Optimal. Leaf size=21 \[ \frac {1}{3} \log (\cos (x))-\frac {1}{24} \log \left (3-4 \cos ^2(x)\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4366, 446, 72} \[ \frac {1}{3} \log (\cos (x))-\frac {1}{24} \log \left (3-4 \cos ^2(x)\right ) \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 4366
Rubi steps
\begin {align*} \int \sec (3 x) \sin ^3(x) \, dx &=-\operatorname {Subst}\left (\int \frac {-1+x^2}{x \left (3-4 x^2\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x}{(3-4 x) x} \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{3 x}+\frac {1}{3 (-3+4 x)}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac {1}{3} \log (\cos (x))-\frac {1}{24} \log \left (3-4 \cos ^2(x)\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.00 \[ \frac {1}{3} \log (\cos (x))-\frac {1}{24} \log \left (1-4 \sin ^2(x)\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sec (3 x) \sin ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.23, size = 19, normalized size = 0.90 \[ -\frac {1}{24} \, \log \left (4 \, \cos \relax (x)^{2} - 3\right ) + \frac {1}{3} \, \log \left (-\cos \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 24, normalized size = 1.14 \[ \frac {1}{6} \, \log \left (-\sin \relax (x)^{2} + 1\right ) - \frac {1}{24} \, \log \left ({\left | 4 \, \sin \relax (x)^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 18, normalized size = 0.86
method | result | size |
default | \(\frac {\ln \left (\cos \relax (x )\right )}{3}-\frac {\ln \left (4 \left (\cos ^{2}\relax (x )\right )-3\right )}{24}\) | \(18\) |
risch | \(-\frac {i x}{4}+\frac {\ln \left (1+{\mathrm e}^{2 i x}\right )}{3}-\frac {\ln \left ({\mathrm e}^{4 i x}-{\mathrm e}^{2 i x}+1\right )}{24}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 81, normalized size = 3.86 \[ -\frac {1}{48} \, \log \left (-2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} - 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + \sin \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1\right ) + \frac {1}{6} \, \log \left (\cos \left (2 \, x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 15, normalized size = 0.71 \[ \frac {\ln \left (\cos \relax (x)\right )}{3}-\frac {\ln \left ({\cos \relax (x)}^2-\frac {3}{4}\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{3}{\relax (x )}}{\cos {\left (3 x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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