Optimal. Leaf size=31 \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0202124, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{1+\pi x+3 x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{-12+\pi ^2-x^2} \, dx,x,\pi +6 x\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{\pi +6 x}{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}}\\ \end{align*}
Mathematica [A] time = 0.0098903, size = 31, normalized size = 1. \[ \frac{2 \tan ^{-1}\left (\frac{6 x+\pi }{\sqrt{12-\pi ^2}}\right )}{\sqrt{12-\pi ^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.045, size = 28, normalized size = 0.9 \begin{align*} 2\,{\frac{1}{\sqrt{-{\pi }^{2}+12}}\arctan \left ({\frac{\pi +6\,x}{\sqrt{-{\pi }^{2}+12}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.10936, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, \arctan \left (\frac{\pi + 6 \, x}{\sqrt{-\pi ^{2} + 12}}\right )}{\sqrt{-\pi ^{2} + 12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.31212, size = 108, normalized size = 3.48 \begin{align*} \frac{2 \, \sqrt{-\pi ^{2} + 12} \arctan \left (\frac{{\left (\pi + 6 \, x\right )} \sqrt{-\pi ^{2} + 12}}{\pi ^{2} - 12}\right )}{\pi ^{2} - 12} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.305933, size = 87, normalized size = 2.81 \begin{align*} - \frac{i \log{\left (x + \frac{\pi }{6} - \frac{2 i}{\sqrt{12 - \pi ^{2}}} + \frac{i \pi ^{2}}{6 \sqrt{12 - \pi ^{2}}} \right )}}{\sqrt{12 - \pi ^{2}}} + \frac{i \log{\left (x + \frac{\pi }{6} - \frac{i \pi ^{2}}{6 \sqrt{12 - \pi ^{2}}} + \frac{2 i}{\sqrt{12 - \pi ^{2}}} \right )}}{\sqrt{12 - \pi ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.24485, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, \arctan \left (\frac{\pi + 6 \, x}{\sqrt{-\pi ^{2} + 12}}\right )}{\sqrt{-\pi ^{2} + 12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]