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 = 0.02, 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 = {5181, 71}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 5181
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 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{2 \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )\, dx\]
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 \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*} 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 ) \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 = -0.02 \begin {gather*} \int {\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (c\,a^2\,x^2+c\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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