Optimal. Leaf size=118 \[ -\frac {1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {4}{3} a^2 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a^2 \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]
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Rubi [A] time = 0.03, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4123, 3767} \[ -\frac {1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-\frac {4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {4}{3} a^2 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-a^2 \sin (x) \cos (x) \sqrt {a \csc ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 4123
Rubi steps
\begin {align*} \int \left (a \csc ^4(x)\right )^{5/2} \, dx &=\left (a^2 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \int \csc ^{10}(x) \, dx\\ &=-\left (\left (a^2 \sqrt {a \csc ^4(x)} \sin ^2(x)\right ) \operatorname {Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,\cot (x)\right )\right )\\ &=-\frac {4}{3} a^2 \cos ^2(x) \cot (x) \sqrt {a \csc ^4(x)}-\frac {6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt {a \csc ^4(x)}-\frac {4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt {a \csc ^4(x)}-\frac {1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt {a \csc ^4(x)}-a^2 \cos (x) \sqrt {a \csc ^4(x)} \sin (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.40 \[ -\frac {1}{315} a^2 \sin (x) \cos (x) \left (35 \csc ^8(x)+40 \csc ^6(x)+48 \csc ^4(x)+64 \csc ^2(x)+128\right ) \sqrt {a \csc ^4(x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 88, normalized size = 0.75 \[ \frac {{\left (128 \, a^{2} \cos \relax (x)^{9} - 576 \, a^{2} \cos \relax (x)^{7} + 1008 \, a^{2} \cos \relax (x)^{5} - 840 \, a^{2} \cos \relax (x)^{3} + 315 \, a^{2} \cos \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{4} - 2 \, \cos \relax (x)^{2} + 1}}}{315 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 51, normalized size = 0.43 \[ -\frac {{\left (315 \, a^{2} \tan \relax (x)^{8} + 420 \, a^{2} \tan \relax (x)^{6} + 378 \, a^{2} \tan \relax (x)^{4} + 180 \, a^{2} \tan \relax (x)^{2} + 35 \, a^{2}\right )} \sqrt {a}}{315 \, \tan \relax (x)^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 41, normalized size = 0.35 \[ -\frac {\left (128 \left (\cos ^{8}\relax (x )\right )-576 \left (\cos ^{6}\relax (x )\right )+1008 \left (\cos ^{4}\relax (x )\right )-840 \left (\cos ^{2}\relax (x )\right )+315\right ) \cos \relax (x ) \sin \relax (x ) \left (\frac {a}{\sin \relax (x )^{4}}\right )^{\frac {5}{2}}}{315} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 48, normalized size = 0.41 \[ -\frac {315 \, a^{\frac {5}{2}} \tan \relax (x)^{8} + 420 \, a^{\frac {5}{2}} \tan \relax (x)^{6} + 378 \, a^{\frac {5}{2}} \tan \relax (x)^{4} + 180 \, a^{\frac {5}{2}} \tan \relax (x)^{2} + 35 \, a^{\frac {5}{2}}}{315 \, \tan \relax (x)^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.42, size = 121, normalized size = 1.03 \[ \frac {128\,a^{5/2}\,\left ({\mathrm {e}}^{x\,46{}\mathrm {i}}\,1{}\mathrm {i}-{\mathrm {e}}^{x\,48{}\mathrm {i}}\,9{}\mathrm {i}+{\mathrm {e}}^{x\,50{}\mathrm {i}}\,36{}\mathrm {i}-{\mathrm {e}}^{x\,52{}\mathrm {i}}\,84{}\mathrm {i}+{\mathrm {e}}^{x\,54{}\mathrm {i}}\,126{}\mathrm {i}\right )}{315\,\left (\frac {{\mathrm {e}}^{-x\,2{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{x\,2{}\mathrm {i}}}{2}-1\right )\,\left ({\mathrm {e}}^{x\,48{}\mathrm {i}}-7\,{\mathrm {e}}^{x\,50{}\mathrm {i}}+21\,{\mathrm {e}}^{x\,52{}\mathrm {i}}-35\,{\mathrm {e}}^{x\,54{}\mathrm {i}}+35\,{\mathrm {e}}^{x\,56{}\mathrm {i}}-21\,{\mathrm {e}}^{x\,58{}\mathrm {i}}+7\,{\mathrm {e}}^{x\,60{}\mathrm {i}}-{\mathrm {e}}^{x\,62{}\mathrm {i}}\right )} \]
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 \[ \int \left (a \csc ^{4}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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