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 = 0.08, antiderivative size = 83, normalized size of antiderivative = 1.00, 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*}
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Mathematica [C] time = 0.29, 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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fricas [A] time = 0.54, size = 66, normalized size = 0.80 \[ \frac {{\left (24 \, a^{4} x^{4} + 24 \, a^{3} x^{3} + 60 \, a^{2} x^{2} + 44 \, a x + 41\right )} e^{\left (\arctan \left (a x\right )\right )}}{85 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 55, normalized size = 0.66 \[ \frac {{\mathrm e}^{\arctan \left (a x \right )} \left (24 a^{4} x^{4}+24 a^{3} x^{3}+60 a^{2} x^{2}+44 a x +41\right )}{85 \left (a^{2} x^{2}+1\right )^{2} a \,c^{3}} \]
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 \[ \int \frac {e^{\left (\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.62, size = 74, normalized size = 0.89 \[ \frac {24\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{85\,a\,c^3}+\frac {12\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{85\,a\,c^3\,\left (a^2\,x^2+1\right )}+\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x+1\right )}{17\,a\,c^3\,{\left (a^2\,x^2+1\right )}^2} \]
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 \[ \begin {cases} \frac {24 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {24 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {60 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {44 a x e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} + \frac {41 e^{\operatorname {atan}{\left (a x \right )}}}{85 a^{5} c^{3} x^{4} + 170 a^{3} c^{3} x^{2} + 85 a c^{3}} & \text {for}\: c \neq 0 \\\tilde {\infty } \int e^{\operatorname {atan}{\left (a x \right )}}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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