Optimal. Leaf size=48 \[ \frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3675, 203} \[ \frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sin (3 x+2) \cos (3 x+2)}{\sin ^2(3 x+2)+\sqrt {2}+1}\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 3675
Rubi steps
\begin {align*} \int \frac {\sec ^2(2+3 x)}{1+2 \tan ^2(2+3 x)} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\tan (2+3 x)\right )\\ &=\frac {x}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\cos (2+3 x) \sin (2+3 x)}{1+\sqrt {2}+\sin ^2(2+3 x)}\right )}{3 \sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 22, normalized size = 0.46 \[ \frac {\tan ^{-1}\left (\sqrt {2} \tan (3 x+2)\right )}{3 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 43, normalized size = 0.90 \[ -\frac {1}{12} \, \sqrt {2} \arctan \left (\frac {3 \, \sqrt {2} \cos \left (3 \, x + 2\right )^{2} - 2 \, \sqrt {2}}{4 \, \cos \left (3 \, x + 2\right ) \sin \left (3 \, x + 2\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.54, size = 16, normalized size = 0.33 \[ \frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} \tan \left (3 \, x + 2\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 17, normalized size = 0.35 \[ \frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, \tan \left (2+3 x \right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 16, normalized size = 0.33 \[ \frac {1}{6} \, \sqrt {2} \arctan \left (\sqrt {2} \tan \left (3 \, x + 2\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.40, size = 16, normalized size = 0.33 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,\mathrm {tan}\left (3\,x+2\right )\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{2}{\left (3 x + 2 \right )}}{2 \tan ^{2}{\left (3 x + 2 \right )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________