Optimal. Leaf size=87 \[ -\frac {\left (\frac {2}{5}-\frac {i}{5}\right ) 2^{\frac {1}{2}+i} (1-i a x)^{\frac {1}{2}-i} \sqrt {1+a^2 x^2} \, _2F_1\left (\frac {1}{2}-i,\frac {1}{2}-i;\frac {3}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 87, 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 = {5184, 5181, 71}
\begin {gather*} -\frac {\left (\frac {2}{5}-\frac {i}{5}\right ) 2^{\frac {1}{2}+i} (1-i a x)^{\frac {1}{2}-i} \sqrt {a^2 x^2+1} \, _2F_1\left (\frac {1}{2}-i,\frac {1}{2}-i;\frac {3}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 5181
Rule 5184
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{-2 \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}-i} (1+i a x)^{-\frac {1}{2}+i} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {\left (\frac {2}{5}-\frac {i}{5}\right ) 2^{\frac {1}{2}+i} (1-i a x)^{\frac {1}{2}-i} \sqrt {1+a^2 x^2} \, _2F_1\left (\frac {1}{2}-i,\frac {1}{2}-i;\frac {3}{2}-i;\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 = 0.02, size = 87, normalized size = 1.00 \begin {gather*} -\frac {\left (\frac {2}{5}-\frac {i}{5}\right ) 2^{\frac {1}{2}+i} (1-i a x)^{\frac {1}{2}-i} \sqrt {1+a^2 x^2} \, _2F_1\left (\frac {1}{2}-i,\frac {1}{2}-i;\frac {3}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{-2 \arctan \left (a x \right )}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{- 2 \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \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 {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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