Optimal. Leaf size=57 \[ -\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 57, 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 = {4122, 192, 191} \[ -\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\csc ^2(x)^{7/2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{9/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6}{7} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{7/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {24}{35} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{5/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16}{35} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right )^{3/2}} \, dx,x,\cot (x)\right )\\ &=-\frac {\cot (x)}{7 \csc ^2(x)^{7/2}}-\frac {6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac {8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac {16 \cot (x)}{35 \sqrt {\csc ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 37, normalized size = 0.65 \[ \frac {(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \csc (x)}{2240 \sqrt {\csc ^2(x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 21, normalized size = 0.37 \[ \frac {1}{7} \, \cos \relax (x)^{7} - \frac {3}{5} \, \cos \relax (x)^{5} + \cos \relax (x)^{3} - \cos \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 78, normalized size = 1.37 \[ -\frac {32 \, {\left (\frac {7 \, {\left (\cos \relax (x) - 1\right )} \mathrm {sgn}\left (\sin \relax (x)\right )}{\cos \relax (x) + 1} - \frac {21 \, {\left (\cos \relax (x) - 1\right )}^{2} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {35 \, {\left (\cos \relax (x) - 1\right )}^{3} \mathrm {sgn}\left (\sin \relax (x)\right )}{{\left (\cos \relax (x) + 1\right )}^{3}} - \mathrm {sgn}\left (\sin \relax (x)\right )\right )}}{35 \, {\left (\frac {\cos \relax (x) - 1}{\cos \relax (x) + 1} - 1\right )}^{7}} + \frac {32}{35} \, \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 44, normalized size = 0.77 \[ \frac {\sin \relax (x ) \left (5 \left (\cos ^{3}\relax (x )\right )-20 \left (\cos ^{2}\relax (x )\right )+29 \cos \relax (x )-16\right ) \sqrt {4}}{70 \left (-1+\cos \relax (x )\right )^{4} \left (-\frac {1}{-1+\cos ^{2}\relax (x )}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 23, normalized size = 0.40 \[ \frac {1}{448} \, \cos \left (7 \, x\right ) - \frac {7}{320} \, \cos \left (5 \, x\right ) + \frac {7}{64} \, \cos \left (3 \, x\right ) - \frac {35}{64} \, \cos \relax (x) \]
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 \[ \int \frac {1}{{\left (\frac {1}{{\sin \relax (x)}^2}\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 138.80, size = 61, normalized size = 1.07 \[ - \frac {16 \cot ^{7}{\relax (x )}}{35 \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {8 \cot ^{5}{\relax (x )}}{5 \left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {2 \cot ^{3}{\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} - \frac {\cot {\relax (x )}}{\left (\csc ^{2}{\relax (x )}\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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