Optimal. Leaf size=123 \[ \frac {9 (a x+1) e^{2 \tan ^{-1}(a x)}}{80 a c^4 \left (a^2 x^2+1\right )}+\frac {3 (2 a x+1) e^{2 \tan ^{-1}(a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}+\frac {(3 a x+1) e^{2 \tan ^{-1}(a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}+\frac {9 e^{2 \tan ^{-1}(a x)}}{160 a c^4} \]
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Rubi [A] time = 0.12, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ \frac {9 (a x+1) e^{2 \tan ^{-1}(a x)}}{80 a c^4 \left (a^2 x^2+1\right )}+\frac {3 (2 a x+1) e^{2 \tan ^{-1}(a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}+\frac {(3 a x+1) e^{2 \tan ^{-1}(a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}+\frac {9 e^{2 \tan ^{-1}(a x)}}{160 a c^4} \]
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 )^4} \, dx &=\frac {e^{2 \tan ^{-1}(a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{4 c}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \tan ^{-1}(a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 \int \frac {e^{2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{20 c^2}\\ &=\frac {e^{2 \tan ^{-1}(a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \tan ^{-1}(a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 e^{2 \tan ^{-1}(a x)} (1+a x)}{80 a c^4 \left (1+a^2 x^2\right )}+\frac {9 \int \frac {e^{2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{80 c^3}\\ &=\frac {9 e^{2 \tan ^{-1}(a x)}}{160 a c^4}+\frac {e^{2 \tan ^{-1}(a x)} (1+3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 e^{2 \tan ^{-1}(a x)} (1+2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 e^{2 \tan ^{-1}(a x)} (1+a x)}{80 a c^4 \left (1+a^2 x^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.27, size = 122, normalized size = 0.99 \[ \frac {8 c (3 a x+1) e^{2 \tan ^{-1}(a x)}+3 \left (a^2 c x^2+c\right ) \left (4 (2 a x+1) e^{2 \tan ^{-1}(a x)}+3 (1-i a x)^i (1+i a x)^{-i} (a x-i) (a x+i) \left (a^2 x^2+2 a x+3\right )\right )}{160 a c^2 \left (a^2 c x^2+c\right )^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.48, size = 95, normalized size = 0.77 \[ \frac {{\left (9 \, a^{6} x^{6} + 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} + 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} + 66 \, a x + 47\right )} e^{\left (2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]
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 = 73, normalized size = 0.59 \[ \frac {{\mathrm e}^{2 \arctan \left (a x \right )} \left (9 a^{6} x^{6}+18 a^{5} x^{5}+45 a^{4} x^{4}+60 a^{3} x^{3}+75 a^{2} x^{2}+66 a x +47\right )}{160 \left (a^{2} x^{2}+1\right )^{3} a \,c^{4}} \]
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 \[ \int \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 111, normalized size = 0.90 \[ \frac {9\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {9\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x+1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}+\frac {3\,{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x+1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]
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 \[ \begin {cases} \frac {9 a^{6} x^{6} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {18 a^{5} x^{5} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {45 a^{4} x^{4} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {60 a^{3} x^{3} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {75 a^{2} x^{2} e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {66 a x e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} + \frac {47 e^{2 \operatorname {atan}{\left (a x \right )}}}{160 a^{7} c^{4} x^{6} + 480 a^{5} c^{4} x^{4} + 480 a^{3} c^{4} x^{2} + 160 a c^{4}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{2 \operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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