Optimal. Leaf size=55 \[ -\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4207, 198, 197}
\begin {gather*} -\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 4207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \csc ^2(x)\right )^{5/2}} \, dx &=-\left (a \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4}{5} \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \text {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\cot (x)\right )}{15 a}\\ &=-\frac {\cot (x)}{5 \left (a \csc ^2(x)\right )^{5/2}}-\frac {4 \cot (x)}{15 a \left (a \csc ^2(x)\right )^{3/2}}-\frac {8 \cot (x)}{15 a^2 \sqrt {a \csc ^2(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 0.65 \begin {gather*} -\frac {(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \sqrt {a \csc ^2(x)} \sin (x)}{240 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 39, normalized size = 0.71
method | result | size |
default | \(\frac {\sin \left (x \right ) \left (3 \left (\cos ^{2}\left (x \right )\right )-9 \cos \left (x \right )+8\right ) \sqrt {4}}{30 \left (\cos \left (x \right )-1\right )^{3} \left (-\frac {a}{\cos ^{2}\left (x \right )-1}\right )^{\frac {5}{2}}}\) | \(39\) |
risch | \(-\frac {i {\mathrm e}^{6 i x}}{160 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i {\mathrm e}^{2 i x}}{16 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}-\frac {5 i}{16 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}+\frac {5 i {\mathrm e}^{-2 i x}}{96 a^{2} \left ({\mathrm e}^{2 i x}-1\right ) \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}}+\frac {11 i \cos \left (4 x \right )}{240 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}-\frac {7 \sin \left (4 x \right )}{120 \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}-1\right ) a^{2}}\) | \(228\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.62, size = 37, normalized size = 0.67 \begin {gather*} -\frac {{\left (3 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}} \sin \left (x\right )}{15 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.60, size = 51, normalized size = 0.93 \begin {gather*} - \frac {8 \cot ^{5}{\left (x \right )}}{15 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {4 \cot ^{3}{\left (x \right )}}{3 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} - \frac {\cot {\left (x \right )}}{\left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 62, normalized size = 1.13 \begin {gather*} \frac {16 \, \mathrm {sgn}\left (\sin \left (x\right )\right )}{15 \, a^{\frac {5}{2}}} - \frac {16 \, {\left (\frac {5 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {10 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - 1\right )}}{15 \, a^{\frac {5}{2}} {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} - 1\right )}^{5} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (\frac {a}{{\sin \left (x\right )}^2}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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