Optimal. Leaf size=65 \[ -\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4207, 201, 223,
212} \begin {gather*} -\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 212
Rule 223
Rule 4207
Rubi steps
\begin {align*} \int \left (a \csc ^2(x)\right )^{5/2} \, dx &=-\left (a \text {Subst}\left (\int \left (a+a x^2\right )^{3/2} \, dx,x,\cot (x)\right )\right )\\ &=-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{4} \left (3 a^2\right ) \text {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}-\frac {1}{8} \left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\cot (x)}{\sqrt {a \csc ^2(x)}}\right )\\ &=-\frac {3}{8} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc ^2(x)}}\right )-\frac {3}{8} a^2 \cot (x) \sqrt {a \csc ^2(x)}-\frac {1}{4} a \cot (x) \left (a \csc ^2(x)\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 51, normalized size = 0.78 \begin {gather*} \frac {1}{64} \left (a \csc ^2(x)\right )^{5/2} \sin (x) \left (-22 \cos (x)+6 \left (\cos (3 x)+4 \left (-\log \left (\cos \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sin ^4(x)\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 79, normalized size = 1.22
method | result | size |
default | \(\frac {\left (3 \left (\cos ^{4}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )+3 \left (\cos ^{3}\left (x \right )\right )-6 \left (\cos ^{2}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )-5 \cos \left (x \right )+3 \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )\right ) \sin \left (x \right ) \left (-\frac {a}{\cos ^{2}\left (x \right )-1}\right )^{\frac {5}{2}} \sqrt {4}}{16}\) | \(79\) |
risch | \(-\frac {i a^{2} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (3 \,{\mathrm e}^{6 i x}-11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}+3\right )}{4 \left ({\mathrm e}^{2 i x}-1\right )^{3}}+\frac {3 a^{2} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{4}-\frac {3 a^{2} \sqrt {-\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{4}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1113 vs.
\(2 (49) = 98\).
time = 0.86, size = 1113, normalized size = 17.12 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.01, size = 80, normalized size = 1.23 \begin {gather*} -\frac {{\left (6 \, a^{2} \cos \left (x\right )^{3} - 10 \, a^{2} \cos \left (x\right ) + 3 \, {\left (a^{2} \cos \left (x\right )^{4} - 2 \, a^{2} \cos \left (x\right )^{2} + a^{2}\right )} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )\right )} \sqrt {-\frac {a}{\cos \left (x\right )^{2} - 1}}}{16 \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\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 \left (a \csc ^{2}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \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. 124 vs.
\(2 (49) = 98\).
time = 0.44, size = 124, normalized size = 1.91 \begin {gather*} \frac {1}{64} \, {\left (12 \, a^{2} \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {8 \, a^{2} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} + \frac {a^{2} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {{\left (a^{2} \mathrm {sgn}\left (\sin \left (x\right )\right ) - \frac {8 \, a^{2} {\left (\cos \left (x\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (x\right )\right )}{\cos \left (x\right ) + 1} + \frac {18 \, a^{2} {\left (\cos \left (x\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (x\right )\right )}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (x\right ) + 1\right )}^{2}}{{\left (\cos \left (x\right ) - 1\right )}^{2}}\right )} \sqrt {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 = -0.02 \begin {gather*} \int {\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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