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 = {5075, 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 5075
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 = 0.04, size = 91, normalized size = 1.44 \[ -\frac {2 i \sqrt {a^2 x^2+1} \left (\sqrt {1+i a x}+\sqrt {1-i a x} \sin ^{-1}\left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{a \sqrt {1-i a x} \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.43, 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 = 0.17, size = 72, normalized size = 1.14 \[ \frac {\log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {c}} - \frac {4}{{\left ({\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} i - \sqrt {c}\right )} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 204, normalized size = 3.24 \[ -\frac {\ln \left (\frac {x \,a^{2} c}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+c}\right )}{\sqrt {a^{2} c}}+\frac {\left (i \sqrt {-a^{2}}+a \right ) \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{a^{3} c \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}-\frac {\left (i \sqrt {-a^{2}}-a \right ) \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{a^{3} c \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )} \]
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 \[ \int \frac {{\left (i \, a x + 1\right )}^{2}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
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 \[ \int \frac {{\left (1+a\,x\,1{}\mathrm {i}\right )}^2}{\sqrt {c\,a^2\,x^2+c}\,\left (a^2\,x^2+1\right )} \,d x \]
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 \[ - \int \frac {a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {2 i a x}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} c x^{2} + c}}\right )\, dx - \int \left (- \frac {1}{a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} + \sqrt {a^{2} c x^{2} + c}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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