Optimal. Leaf size=14 \[ -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4207, 197}
\begin {gather*} -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 4207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \csc ^2(x)}} \, dx &=-\left (a \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\cot (x)}{\sqrt {a \csc ^2(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(26\) vs.
\(2(12)=24\).
time = 0.08, size = 27, normalized size = 1.93
method | result | size |
default | \(\frac {\sin \left (x \right ) \sqrt {4}}{2 \sqrt {-\frac {a}{\cos ^{2}\left (x \right )-1}}\, \left (\cos \left (x \right )-1\right )}\) | \(27\) |
risch | \(-\frac {i {\mathrm e}^{2 i x}}{2 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i}{2 \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{\sqrt {\tan \left (x\right )^{2} + 1} \sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 22, normalized size = 1.57 \begin {gather*} -\frac {\sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \cos \left (x\right ) \sin \left (x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 14, normalized size = 1.00 \begin {gather*} - \frac {\cot {\left (x \right )}}{\sqrt {a \csc ^{2}{\left (x \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 34 vs.
\(2 (12) = 24\).
time = 0.46, size = 34, normalized size = 2.43 \begin {gather*} \frac {2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{\sqrt {a}} + \frac {2}{\sqrt {a} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 15, normalized size = 1.07 \begin {gather*} -\frac {\sin \left (2\,x\right )}{2\,\sqrt {a}\,\sqrt {{\sin \left (x\right )}^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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