Optimal. Leaf size=77 \[ \frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 8}
\begin {gather*} \frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3739
Rubi steps
\begin {align*} \int \frac {1}{\left (a \cot ^4(x)\right )^{3/2}} \, dx &=\frac {\cot ^2(x) \int \tan ^6(x) \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\cot ^2(x) \int \tan ^4(x) \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}+\frac {\cot ^2(x) \int \tan ^2(x) \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}-\frac {\cot ^2(x) \int 1 \, dx}{a \sqrt {a \cot ^4(x)}}\\ &=\frac {\cot (x)}{a \sqrt {a \cot ^4(x)}}-\frac {x \cot ^2(x)}{a \sqrt {a \cot ^4(x)}}-\frac {\tan (x)}{3 a \sqrt {a \cot ^4(x)}}+\frac {\tan ^3(x)}{5 a \sqrt {a \cot ^4(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 42, normalized size = 0.55 \begin {gather*} \frac {23 \cot (x)-15 x \cot ^2(x)+\csc (x) \sec (x) \left (-11+3 \sec ^2(x)\right )}{15 a \sqrt {a \cot ^4(x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 42, normalized size = 0.55
method | result | size |
derivativedivides | \(\frac {\cot \left (x \right ) \left (15 \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (x \right )\right )\right ) \left (\cot ^{5}\left (x \right )\right )+15 \left (\cot ^{4}\left (x \right )\right )-5 \left (\cot ^{2}\left (x \right )\right )+3\right )}{15 \left (a \left (\cot ^{4}\left (x \right )\right )\right )^{\frac {3}{2}}}\) | \(42\) |
default | \(\frac {\cot \left (x \right ) \left (15 \left (\frac {\pi }{2}-\mathrm {arccot}\left (\cot \left (x \right )\right )\right ) \left (\cot ^{5}\left (x \right )\right )+15 \left (\cot ^{4}\left (x \right )\right )-5 \left (\cot ^{2}\left (x \right )\right )+3\right )}{15 \left (a \left (\cot ^{4}\left (x \right )\right )\right )^{\frac {3}{2}}}\) | \(42\) |
risch | \(\frac {\left ({\mathrm e}^{2 i x}+1\right )^{2} x}{a \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}-\frac {2 i \left (45 \,{\mathrm e}^{8 i x}+90 \,{\mathrm e}^{6 i x}+140 \,{\mathrm e}^{4 i x}+70 \,{\mathrm e}^{2 i x}+23\right )}{15 a \left ({\mathrm e}^{2 i x}+1\right )^{3} \left ({\mathrm e}^{2 i x}-1\right )^{2} \sqrt {\frac {a \left ({\mathrm e}^{2 i x}+1\right )^{4}}{\left ({\mathrm e}^{2 i x}-1\right )^{4}}}}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 29, normalized size = 0.38 \begin {gather*} \frac {3 \, \tan \left (x\right )^{5} - 5 \, \tan \left (x\right )^{3} + 15 \, \tan \left (x\right )}{15 \, a^{\frac {3}{2}}} - \frac {x}{a^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs.
\(2 (65) = 130\).
time = 2.81, size = 142, normalized size = 1.84 \begin {gather*} \frac {{\left (15 \, x \cos \left (2 \, x\right )^{4} + 30 \, x \cos \left (2 \, x\right )^{3} - 30 \, x \cos \left (2 \, x\right ) - {\left (23 \, \cos \left (2 \, x\right )^{3} + \cos \left (2 \, x\right )^{2} - 11 \, \cos \left (2 \, x\right ) - 13\right )} \sin \left (2 \, x\right ) - 15 \, x\right )} \sqrt {\frac {a \cos \left (2 \, x\right )^{2} + 2 \, a \cos \left (2 \, x\right ) + a}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{15 \, {\left (a^{2} \cos \left (2 \, x\right )^{4} + 4 \, a^{2} \cos \left (2 \, x\right )^{3} + 6 \, a^{2} \cos \left (2 \, x\right )^{2} + 4 \, a^{2} \cos \left (2 \, x\right ) + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a \cot ^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 43, normalized size = 0.56 \begin {gather*} -\frac {\frac {15 \, x}{\sqrt {a}} - \frac {3 \, a^{2} \tan \left (x\right )^{5} - 5 \, a^{2} \tan \left (x\right )^{3} + 15 \, a^{2} \tan \left (x\right )}{a^{\frac {5}{2}}}}{15 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (a\,{\mathrm {cot}\left (x\right )}^4\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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