Optimal. Leaf size=53 \[ \frac {(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}+\frac {e^{2 \tan ^{-1}(a x)}}{8 a c^2} \]
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Rubi [A] time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ \frac {(a x+1) e^{2 \tan ^{-1}(a x)}}{4 a c^2 \left (a^2 x^2+1\right )}+\frac {e^{2 \tan ^{-1}(a x)}}{8 a c^2} \]
Antiderivative was successfully verified.
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Rule 5070
Rule 5071
Rubi steps
\begin {align*} \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx &=\frac {e^{2 \tan ^{-1}(a x)} (1+a x)}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {\int \frac {e^{2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{4 c}\\ &=\frac {e^{2 \tan ^{-1}(a x)}}{8 a c^2}+\frac {e^{2 \tan ^{-1}(a x)} (1+a x)}{4 a c^2 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 55, normalized size = 1.04 \[ \frac {(1-i a x)^i (1+i a x)^{-i} \left (a^2 x^2+2 a x+3\right )}{8 c^2 \left (a^3 x^2+a\right )} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 40, normalized size = 0.75 \[ \frac {{\left (a^{2} x^{2} + 2 \, a x + 3\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{8 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \]
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 [A] time = 0.04, size = 40, normalized size = 0.75 \[ \frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (a^{2} x^{2}+2 a x +3\right )}{8 \left (a^{2} x^{2}+1\right ) a \,c^{2}} \]
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 \[ \int \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 46, normalized size = 0.87 \[ \frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {3}{8\,a^3\,c^2}+\frac {x}{4\,a^2\,c^2}+\frac {x^2}{8\,a\,c^2}\right )}{\frac {1}{a^2}+x^2} \]
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 \[ \begin {cases} \frac {a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{8 a^{3} c^{2} x^{2} + 8 a c^{2}} + \frac {2 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{8 a^{3} c^{2} x^{2} + 8 a c^{2}} + \frac {3 e^{2 \operatorname {atan}{\left (a x \right )}}}{8 a^{3} c^{2} x^{2} + 8 a c^{2}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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