Optimal. Leaf size=19 \[ -\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 260} \[ -\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 260
Rubi steps
\begin {align*} \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx &=\operatorname {Subst}\left (\int \frac {2 x}{a^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x}{a^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ -\frac {\log \left (a^2-b^2 \sin ^2(x)\right )}{b^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin (2 x)}{a^2-b^2 \sin ^2(x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.10, size = 23, normalized size = 1.21 \[ -\frac {\log \left (b^{2} \cos \relax (x)^{2} + a^{2} - b^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 21, normalized size = 1.11 \[ -\frac {\log \left ({\left | b^{2} \sin \relax (x)^{2} - a^{2} \right |}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 20, normalized size = 1.05
method | result | size |
derivativedivides | \(-\frac {\ln \left (a^{2}-b^{2} \left (\sin ^{2}\relax (x )\right )\right )}{b^{2}}\) | \(20\) |
default | \(-\frac {\ln \left (a^{2}-b^{2} \left (\sin ^{2}\relax (x )\right )\right )}{b^{2}}\) | \(20\) |
risch | \(\frac {2 i x}{b^{2}}-\frac {\ln \left ({\mathrm e}^{4 i x}+\frac {2 \left (2 a^{2}-b^{2}\right ) {\mathrm e}^{2 i x}}{b^{2}}+1\right )}{b^{2}}\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 20, normalized size = 1.05 \[ -\frac {\log \left (b^{2} \sin \relax (x)^{2} - a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 48, normalized size = 2.53 \[ \frac {\mathrm {atan}\left (\frac {b^2\,{\sin \relax (x)}^2}{a^2\,{\cos \relax (x)}^2\,2{}\mathrm {i}+a^2\,{\sin \relax (x)}^2\,2{}\mathrm {i}-b^2\,{\sin \relax (x)}^2\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.32, size = 34, normalized size = 1.79 \[ 2 \left (\begin {cases} - \frac {\cos ^{2}{\relax (x )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\- \frac {\log {\left (a^{2} - b^{2} \sin ^{2}{\relax (x )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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