Optimal. Leaf size=7 \[ \tan (x)-\cot (x) \]
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Rubi [A] time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2620, 14} \[ \tan (x)-\cot (x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 2620
Rubi steps
\begin {align*} \int \csc ^2(x) \sec ^2(x) \, dx &=\operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (x)\right )\\ &=-\cot (x)+\tan (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 6, normalized size = 0.86 \[ -2 \cot (2 x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^2(x) \sec ^2(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.34, size = 18, normalized size = 2.57 \[ -\frac {2 \, \cos \relax (x)^{2} - 1}{\cos \relax (x) \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 9, normalized size = 1.29 \[ -\frac {1}{\tan \relax (x)} + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 15, normalized size = 2.14
method | result | size |
default | \(\frac {1}{\sin \relax (x ) \cos \relax (x )}-2 \cot \relax (x )\) | \(15\) |
risch | \(-\frac {4 i}{\left (1+{\mathrm e}^{2 i x}\right ) \left ({\mathrm e}^{2 i x}-1\right )}\) | \(22\) |
norman | \(\frac {\frac {1}{2}-3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\frac {\left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}}{\left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 9, normalized size = 1.29 \[ -\frac {1}{\tan \relax (x)} + \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 6, normalized size = 0.86 \[ -2\,\mathrm {cot}\left (2\,x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.07, size = 12, normalized size = 1.71 \[ - \frac {2 \cos {\left (2 x \right )}}{\sin {\left (2 x \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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