Optimal. Leaf size=124 \[ -\frac {(1-3 a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}-\frac {9 (1-a x) e^{-2 \tan ^{-1}(a x)}}{80 a c^4 \left (a^2 x^2+1\right )}-\frac {3 (1-2 a x) e^{-2 \tan ^{-1}(a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}-\frac {9 e^{-2 \tan ^{-1}(a x)}}{160 a c^4} \]
[Out]
________________________________________________________________________________________
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} \[ -\frac {(1-3 a x) e^{-2 \tan ^{-1}(a x)}}{20 a c^4 \left (a^2 x^2+1\right )^3}-\frac {9 (1-a x) e^{-2 \tan ^{-1}(a x)}}{80 a c^4 \left (a^2 x^2+1\right )}-\frac {3 (1-2 a x) e^{-2 \tan ^{-1}(a x)}}{40 a c^4 \left (a^2 x^2+1\right )^2}-\frac {9 e^{-2 \tan ^{-1}(a x)}}{160 a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5070
Rule 5071
Rubi steps
\begin {align*} \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^4} \, dx &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}+\frac {3 \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx}{4 c}\\ &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}+\frac {9 \int \frac {e^{-2 \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{20 c^2}\\ &=-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \tan ^{-1}(a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )}+\frac {9 \int \frac {e^{-2 \tan ^{-1}(a x)}}{c+a^2 c x^2} \, dx}{80 c^3}\\ &=-\frac {9 e^{-2 \tan ^{-1}(a x)}}{160 a c^4}-\frac {e^{-2 \tan ^{-1}(a x)} (1-3 a x)}{20 a c^4 \left (1+a^2 x^2\right )^3}-\frac {3 e^{-2 \tan ^{-1}(a x)} (1-2 a x)}{40 a c^4 \left (1+a^2 x^2\right )^2}-\frac {9 e^{-2 \tan ^{-1}(a x)} (1-a x)}{80 a c^4 \left (1+a^2 x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.37, size = 121, normalized size = 0.98 \[ \frac {8 c (3 a x-1) e^{-2 \tan ^{-1}(a x)}-3 \left (a^2 c x^2+c\right ) \left ((4-8 a x) e^{-2 \tan ^{-1}(a x)}+3 (1-i a x)^{-i} (1+i a x)^i (a x-i) (a x+i) \left (a^2 x^2-2 a x+3\right )\right )}{160 a c^2 \left (a^2 c x^2+c\right )^3} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.49, size = 95, normalized size = 0.77 \[ -\frac {{\left (9 \, a^{6} x^{6} - 18 \, a^{5} x^{5} + 45 \, a^{4} x^{4} - 60 \, a^{3} x^{3} + 75 \, a^{2} x^{2} - 66 \, a x + 47\right )} e^{\left (-2 \, \arctan \left (a x\right )\right )}}{160 \, {\left (a^{7} c^{4} x^{6} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 75, normalized size = 0.60 \[ -\frac {\left (9 a^{6} x^{6}-18 a^{5} x^{5}+45 a^{4} x^{4}-60 a^{3} x^{3}+75 a^{2} x^{2}-66 a x +47\right ) {\mathrm e}^{-2 \arctan \left (a x \right )}}{160 \left (a^{2} x^{2}+1\right )^{3} c^{4} a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (-2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.67, size = 111, normalized size = 0.90 \[ \frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (a\,x-1\right )}{80\,a\,c^4\,\left (a^2\,x^2+1\right )}-\frac {9\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}}{160\,a\,c^4}+\frac {3\,{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (2\,a\,x-1\right )}{40\,a\,c^4\,{\left (a^2\,x^2+1\right )}^2}+\frac {{\mathrm {e}}^{-2\,\mathrm {atan}\left (a\,x\right )}\,\left (3\,a\,x-1\right )}{20\,a\,c^4\,{\left (a^2\,x^2+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________