Optimal. Leaf size=15 \[ \frac {1}{12} \tan ^{-1}\left (\frac {1}{3} \sin ^2(4 x)\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 275, 203} \[ \frac {1}{12} \tan ^{-1}\left (\frac {1}{3} \sin ^2(4 x)\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 275
Rubi steps
\begin {align*} \int \frac {\sin (8 x)}{9+\sin ^4(4 x)} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {2 x}{9+x^4} \, dx,x,\sin (4 x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{9+x^4} \, dx,x,\sin (4 x)\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{9+x^2} \, dx,x,\sin ^2(4 x)\right )\\ &=\frac {1}{12} \tan ^{-1}\left (\frac {1}{3} \sin ^2(4 x)\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 1.00 \[ \frac {1}{12} \tan ^{-1}\left (\frac {1}{3} \sin ^2(4 x)\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 13, normalized size = 0.87 \[ -\frac {1}{12} \, \arctan \left (\frac {1}{3} \, \cos \left (4 \, x\right )^{2} - \frac {1}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.66, size = 15, normalized size = 1.00 \[ \frac {1}{12} \, \arctan \left (\frac {3}{\cos \left (4 \, x\right )^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 12, normalized size = 0.80 \[ \frac {\arctan \left (\frac {\left (\sin ^{2}\left (4 x \right )\right )}{3}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (8 \, x\right )}{\sin \left (4 \, x\right )^{4} + 9}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.96, size = 13, normalized size = 0.87 \[ \frac {\mathrm {atan}\left (\frac {10\,{\mathrm {tan}\left (4\,x\right )}^2}{3}+3\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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