Optimal. Leaf size=8 \[ 2 \tan (x)-x \]
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Rubi [A] time = 0.0414272, antiderivative size = 8, normalized size of antiderivative = 1., 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 3171
Rule 3175
Rule 3767
Rule 8
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*}
Mathematica [A] time = 0.0079257, size = 8, normalized size = 1. \[ 2 \tan (x)-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 9, normalized size = 1.1 \begin{align*} -x+2\,\tan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49172, size = 11, normalized size = 1.38 \begin{align*} -x + 2 \, \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21415, size = 42, normalized size = 5.25 \begin{align*} -\frac{x \cos \left (x\right ) - 2 \, \sin \left (x\right )}{\cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.50302, size = 41, normalized size = 5.12 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11664, size = 11, normalized size = 1.38 \begin{align*} -x + 2 \, \tan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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