Optimal. Leaf size=124 \[ -\frac {(1-6 a x) e^{-\tan ^{-1}(a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}-\frac {72 (1-2 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )}-\frac {30 (1-4 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}-\frac {144 e^{-\tan ^{-1}(a x)}}{629 a c^4} \]
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Rubi [A] time = 0.12, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5070, 5071} \[ -\frac {(1-6 a x) e^{-\tan ^{-1}(a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}-\frac {72 (1-2 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )}-\frac {30 (1-4 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}-\frac {144 e^{-\tan ^{-1}(a x)}}{629 a c^4} \]
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 )^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] time = 0.39, size = 127, normalized size = 1.02 \[ \frac {17 c (6 a x-1) e^{-\tan ^{-1}(a x)}-6 \left (a^2 c x^2+c\right ) \left (5 (1-4 a x) e^{-\tan ^{-1}(a x)}+12 (1-i a x)^{-\frac {i}{2}} (1+i a x)^{\frac {i}{2}} (a x-i) (a x+i) \left (2 a^2 x^2-2 a x+3\right )\right )}{629 a c^2 \left (a^2 c x^2+c\right )^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 95, normalized size = 0.77 \[ -\frac {{\left (144 \, a^{6} x^{6} - 144 \, a^{5} x^{5} + 504 \, a^{4} x^{4} - 408 \, a^{3} x^{3} + 606 \, a^{2} x^{2} - 366 \, a x + 263\right )} e^{\left (-\arctan \left (a x\right )\right )}}{629 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]
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 = 73, normalized size = 0.59 \[ -\frac {\left (144 a^{6} x^{6}-144 a^{5} x^{5}+504 a^{4} x^{4}-408 a^{3} x^{3}+606 a^{2} x^{2}-366 a x +263\right ) {\mathrm e}^{-\arctan \left (a x \right )}}{629 \left (a^{2} x^{2}+1\right )^{3} c^{4} a} \]
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 \[ \int \frac {e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 112, normalized size = 0.90 \[ \frac {72\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{629\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {144\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}}{629\,a\,c^4}+\frac {30\,{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (4\,a\,x-1\right )}{629\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-\mathrm {atan}\left (a\,x\right )}\,\left (6\,a\,x-1\right )}{37\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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