Optimal. Leaf size=50 \[ -\frac {5}{16} \sinh ^{-1}(\cot (x))-\frac {5}{16} \cot (x) \sqrt {\csc ^2(x)}-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4207, 201, 221}
\begin {gather*} -\frac {1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {5}{16} \cot (x) \sqrt {\csc ^2(x)}-\frac {5}{16} \sinh ^{-1}(\cot (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 4207
Rubi steps
\begin {align*} \int \csc ^2(x)^{7/2} \, dx &=-\text {Subst}\left (\int \left (1+x^2\right )^{5/2} \, dx,x,\cot (x)\right )\\ &=-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac {5}{6} \text {Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,\cot (x)\right )\\ &=-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac {5}{8} \text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac {5}{16} \cot (x) \sqrt {\csc ^2(x)}-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac {5}{16} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\cot (x)\right )\\ &=-\frac {5}{16} \sinh ^{-1}(\cot (x))-\frac {5}{16} \cot (x) \sqrt {\csc ^2(x)}-\frac {5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac {1}{6} \cot (x) \csc ^2(x)^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 92, normalized size = 1.84 \begin {gather*} \frac {1}{384} \sqrt {\csc ^2(x)} \left (-30 \csc ^2\left (\frac {x}{2}\right )-6 \csc ^4\left (\frac {x}{2}\right )-\csc ^6\left (\frac {x}{2}\right )-120 \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right )+30 \sec ^2\left (\frac {x}{2}\right )+6 \sec ^4\left (\frac {x}{2}\right )+\sec ^6\left (\frac {x}{2}\right )\right ) \sin (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. \(100\) vs.
\(2(36)=72\).
time = 0.13, size = 101, normalized size = 2.02
method | result | size |
default | \(-\frac {\left (15 \left (\cos ^{6}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )+15 \left (\cos ^{5}\left (x \right )\right )-45 \left (\cos ^{4}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )-40 \left (\cos ^{3}\left (x \right )\right )+45 \left (\cos ^{2}\left (x \right )\right ) \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )+33 \cos \left (x \right )-15 \ln \left (-\frac {\cos \left (x \right )-1}{\sin \left (x \right )}\right )\right ) \sin \left (x \right ) \left (-\frac {1}{\cos ^{2}\left (x \right )-1}\right )^{\frac {7}{2}} \sqrt {4}}{96}\) | \(101\) |
risch | \(-\frac {i \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left (15 \,{\mathrm e}^{10 i x}-85 \,{\mathrm e}^{8 i x}+198 \,{\mathrm e}^{6 i x}+198 \,{\mathrm e}^{4 i x}-85 \,{\mathrm e}^{2 i x}+15\right )}{24 \left ({\mathrm e}^{2 i x}-1\right )^{5}}+\frac {5 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )}{8}-\frac {5 \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )}{8}\) | \(129\) |
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. 1669 vs.
\(2 (36) = 72\).
time = 0.61, size = 1669, normalized size = 33.38 \begin {gather*} \text {Too large to display} \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. 93 vs.
\(2 (36) = 72\).
time = 4.05, size = 93, normalized size = 1.86 \begin {gather*} \frac {30 \, \cos \left (x\right )^{5} - 80 \, \cos \left (x\right )^{3} - 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 15 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 66 \, \cos \left (x\right )}{96 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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. 129 vs.
\(2 (36) = 72\).
time = 0.47, size = 129, normalized size = 2.58 \begin {gather*} -\frac {\frac {45 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {9 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {{\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}}}{384 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} - \frac {{\left (\frac {9 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - \frac {45 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {110 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}^{3}}{384 \, {\left (\cos \left (x\right ) - 1\right )}^{3} \mathrm {sgn}\left (\sin \left (x\right )\right )} + \frac {5 \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right )}{32 \, \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 {\left (\frac {1}{{\sin \left (x\right )}^2}\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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