Optimal. Leaf size=7 \[ \tan (x)-\cos (x) \]
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Rubi [A] time = 0.04, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4401, 3767, 8, 2638} \[ \tan (x)-\cos (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2638
Rule 3767
Rule 4401
Rubi steps
\begin {align*} \int \cos (x) \left (\sec ^3(x)+\tan (x)\right ) \, dx &=\int \left (\sec ^2(x)+\sin (x)\right ) \, dx\\ &=\int \sec ^2(x) \, dx+\int \sin (x) \, dx\\ &=-\cos (x)-\operatorname {Subst}(\int 1 \, dx,x,-\tan (x))\\ &=-\cos (x)+\tan (x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 7, normalized size = 1.00 \[ \tan (x)-\cos (x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.74, size = 15, normalized size = 2.14 \[ -\frac {\cos \relax (x)^{2} - \sin \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 30, normalized size = 4.29 \[ -\frac {2 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right ) - 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 8, normalized size = 1.14 \[ -\cos \relax (x )+\tan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 7, normalized size = 1.00 \[ -\cos \relax (x) + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.97, size = 12, normalized size = 1.71 \[ \frac {\sin \relax (x)}{\cos \relax (x)}-\cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.81, size = 8, normalized size = 1.14 \[ \frac {\sin {\relax (x )}}{\cos {\relax (x )}} - \cos {\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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