Optimal. Leaf size=116 \[ \frac {144 e^{\text {ArcTan}(a x)}}{629 a c^4}+\frac {e^{\text {ArcTan}(a x)} (1+6 a x)}{37 a c^4 \left (1+a^2 x^2\right )^3}+\frac {30 e^{\text {ArcTan}(a x)} (1+4 a x)}{629 a c^4 \left (1+a^2 x^2\right )^2}+\frac {72 e^{\text {ArcTan}(a x)} (1+2 a x)}{629 a c^4 \left (1+a^2 x^2\right )} \]
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Rubi [A]
time = 0.08, antiderivative size = 116, normalized size of antiderivative = 1.00, 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 = {5178, 5179}
\begin {gather*} \frac {72 (2 a x+1) e^{\text {ArcTan}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )}+\frac {30 (4 a x+1) e^{\text {ArcTan}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}+\frac {(6 a x+1) e^{\text {ArcTan}(a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}+\frac {144 e^{\text {ArcTan}(a x)}}{629 a c^4} \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 )^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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.20, size = 123, normalized size = 1.06 \begin {gather*} \frac {17 c e^{\text {ArcTan}(a x)} (1+6 a x)+6 \left (c+a^2 c x^2\right ) \left (5 e^{\text {ArcTan}(a x)} (1+4 a x)+12 (1-i a x)^{\frac {i}{2}} (1+i a x)^{-\frac {i}{2}} (-i+a x) (i+a x) \left (3+2 a x+2 a^2 x^2\right )\right )}{629 a c^2 \left (c+a^2 c x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 71, normalized size = 0.61
method | result | size |
gosper | \(\frac {{\mathrm e}^{\arctan \left (a x \right )} \left (144 a^{6} x^{6}+144 a^{5} x^{5}+504 a^{4} x^{4}+408 a^{3} x^{3}+606 a^{2} x^{2}+366 a x +263\right )}{629 \left (a^{2} x^{2}+1\right )^{3} c^{4} a}\) | \(71\) |
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 = 2.61, size = 93, normalized size = 0.80 \begin {gather*} \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 {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. 398 vs.
\(2 (107) = 214\).
time = 10.80, size = 398, normalized size = 3.43 \begin {gather*} \begin {cases} \frac {144 a^{6} x^{6} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {144 a^{5} x^{5} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {504 a^{4} x^{4} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {408 a^{3} x^{3} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {606 a^{2} x^{2} e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {366 a x e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} + \frac {263 e^{\operatorname {atan}{\left (a x \right )}}}{629 a^{7} c^{4} x^{6} + 1887 a^{5} c^{4} x^{4} + 1887 a^{3} c^{4} x^{2} + 629 a c^{4}} & \text {for}\: a \neq 0 \\\frac {x}{c^{4}} & \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 = 0.69, size = 104, normalized size = 0.90 \begin {gather*} \frac {144\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}}{629\,a\,c^4}+\frac {72\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x+1\right )}{629\,a\,c^4\,\left (a^2\,x^2+1\right )}+\frac {30\,{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x+1\right )}{629\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{\mathrm {atan}\left (a\,x\right )}\,\left (6\,a\,x+1\right )}{37\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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