Optimal. Leaf size=86 \[ -\frac {2 \sqrt {1+a^2 x^2}}{a (i-a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{a \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 45}
\begin {gather*} \frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{a (-a x+i) \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 5181
Rule 5184
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {1-i a x}{(1+i a x)^2} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \left (-\frac {2}{(-i+a x)^2}+\frac {i}{-i+a x}\right ) \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{a (i-a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 60, normalized size = 0.70 \begin {gather*} \frac {\sqrt {1+a^2 x^2} \left (-\frac {2}{a (i-a x)}+\frac {i \log (i-a x)}{a}\right )}{\sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 66, normalized size = 0.77
method | result | size |
default | \(\frac {\left (-i \ln \left (-a x +i\right ) a x -\ln \left (-a x +i\right )-2\right ) \sqrt {c \left (a^{2} x^{2}+1\right )}}{\sqrt {a^{2} x^{2}+1}\, c \left (-a x +i\right ) a}\) | \(66\) |
risch | \(\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a \left (a x -i\right )}+\frac {i \sqrt {a^{2} x^{2}+1}\, \ln \left (a x -i\right )}{\sqrt {c \left (a^{2} x^{2}+1\right )}\, a}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 35, normalized size = 0.41 \begin {gather*} \frac {i \, \log \left (i \, a x + 1\right )}{a \sqrt {c}} + \frac {2}{a^{2} \sqrt {c} x - i \, a \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 357 vs. \(2 (71) = 142\).
time = 4.54, size = 357, normalized size = 4.15 \begin {gather*} \frac {{\left (-i \, a^{3} c x^{3} - a^{2} c x^{2} - i \, a c x - c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2} + a x - i\right )}}\right ) + {\left (i \, a^{3} c x^{3} + a^{2} c x^{2} + i \, a c x + c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, {\left (a^{3} x^{3} - i \, a^{2} x^{2} + a x - i\right )}}\right ) - 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, {\left (a^{3} c x^{3} - i \, a^{2} c x^{2} + a c x - i \, c\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*} i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} \sqrt {a^{2} c x^{2} + c} - 3 i a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 3 a x \sqrt {a^{2} c x^{2} + c} + i \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} \sqrt {a^{2} c x^{2} + c} - 3 i a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 3 a x \sqrt {a^{2} c x^{2} + c} + i \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a^2\,x^2+1\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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