Optimal. Leaf size=11 \[ \frac {\tan (x)}{\sqrt {\sec ^2(x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 197}
\begin {gather*} \frac {\tan (x)}{\sqrt {\sec ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 4207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sec ^2(x)}} \, dx &=\text {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{\sqrt {\sec ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 11, normalized size = 1.00 \begin {gather*} \frac {\tan (x)}{\sqrt {\sec ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 14, normalized size = 1.27
method | result | size |
default | \(\frac {\sin \left (x \right ) \sqrt {2}}{2 \sqrt {\frac {1}{\cos \left (2 x \right )+1}}\, \cos \left (x \right )}\) | \(14\) |
risch | \(-\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}+\frac {i}{2 \sqrt {\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 11, normalized size = 1.00 \begin {gather*} \frac {\tan \left (x\right )}{\sqrt {\tan \left (x\right )^{2} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.89, size = 4, normalized size = 0.36 \begin {gather*} -\sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 10, normalized size = 0.91 \begin {gather*} \frac {\tan {\left (x \right )}}{\sqrt {\sec ^{2}{\left (x \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 6, normalized size = 0.55 \begin {gather*} \mathrm {sgn}\left (\cos \left (x\right )\right ) \sin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 12, normalized size = 1.09 \begin {gather*} \frac {\sqrt {2}\,\sin \left (2\,x\right )}{2\,\sqrt {2\,{\cos \left (x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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