Optimal. Leaf size=17 \[ \frac {\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \]
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Rubi [A]
time = 0.03, 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, 266}
\begin {gather*} \frac {\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 266
Rubi steps
\begin {align*} \int \frac {\sin (2 x)}{a^2+b^2 \sin ^2(x)} \, dx &=\text {Subst}\left (\int \frac {2 x}{a^2+b^2 x^2} \, dx,x,\sin (x)\right )\\ &=2 \text {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 \begin {gather*} \frac {\log \left (a^2+b^2 \sin ^2(x)\right )}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 18, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\ln \left (a^{2}+b^{2} \left (\sin ^{2}\left (x \right )\right )\right )}{b^{2}}\) | \(18\) |
default | \(\frac {\ln \left (a^{2}+b^{2} \left (\sin ^{2}\left (x \right )\right )\right )}{b^{2}}\) | \(18\) |
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 = 1.85, size = 17, normalized size = 1.00 \begin {gather*} \frac {\log \left (b^{2} \sin \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.26, size = 21, normalized size = 1.24 \begin {gather*} \frac {\log \left (-b^{2} \cos \left (x\right )^{2} + a^{2} + b^{2}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.38, size = 32, normalized size = 1.88 \begin {gather*} 2 \left (\begin {cases} - \frac {\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\\frac {\log {\left (a^{2} + b^{2} \sin ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.89, size = 17, normalized size = 1.00 \begin {gather*} \frac {\log \left (b^{2} \sin \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end {gather*}
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 \begin {gather*} \frac {\mathrm {atan}\left (\frac {b^2\,{\sin \left (x\right )}^2}{a^2\,{\cos \left (x\right )}^2\,2{}\mathrm {i}+a^2\,{\sin \left (x\right )}^2\,2{}\mathrm {i}+b^2\,{\sin \left (x\right )}^2\,1{}\mathrm {i}}\right )\,2{}\mathrm {i}}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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