Optimal. Leaf size=8 \[ 2 \tan (x)-x \]
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Rubi [A] time = 0.04, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3171, 3175, 3767, 8} \[ 2 \tan (x)-x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3171
Rule 3175
Rule 3767
Rubi steps
\begin {align*} \int \frac {1+\sin ^2(x)}{1-\sin ^2(x)} \, dx &=-x+2 \int \frac {1}{1-\sin ^2(x)} \, dx\\ &=-x+2 \int \sec ^2(x) \, dx\\ &=-x-2 \operatorname {Subst}(\int 1 \, dx,x,-\tan (x))\\ &=-x+2 \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 8, normalized size = 1.00 \[ 2 \tan (x)-x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 15, normalized size = 1.88 \[ -\frac {x \cos \relax (x) - 2 \, \sin \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 8, normalized size = 1.00 \[ -x + 2 \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 9, normalized size = 1.12 \[ -x +2 \tan \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 8, normalized size = 1.00 \[ -x + 2 \, \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.31, size = 8, normalized size = 1.00 \[ 2\,\mathrm {tan}\relax (x)-x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.97, size = 41, normalized size = 5.12 \[ - \frac {x \tan ^{2}{\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} + \frac {x}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {4 \tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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