Optimal. Leaf size=83 \[ \frac{12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac{(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3} \]
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Rubi [A] time = 0.0777701, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac{12 (2 a x+1) e^{\tan ^{-1}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac{(4 a x+1) e^{\tan ^{-1}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3} \]
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 )^3} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{17 c}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 e^{\tan ^{-1}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}+\frac{24 \int \frac{e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{85 c^2}\\ &=\frac{24 e^{\tan ^{-1}(a x)}}{85 a c^3}+\frac{e^{\tan ^{-1}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac{12 e^{\tan ^{-1}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )}\\ \end{align*}
Mathematica [C] time = 0.248842, size = 114, normalized size = 1.37 \[ \frac{\frac{5 (4 a x+1) e^{\tan ^{-1}(a x)}}{\left (a^2 x^2+1\right )^2}+\frac{24 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x+(1-i))}{a x-i}+(12-24 i) (1-i a x)^{-1+\frac{i}{2}} (1+i a x)^{-1-\frac{i}{2}}}{85 a c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.036, size = 55, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{\arctan \left ( ax \right ) }} \left ( 24\,{a}^{4}{x}^{4}+24\,{a}^{3}{x}^{3}+60\,{a}^{2}{x}^{2}+44\,ax+41 \right ) }{85\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}a{c}^{3}}} \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02264, size = 153, normalized size = 1.84 \begin{align*} \frac{{\left (24 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} + 44 \, a x + 41\right )} e^{\left (\arctan \left (a x\right )\right )}}{85 \,{\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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