Optimal. Leaf size=149 \[ \frac{4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac{336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac{56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac{(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5} \]
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Rubi [A] time = 0.140465, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5070, 5071} \[ \frac{4032 (2 a x+1) e^{\tan ^{-1}(a x)}}{40885 a c^5 \left (a^2 x^2+1\right )}+\frac{336 (4 a x+1) e^{\tan ^{-1}(a x)}}{8177 a c^5 \left (a^2 x^2+1\right )^2}+\frac{56 (6 a x+1) e^{\tan ^{-1}(a x)}}{2405 a c^5 \left (a^2 x^2+1\right )^3}+\frac{(8 a x+1) e^{\tan ^{-1}(a x)}}{65 a c^5 \left (a^2 x^2+1\right )^4}+\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5} \]
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 )^5} \, dx &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx}{65 c}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{481 c^2}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac{4032 \int \frac{e^{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{8177 c^3}\\ &=\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac{4032 e^{\tan ^{-1}(a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}+\frac{8064 \int \frac{e^{\tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{40885 c^4}\\ &=\frac{8064 e^{\tan ^{-1}(a x)}}{40885 a c^5}+\frac{e^{\tan ^{-1}(a x)} (1+8 a x)}{65 a c^5 \left (1+a^2 x^2\right )^4}+\frac{56 e^{\tan ^{-1}(a x)} (1+6 a x)}{2405 a c^5 \left (1+a^2 x^2\right )^3}+\frac{336 e^{\tan ^{-1}(a x)} (1+4 a x)}{8177 a c^5 \left (1+a^2 x^2\right )^2}+\frac{4032 e^{\tan ^{-1}(a x)} (1+2 a x)}{40885 a c^5 \left (1+a^2 x^2\right )}\\ \end{align*}
Mathematica [C] time = 0.257846, size = 153, normalized size = 1.03 \[ \frac{629 (8 a x+1) e^{\tan ^{-1}(a x)}+\frac{56 \left (a^2 c x^2+c\right ) \left (17 c (6 a x+1) e^{\tan ^{-1}(a x)}+6 \left (a^2 c x^2+c\right ) \left (5 (4 a x+1) e^{\tan ^{-1}(a x)}+12 (1-i a x)^{\frac{i}{2}} (1+i a x)^{-\frac{i}{2}} (a x-i) (a x+i) \left (2 a^2 x^2+2 a x+3\right )\right )\right )}{c^2}}{40885 a c \left (a^2 c x^2+c\right )^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.037, size = 87, normalized size = 0.6 \begin{align*}{\frac{{{\rm e}^{\arctan \left ( ax \right ) }} \left ( 8064\,{a}^{8}{x}^{8}+8064\,{a}^{7}{x}^{7}+36288\,{a}^{6}{x}^{6}+30912\,{a}^{5}{x}^{5}+62160\,{a}^{4}{x}^{4}+43344\,{a}^{3}{x}^{3}+48664\,{a}^{2}{x}^{2}+25528\,ax+15357 \right ) }{40885\, \left ({a}^{2}{x}^{2}+1 \right ) ^{4}{c}^{5}a}} \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 )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97205, size = 304, normalized size = 2.04 \begin{align*} \frac{{\left (8064 \, a^{8} x^{8} + 8064 \, a^{7} x^{7} + 36288 \, a^{6} x^{6} + 30912 \, a^{5} x^{5} + 62160 \, a^{4} x^{4} + 43344 \, a^{3} x^{3} + 48664 \, a^{2} x^{2} + 25528 \, a x + 15357\right )} e^{\left (\arctan \left (a x\right )\right )}}{40885 \,{\left (a^{9} c^{5} x^{8} + 4 \, a^{7} c^{5} x^{6} + 6 \, a^{5} c^{5} x^{4} + 4 \, a^{3} c^{5} x^{2} + a c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )}^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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