Optimal. Leaf size=54 \[ \frac {2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5182, 667, 197}
\begin {gather*} \frac {x}{3 c \sqrt {a^2 c x^2+c}}+\frac {2 i (1-i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 667
Rule 5182
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1-i a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {1}{3} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {x}{3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 78, normalized size = 1.44 \begin {gather*} \frac {\sqrt {1-i a x} (2+i a x) \sqrt {1+a^2 x^2}}{3 a c \sqrt {1+i a x} (-i+a x) \sqrt {c+a^2 c x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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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. 136 vs. \(2 (44 ) = 88\).
time = 0.07, size = 137, normalized size = 2.54
method | result | size |
default | \(-\frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}-\frac {2 i \left (\frac {i}{3 a c \left (x -\frac {i}{a}\right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}+\frac {i \left (2 a^{2} c \left (x -\frac {i}{a}\right )+2 i a c \right )}{3 a \,c^{2} \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}\right )}{a}\) | \(137\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 59, normalized size = 1.09 \begin {gather*} \frac {x}{3 \, \sqrt {a^{2} c x^{2} + c} c} + \frac {2 i}{3 i \, \sqrt {a^{2} c x^{2} + c} a^{2} c x + 3 \, \sqrt {a^{2} c x^{2} + c} a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.79, size = 47, normalized size = 0.87 \begin {gather*} \frac {\sqrt {a^{2} c x^{2} + c} {\left (a x - 2 i\right )}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 i \, a^{2} c^{2} x - a c^{2}\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 {a^{2} x^{2}}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} - 2 i a^{3} c x^{3} \sqrt {a^{2} c x^{2} + c} - 2 i a c x \sqrt {a^{2} c x^{2} + c} - c \sqrt {a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a^{4} c x^{4} \sqrt {a^{2} c x^{2} + c} - 2 i a^{3} c x^{3} \sqrt {a^{2} c x^{2} + c} - 2 i a c x \sqrt {a^{2} c x^{2} + c} - c \sqrt {a^{2} c x^{2} + c}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 76, normalized size = 1.41 \begin {gather*} -\frac {2 \, \sqrt {a^{2} c} {\left (3 \, \sqrt {a^{2} c} x - 3 \, \sqrt {a^{2} c x^{2} + c} - i \, \sqrt {c}\right )}}{3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c} - i \, \sqrt {c}\right )}^{3} a^{2} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 32, normalized size = 0.59 \begin {gather*} \frac {a^3\,x^3+3\,a\,x+2{}\mathrm {i}}{3\,a\,{\left (c\,\left (a^2\,x^2+1\right )\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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