Optimal. Leaf size=17 \[ \frac {\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \[ \frac {\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \]
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 = 17, 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 = 1.04, size = 21, normalized size = 1.24 \[ \frac {\log \left (-b^{2} \cos \relax (x)^{2} + a^{2} + b^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.04, size = 17, normalized size = 1.00 \[ \frac {\log \left (b^{2} \sin \relax (x)^{2} + a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 18, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\ln \left (a^{2}+b^{2} \left (\sin ^{2}\relax (x )\right )\right )}{b^{2}}\) | \(18\) |
default | \(\frac {\ln \left (a^{2}+b^{2} \left (\sin ^{2}\relax (x )\right )\right )}{b^{2}}\) | \(18\) |
risch | \(-\frac {2 i x}{b^{2}}+\frac {\ln \left ({\mathrm e}^{4 i x}-\frac {2 \left (2 a^{2}+b^{2}\right ) {\mathrm e}^{2 i x}}{b^{2}}+1\right )}{b^{2}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 17, normalized size = 1.00 \[ \frac {\log \left (b^{2} \sin \relax (x)^{2} + a^{2}\right )}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 48, normalized size = 2.82 \[ \frac {\mathrm {atan}\left (\frac {b^2\,{\sin \relax (x)}^2}{a^2\,{\cos \relax (x)}^2\,2{}\mathrm {i}+a^2\,{\sin \relax (x)}^2\,2{}\mathrm {i}+b^2\,{\sin \relax (x)}^2\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.98, size = 32, normalized size = 1.88 \[ 2 \left (\begin {cases} - \frac {\cos ^{2}{\relax (x )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\\frac {\log {\left (a^{2} + b^{2} \sin ^{2}{\relax (x )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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