sqrt {1+a^2 x^2}}{x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {1}{8} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+a^2 x}} \, dx,x,x^2\right )+\left (4 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )+a^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {1+a^2 x^2}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-4 a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+\left (2 a^2\right ) \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}{16} \left (3 a^4\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}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-6 a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )+\frac {1}{8} \left (3 a^2\right ) \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}}{4 x^4}+\frac {i a \sqrt {1+a^2 x^2}}{x^3}+\frac {19 a^2 \sqrt {1+a^2 x^2}}{8 x^2}-\frac {6 i a^3 \sqrt {1+a^2 x^2}}{x}+\frac {4 i a^4 \sqrt {1+a^2 x^2}}{i-a x}-\frac {51}{8} a^4 \tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 95, normalized size = 0.68 \[ \frac {1}{8} \left (51 a^4 \log (x)-51 a^4 \log \left (\sqrt {a^2 x^2+1}+1\right )+\frac {\sqrt {a^2 x^2+1} \left (-80 i a^4 x^4-29 a^3 x^3-11 i a^2 x^2+6 a x+2 i\right )}{x^4 (a x-i)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 146, normalized size = 1.05 \[ \frac {-80 i \, a^{5} x^{5} - 80 \, a^{4} x^{4} - 51 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + 51 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + {\left (-80 i \, a^{4} x^{4} - 29 \, a^{3} x^{3} - 11 i \, a^{2} x^{2} + 6 \, a x + 2 i\right )} \sqrt {a^{2} x^{2} + 1}}{8 \, a x^{5} - 8 i \, x^{4}} \]
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 = 0.19, size = 416, normalized size = 2.99 \[ \frac {3 a^{2} \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{\left (x -\frac {i}{a}\right )^{2}}+\frac {51 a^{4} \sqrt {a^{2} x^{2}+1}}{8}-8 a^{4} \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}+\frac {12 i a^{5} \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+12 i a^{5} x \sqrt {a^{2} x^{2}+1}+8 i a^{5} x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}-\frac {8 i a^{3} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x}+\frac {17 a^{4} \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{8}+\frac {i a \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{\left (x -\frac {i}{a}\right )^{3}}-\frac {12 i a^{5} \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}+\frac {i a \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{3}}-\frac {51 a^{4} \arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )}{8}-\frac {\left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{4 x^{4}}-12 i a^{5} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x +\frac {23 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}{8 x^{2}} \]
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 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 139, normalized size = 1.00 \[ \frac {a^4\,\mathrm {atan}\left (\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}\right )\,51{}\mathrm {i}}{8}-\frac {\sqrt {a^2\,x^2+1}}{4\,x^4}+\frac {a\,\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{x^3}+\frac {19\,a^2\,\sqrt {a^2\,x^2+1}}{8\,x^2}-\frac {a^3\,\sqrt {a^2\,x^2+1}\,6{}\mathrm {i}}{x}+\frac {a^5\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{8} - 3 i a^{2} x^{7} - 3 a x^{6} + i x^{5}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{8} - 3 i a^{2} x^{7} - 3 a x^{6} + i x^{5}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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