Optimal. Leaf size=83 \[ \frac {24 e^{\text {ArcTan}(a x)}}{85 a c^3}+\frac {e^{\text {ArcTan}(a x)} (1+4 a x)}{17 a c^3 \left (1+a^2 x^2\right )^2}+\frac {12 e^{\text {ArcTan}(a x)} (1+2 a x)}{85 a c^3 \left (1+a^2 x^2\right )} \]
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Rubi [A]
time = 0.06, 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 = {5178, 5179}
\begin {gather*} \frac {12 (2 a x+1) e^{\text {ArcTan}(a x)}}{85 a c^3 \left (a^2 x^2+1\right )}+\frac {(4 a x+1) e^{\text {ArcTan}(a x)}}{17 a c^3 \left (a^2 x^2+1\right )^2}+\frac {24 e^{\text {ArcTan}(a x)}}{85 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 5178
Rule 5179
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] Result contains complex when optimal does not.
time = 0.21, size = 114, normalized size = 1.37 \begin {gather*} \frac {(12-24 i) (1-i a x)^{-1+\frac {i}{2}} (1+i a x)^{-1-\frac {i}{2}}+\frac {24 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} ((1-i)+a x)}{-i+a x}+\frac {5 e^{\text {ArcTan}(a x)} (1+4 a x)}{\left (1+a^2 x^2\right )^2}}{85 a c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 55, normalized size = 0.66
method | result | size |
gosper | \(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (24 a^{4} x^{4}+24 a^{3} x^{3}+60 a^{2} x^{2}+44 a x +41\right )}{85 \left (a^{2} x^{2}+1\right )^{2} c^{3} a}\) | \(55\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.72, size = 66, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 223 vs.
\(2 (75) = 150\).
time = 4.12, size = 223, normalized size = 2.69 \begin {gather*} \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}\: a \neq 0 \\\frac {x}{c^{3}} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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