Optimal. Leaf size=88 \[ \frac {3 (a x+1) e^{2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}+\frac {(2 a x+1) e^{2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 e^{2 \tan ^{-1}(a x)}}{40 a c^3} \]
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Rubi [A] time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ \frac {3 (a x+1) e^{2 \tan ^{-1}(a x)}}{20 a c^3 \left (a^2 x^2+1\right )}+\frac {(2 a x+1) e^{2 \tan ^{-1}(a x)}}{10 a c^3 \left (a^2 x^2+1\right )^2}+\frac {3 e^{2 \tan ^{-1}(a x)}}{40 a c^3} \]
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 )^3} \, dx &=\frac {e^{2 \tan ^{-1}(a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{5 c}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 e^{2 \tan ^{-1}(a x)} (1+a x)}{20 a c^3 \left (1+a^2 x^2\right )}+\frac {3 \int \frac {e^{2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{20 c^2}\\ &=\frac {3 e^{2 \tan ^{-1}(a x)}}{40 a c^3}+\frac {e^{2 \tan ^{-1}(a x)} (1+2 a x)}{10 a c^3 \left (1+a^2 x^2\right )^2}+\frac {3 e^{2 \tan ^{-1}(a x)} (1+a x)}{20 a c^3 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 86, normalized size = 0.98 \[ \frac {4 (2 a x+1) e^{2 \tan ^{-1}(a x)}+3 (1-i a x)^i (1+i a x)^{-i} \left (a^2 x^2+1\right ) \left (a^2 x^2+2 a x+3\right )}{40 a c^3 \left (a^2 x^2+1\right )^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 68, normalized size = 0.77 \[ \frac {{\left (3 \, a^{4} x^{4} + 6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 14 \, a x + 13\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{40 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 57, normalized size = 0.65 \[ \frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (3 a^{4} x^{4}+6 a^{3} x^{3}+12 a^{2} x^{2}+14 a x +13\right )}{40 \left (a^{2} x^{2}+1\right )^{2} a \,c^{3}} \]
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 \[ \int \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 79, normalized size = 0.90 \[ \frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}}{40\,a\,c^3}+\frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x+1\right )}{20\,a\,c^3\,\left (a^2\,x^2+1\right )}+\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{10\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
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 \[ \begin {cases} \frac {3 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {6 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {12 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {14 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} + \frac {13 e^{2 \operatorname {atan}{\left (a x \right )}}}{40 a^{5} c^{3} x^{4} + 80 a^{3} c^{3} x^{2} + 40 a c^{3}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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