Optimal. Leaf size=51 \[ -\frac {\left (\frac {1}{5}-\frac {2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac {1}{2} (1-i a x)\right )}{a} \]
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Rubi [A] time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {5073, 69} \[ -\frac {\left (\frac {1}{5}-\frac {2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac {1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rubi steps
\begin {align*} \int e^{-2 \tan ^{-1}(a x)} \left (c+a^2 c x^2\right ) \, dx &=c \int (1-i a x)^{1-i} (1+i a x)^{1+i} \, dx\\ &=-\frac {\left (\frac {1}{5}-\frac {2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac {1}{2} (1-i a x)\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.00 \[ -\frac {\left (\frac {1}{5}-\frac {2 i}{5}\right ) 2^{1+i} c (1-i a x)^{2-i} \, _2F_1\left (-1-i,2-i;3-i;\frac {1}{2} (1-i a x)\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] 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 = 0.07, size = 0, normalized size = 0.00 \[ \int \left (a^{2} c \,x^{2}+c \right ) {\mathrm e}^{-2 \arctan \left (a x \right )}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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 \[ c \left (\int a^{2} x^{2} e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx + \int e^{- 2 \operatorname {atan}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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