Optimal. Leaf size=67 \[ -\frac {i (1+i a x)^{3/2}}{5 a (1-i a x)^{5/2}}-\frac {i (1+i a x)^{3/2}}{15 a (1-i a x)^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5181, 47, 37}
\begin {gather*} -\frac {i (1+i a x)^{3/2}}{15 a (1-i a x)^{3/2}}-\frac {i (1+i a x)^{3/2}}{5 a (1-i a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{4 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac {\sqrt {1+i a x}}{(1-i a x)^{7/2}} \, dx\\ &=-\frac {i (1+i a x)^{3/2}}{5 a (1-i a x)^{5/2}}+\frac {1}{5} \int \frac {\sqrt {1+i a x}}{(1-i a x)^{5/2}} \, dx\\ &=-\frac {i (1+i a x)^{3/2}}{5 a (1-i a x)^{5/2}}-\frac {i (1+i a x)^{3/2}}{15 a (1-i a x)^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 0.70 \begin {gather*} \frac {(1+i a x)^{3/2} (4 i+a x)}{15 a \sqrt {1-i a x} (i+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 268 vs. \(2 (49 ) = 98\).
time = 0.09, size = 269, normalized size = 4.01
method | result | size |
trager | \(\frac {-a^{5} x^{5}-10 a^{3} x^{3}+20 i a^{2} x^{2}+15 a x -4 i}{15 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} a}\) | \(49\) |
meijerg | \(\frac {x \left (8 a^{4} x^{4}+20 a^{2} x^{2}+15\right )}{15 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {16 i \left (\frac {3 \sqrt {\pi }}{4}-\frac {3 \sqrt {\pi }}{4 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{15 a \sqrt {\pi }}-\frac {2 a^{2} x^{3} \left (2 a^{2} x^{2}+5\right )}{5 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {16 i \left (\frac {\sqrt {\pi }}{2}-\frac {\sqrt {\pi }\, \left (20 a^{2} x^{2}+8\right )}{16 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )}{15 a \sqrt {\pi }}+\frac {a^{4} x^{5}}{5 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\) | \(154\) |
default | \(\frac {x}{5 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {4 x}{15 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 x}{15 \sqrt {a^{2} x^{2}+1}}+a^{4} \left (-\frac {x^{3}}{2 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {-\frac {3 x}{8 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {3 \left (\frac {x}{5 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {4 x}{15 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 x}{15 \sqrt {a^{2} x^{2}+1}}\right )}{8 a^{2}}}{a^{2}}\right )-4 i a^{3} \left (-\frac {x^{2}}{3 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {2}{15 a^{4} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\right )-6 a^{2} \left (-\frac {x}{4 a^{2} \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {\frac {x}{5 \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}+\frac {4 x}{15 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 x}{15 \sqrt {a^{2} x^{2}+1}}}{4 a^{2}}\right )-\frac {4 i}{5 a \left (a^{2} x^{2}+1\right )^{\frac {5}{2}}}\) | \(269\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 95 vs. \(2 (43) = 86\).
time = 0.25, size = 95, normalized size = 1.42 \begin {gather*} -\frac {a^{2} x^{3}}{2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} - \frac {x}{15 \, \sqrt {a^{2} x^{2} + 1}} + \frac {4 i \, a x^{2}}{3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} - \frac {x}{30 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {11 \, x}{10 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} - \frac {4 i}{15 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.28, size = 75, normalized size = 1.12 \begin {gather*} -\frac {a^{3} x^{3} + 3 i \, a^{2} x^{2} - 3 \, a x + {\left (a^{2} x^{2} + 3 i \, a x + 4\right )} \sqrt {a^{2} x^{2} + 1} - i}{15 \, {\left (a^{4} x^{3} + 3 i \, a^{3} x^{2} - 3 \, a^{2} x - i \, a\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\left (a x - i\right )^{4}}{\left (a^{2} x^{2} + 1\right )^{\frac {7}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 111 vs. \(2 (43) = 86\).
time = 0.42, size = 111, normalized size = 1.66 \begin {gather*} \frac {2 \, {\left (4 \, a^{4} - 25 \, a^{2} {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{2} + 15 i \, a {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{3} + 15 \, {\left (\sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{4} - 5 \, a^{3} {\left (i \, \sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {i}{x}\right )}\right )}}{15 \, {\left (i \, a + \sqrt {a^{2} + \frac {1}{x^{2}}} - \frac {1}{x}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 41, normalized size = 0.61 \begin {gather*} \frac {\sqrt {a^2\,x^2+1}\,\left (a^2\,x^2\,1{}\mathrm {i}-3\,a\,x+4{}\mathrm {i}\right )}{15\,a\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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