Optimal. Leaf size=124 \[ -\frac{(1-6 a x) e^{-\tan ^{-1}(a x)}}{37 a c^4 \left (a^2 x^2+1\right )^3}-\frac{72 (1-2 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )}-\frac{30 (1-4 a x) e^{-\tan ^{-1}(a x)}}{629 a c^4 \left (a^2 x^2+1\right )^2}-\frac{144 e^{-\tan ^{-1}(a x)}}{629 a c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.119277, antiderivative size = 124, normalized size of antiderivative = 1., 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.
[In]
[Out]
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*}
Mathematica [C] time = 0.346536, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.039, size = 73, normalized size = 0.6 \begin{align*} -{\frac{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\,ax+263}{629\,{c}^{4} \left ({a}^{2}{x}^{2}+1 \right ) ^{3}{{\rm e}^{\arctan \left ( ax \right ) }}a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.98611, size = 223, normalized size = 1.8 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (-\arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]