Optimal. Leaf size=18 \[ -\frac {\log \left (a^2+b^2 \cos ^2(x)\right )}{b^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 260} \[ -\frac {\log \left (a^2-b^2 \sin ^2(x)+b^2\right )}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 260
Rubi steps
\begin {align*} \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {2 x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {\log \left (a^2+b^2-b^2 \sin ^2(x)\right )}{b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 22, normalized size = 1.22 \[ -\frac {\log \left (a^2-b^2 \sin ^2(x)+b^2\right )}{b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 18, normalized size = 1.00 \[ -\frac {\log \left (b^{2} \cos \relax (x)^{2} + a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.92, size = 18, normalized size = 1.00 \[ -\frac {\log \left (b^{2} \cos \relax (x)^{2} + a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.16, size = 19, normalized size = 1.06
method | result | size |
derivativedivides | \(-\frac {\ln \left (a^{2}+b^{2} \left (\cos ^{2}\relax (x )\right )\right )}{b^{2}}\) | \(19\) |
default | \(-\frac {\ln \left (a^{2}+b^{2} \left (\cos ^{2}\relax (x )\right )\right )}{b^{2}}\) | \(19\) |
risch | \(\frac {2 i x}{b^{2}}-\frac {\ln \left ({\mathrm e}^{4 i x}+\frac {2 \left (2 a^{2}+b^{2}\right ) {\mathrm e}^{2 i x}}{b^{2}}+1\right )}{b^{2}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 18, normalized size = 1.00 \[ -\frac {\log \left (b^{2} \cos \relax (x)^{2} + a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 58, normalized size = 3.22 \[ \frac {2\,\mathrm {atanh}\left (\frac {b^2}{2\,a^2+b^2\,{\cos \relax (x)}^2+b^2}-\frac {b^2\,{\cos \relax (x)}^2}{2\,a^2+b^2\,{\cos \relax (x)}^2+b^2}\right )}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.00, size = 34, normalized size = 1.89 \[ 2 \left (\begin {cases} - \frac {\cos ^{2}{\relax (x )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\- \frac {\log {\left (a^{2} + b^{2} \cos ^{2}{\relax (x )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________