Optimal. Leaf size=63 \[ -\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a \sqrt {c}}+\frac {2 i (1-i a x)}{a \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {5074, 653, 217, 206} \[ -\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a \sqrt {c}}+\frac {2 i (1-i a x)}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 653
Rule 5074
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=c \int \frac {(1-i a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 i (1-i a x)}{a \sqrt {c+a^2 c x^2}}-\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {2 i (1-i a x)}{a \sqrt {c+a^2 c x^2}}-\operatorname {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )\\ &=\frac {2 i (1-i a x)}{a \sqrt {c+a^2 c x^2}}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 117, normalized size = 1.86 \[ \frac {2 \sqrt {a^2 x^2+1} \left ((1-i a x) \sqrt {1+i a x}-i \sqrt {1-i a x} (a x-i) \sin ^{-1}\left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-i a x} (a x-i) \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.55, size = 152, normalized size = 2.41 \[ -\frac {{\left (a^{2} c x - i \, a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x + \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - {\left (a^{2} c x - i \, a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x - \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - 4 \, \sqrt {a^{2} c x^{2} + c}}{2 \, {\left (a^{2} c x - i \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 70, normalized size = 1.11 \[ \frac {\log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {c}} + \frac {4}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} i + \sqrt {c}\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 87, normalized size = 1.38 \[ -\frac {\ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+c}\right )}{\sqrt {a^{2} c}}+\frac {2 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2} c +2 i a c \left (x -\frac {i}{a}\right )}}{a^{2} c \left (x -\frac {i}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 40, normalized size = 0.63 \[ \frac {2 i \, \sqrt {a^{2} c x^{2} + c}}{i \, a^{2} c x + a c} - \frac {\operatorname {arsinh}\left (a x\right )}{a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a^2\,x^2+1}{\sqrt {c\,a^2\,x^2+c}\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \int \frac {a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 2 i a x \sqrt {a^{2} c x^{2} + c} - \sqrt {a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 2 i a x \sqrt {a^{2} c x^{2} + c} - \sqrt {a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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