Optimal. Leaf size=35 \[ -\frac {(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )}-\frac {2 e^{\cot ^{-1}(x)}}{5 a^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5115, 5113} \[ -\frac {(1-2 x) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )}-\frac {2 e^{\cot ^{-1}(x)}}{5 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5113
Rule 5115
Rubi steps
\begin {align*} \int \frac {e^{\cot ^{-1}(x)}}{\left (a+a x^2\right )^2} \, dx &=-\frac {e^{\cot ^{-1}(x)} (1-2 x)}{5 a^2 \left (1+x^2\right )}+\frac {2 \int \frac {e^{\cot ^{-1}(x)}}{a+a x^2} \, dx}{5 a}\\ &=-\frac {2 e^{\cot ^{-1}(x)}}{5 a^2}-\frac {e^{\cot ^{-1}(x)} (1-2 x)}{5 a^2 \left (1+x^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 28, normalized size = 0.80 \[ -\frac {\left (2 x^2-2 x+3\right ) e^{\cot ^{-1}(x)}}{5 a^2 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 28, normalized size = 0.80 \[ -\frac {{\left (2 \, x^{2} - 2 \, x + 3\right )} e^{\operatorname {arccot}\relax (x)}}{5 \, {\left (a^{2} x^{2} + a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 26, normalized size = 0.74 \[ -\frac {{\mathrm e}^{\mathrm {arccot}\relax (x )} \left (2 x^{2}-2 x +3\right )}{5 \left (x^{2}+1\right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {arccot}\relax (x)}}{{\left (a x^{2} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.64, size = 25, normalized size = 0.71 \[ -\frac {{\mathrm {e}}^{\mathrm {acot}\relax (x)}\,\left (2\,x^2-2\,x+3\right )}{5\,a^2\,\left (x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.48, size = 65, normalized size = 1.86 \[ - \frac {2 x^{2} e^{\operatorname {acot}{\relax (x )}}}{5 a^{2} x^{2} + 5 a^{2}} + \frac {2 x e^{\operatorname {acot}{\relax (x )}}}{5 a^{2} x^{2} + 5 a^{2}} - \frac {3 e^{\operatorname {acot}{\relax (x )}}}{5 a^{2} x^{2} + 5 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________