Optimal. Leaf size=42 \[ \frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{a \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.06, antiderivative size = 42, 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 = {5076, 5073, 31} \[ \frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{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}{1-i a x} \, dx}{\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 = 42, normalized size = 1.00 \[ \frac {i \sqrt {a^2 x^2+1} \log (a x+i)}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 255, normalized size = 6.07 \[ \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, a^{3} x^{3} + 8 i \, a^{2} x^{2} + 8 \, a x + 8 i}\right ) - \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (i \, a^{6} x^{2} - 2 \, a^{5} x - 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, a^{3} x^{3} + 8 i \, a^{2} x^{2} + 8 \, a x + 8 i}\right ) \]
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 \[ \int \frac {i \, a x + 1}{\sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 53, normalized size = 1.26 \[ \frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (i \ln \left (a^{2} x^{2}+1\right )+2 \arctan \left (a x \right )\right )}{2 \sqrt {a^{2} x^{2}+1}\, c a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {c\,a^2\,x^2+c}\,\sqrt {a^2\,x^2+1}} \,d x \]
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 \[ i \left (\int \left (- \frac {i}{\sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a x}{\sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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