Optimal. Leaf size=87 \[ -\frac {\left (\frac {2}{13}-\frac {3 i}{13}\right ) 2^{\frac {3}{2}+i} (1-i a x)^{\frac {3}{2}-i} \sqrt {a^2 c x^2+c} \, _2F_1\left (-\frac {1}{2}-i,\frac {3}{2}-i;\frac {5}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 x^2+1}} \]
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Rubi [A] time = 0.07, 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 = {5076, 5073, 69} \[ -\frac {\left (\frac {2}{13}-\frac {3 i}{13}\right ) 2^{\frac {3}{2}+i} (1-i a x)^{\frac {3}{2}-i} \sqrt {a^2 c x^2+c} \, _2F_1\left (-\frac {1}{2}-i,\frac {3}{2}-i;\frac {5}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int e^{-2 \tan ^{-1}(a x)} \sqrt {c+a^2 c x^2} \, dx &=\frac {\sqrt {c+a^2 c x^2} \int e^{-2 \tan ^{-1}(a x)} \sqrt {1+a^2 x^2} \, dx}{\sqrt {1+a^2 x^2}}\\ &=\frac {\sqrt {c+a^2 c x^2} \int (1-i a x)^{\frac {1}{2}-i} (1+i a x)^{\frac {1}{2}+i} \, dx}{\sqrt {1+a^2 x^2}}\\ &=-\frac {\left (\frac {2}{13}-\frac {3 i}{13}\right ) 2^{\frac {3}{2}+i} (1-i a x)^{\frac {3}{2}-i} \sqrt {c+a^2 c x^2} \, _2F_1\left (-\frac {1}{2}-i,\frac {3}{2}-i;\frac {5}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 87, normalized size = 1.00 \[ -\frac {\left (\frac {2}{13}-\frac {3 i}{13}\right ) 2^{\frac {3}{2}+i} (1-i a x)^{\frac {3}{2}-i} \sqrt {a^2 c x^2+c} \, _2F_1\left (-\frac {1}{2}-i,\frac {3}{2}-i;\frac {5}{2}-i;\frac {1}{2} (1-i a x)\right )}{a \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a^{2} c x^{2} + c} e^{\left (-2 \, \arctan \left (a x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \sqrt {a^{2} c \,x^{2}+c}\, {\mathrm e}^{-2 \arctan \left (a x \right )}\, dx \]
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 \[ \int \sqrt {a^{2} c x^{2} + c} e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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 \[ \int \sqrt {c \left (a^{2} x^{2} + 1\right )} e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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