Optimal. Leaf size=116 \[ \frac{72 (2 a x+1) e^{\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )}+\frac{30 (4 a x+1) e^{\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}+\frac{(6 a x+1) e^{\tan ^{-1}(a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}+\frac{144 e^{\tan ^{-1}(a x)}}{629 a c^4} \]
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Rubi [A] time = 0.109743, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac{72 (2 a x+1) e^{\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )}+\frac{30 (4 a x+1) e^{\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}+\frac{(6 a x+1) e^{\tan ^{-1}(a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}+\frac{144 e^{\tan ^{-1}(a x)}}{629 a c^4} \]
Antiderivative was successfully verified.
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Rule 5070
Rule 5071
Rubi steps
\begin{align*} \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac{30 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{37 c}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac{30 e^{\tan ^{-1}(a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac{360 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{629 c^2}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac{30 e^{\tan ^{-1}(a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac{72 e^{\tan ^{-1}(a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )}+\frac{144 \int \frac{e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{629 c^3}\\ &=\frac{144 e^{\tan ^{-1}(a x)}}{629 a c^4}+\frac{e^{\tan ^{-1}(a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac{30 e^{\tan ^{-1}(a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac{72 e^{\tan ^{-1}(a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )}\\ \end{align*}
Mathematica [C] time = 0.263634, size = 123, normalized size = 1.06 \[ \frac{17 c (6 a x+1) e^{\tan ^{-1}(a x)}+6 \left (a^2 c x^2+c\right ) \left (5 (4 a x+1) e^{\tan ^{-1}(a x)}+12 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x-i) (a x+i) \left (2 a^2 x^2+2 a x+3\right )\right )}{629 a c^2 \left (a^2 c x^2+c\right )^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.038, size = 71, normalized size = 0.6 \begin{align*}{\frac{{{\rm e}^{\arctan \left ( ax \right ) }} \left ( 144\,{a}^{6}{x}^{6}+144\,{a}^{5}{x}^{5}+504\,{a}^{4}{x}^{4}+408\,{a}^{3}{x}^{3}+606\,{a}^{2}{x}^{2}+366\,ax+263 \right ) }{629\, \left ({a}^{2}{x}^{2}+1 \right ) ^{3}a{c}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96869, size = 220, normalized size = 1.9 \begin{align*} \frac{{\left (144 \, a^{6} x^{6} + 144 \, a^{5} x^{5} + 504 \, a^{4} x^{4} + 408 \, a^{3} x^{3} + 606 \, a^{2} x^{2} + 366 \, a x + 263\right )} e^{\left (\arctan \left (a x\right )\right )}}{629 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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