Optimal. Leaf size=31 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1107, 618, 204} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 1107
Rubi steps
\begin {align*} \int \frac {x}{a+b+2 a x^2+a x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{a+b+2 a x+a x^2} \, dx,x,x^2\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{-4 a b-x^2} \, dx,x,2 a \left (1+x^2\right )\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 31, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a} \left (x^2+1\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.90, size = 91, normalized size = 2.94 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {a x^{4} + 2 \, a x^{2} - 2 \, \sqrt {-a b} {\left (x^{2} + 1\right )} + a - b}{a x^{4} + 2 \, a x^{2} + a + b}\right )}{4 \, a b}, -\frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a x^{2} + a}\right )}{2 \, a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 21, normalized size = 0.68 \[ \frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 26, normalized size = 0.84 \[ \frac {\arctan \left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 2.94, size = 21, normalized size = 0.68 \[ \frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {a b}}\right )}{2 \, \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 24, normalized size = 0.77 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {a}+\sqrt {a}\,x^2}{\sqrt {b}}\right )}{2\,\sqrt {a}\,\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.46, size = 60, normalized size = 1.94 \[ - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- b \sqrt {- \frac {1}{a b}} + x^{2} + 1 \right )}}{4} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (b \sqrt {- \frac {1}{a b}} + x^{2} + 1 \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________