Optimal. Leaf size=93 \[ -\frac {(1-i) 2^{-\frac {1}{2}+\frac {i}{2}} (1-i a x)^{\frac {1}{2}-\frac {i}{2}} \sqrt {a^2 x^2+1} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},\frac {1}{2}-\frac {i}{2};\frac {3}{2}-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.08, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5076, 5073, 69} \[ -\frac {(1-i) 2^{-\frac {1}{2}+\frac {i}{2}} (1-i a x)^{\frac {1}{2}-\frac {i}{2}} \sqrt {a^2 x^2+1} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},\frac {1}{2}-\frac {i}{2};\frac {3}{2}-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{-\tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{-\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 (1-i a x)^{-\frac {1}{2}-\frac {i}{2}} (1+i a x)^{-\frac {1}{2}+\frac {i}{2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {(1-i) 2^{-\frac {1}{2}+\frac {i}{2}} (1-i a x)^{\frac {1}{2}-\frac {i}{2}} \sqrt {1+a^2 x^2} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},\frac {1}{2}-\frac {i}{2};\frac {3}{2}-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 93, normalized size = 1.00 \[ -\frac {(1-i) 2^{-\frac {1}{2}+\frac {i}{2}} (1-i a x)^{\frac {1}{2}-\frac {i}{2}} \sqrt {a^2 x^2+1} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},\frac {1}{2}-\frac {i}{2};\frac {3}{2}-\frac {i}{2};\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (-\arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{-\arctan \left (a x \right )}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {e^{\left (-\arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {e^{- \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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