Optimal. Leaf size=76 \[ -\frac {35}{16} \sqrt {\cot ^2(x)}+\frac {1}{6} \cos ^6(x) \sqrt {\cot ^2(x)}+\frac {7}{24} \cos ^4(x) \sqrt {\cot ^2(x)}+\frac {35}{48} \cos ^2(x) \sqrt {\cot ^2(x)}-\frac {35}{16} x \tan (x) \sqrt {\cot ^2(x)} \]
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Rubi [A] time = 0.16, antiderivative size = 84, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3175, 4360, 25, 266, 47, 50, 63, 203} \[ -\frac {35}{16} \sqrt {\csc ^2(x)-1}+\frac {1}{6} \sin ^6(x) \left (\csc ^2(x)-1\right )^{7/2}+\frac {7}{24} \sin ^4(x) \left (\csc ^2(x)-1\right )^{5/2}+\frac {35}{48} \sin ^2(x) \left (\csc ^2(x)-1\right )^{3/2}+\frac {35}{16} \tan ^{-1}\left (\sqrt {\csc ^2(x)-1}\right ) \]
Antiderivative was successfully verified.
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Rule 25
Rule 47
Rule 50
Rule 63
Rule 203
Rule 266
Rule 3175
Rule 4360
Rubi steps
\begin {align*} \int \cot (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx &=\int \cos ^6(x) \cot (x) \sqrt {-1+\csc ^2(x)} \, dx\\ &=\operatorname {Subst}\left (\int \frac {\sqrt {-1+\frac {1}{x^2}} \left (1-x^2\right )^3}{x} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (-1+\frac {1}{x^2}\right )^{7/2} x^5 \, dx,x,\sin (x)\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {(-1+x)^{7/2}}{x^4} \, dx,x,\csc ^2(x)\right )\right )\\ &=\frac {1}{6} \cot ^2(x)^{7/2} \sin ^6(x)-\frac {7}{12} \operatorname {Subst}\left (\int \frac {(-1+x)^{5/2}}{x^3} \, dx,x,\csc ^2(x)\right )\\ &=\frac {7}{24} \cot ^2(x)^{5/2} \sin ^4(x)+\frac {1}{6} \cot ^2(x)^{7/2} \sin ^6(x)-\frac {35}{48} \operatorname {Subst}\left (\int \frac {(-1+x)^{3/2}}{x^2} \, dx,x,\csc ^2(x)\right )\\ &=\frac {35}{48} \cot ^2(x)^{3/2} \sin ^2(x)+\frac {7}{24} \cot ^2(x)^{5/2} \sin ^4(x)+\frac {1}{6} \cot ^2(x)^{7/2} \sin ^6(x)-\frac {35}{32} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,\csc ^2(x)\right )\\ &=-\frac {35}{16} \sqrt {\cot ^2(x)}+\frac {35}{48} \cot ^2(x)^{3/2} \sin ^2(x)+\frac {7}{24} \cot ^2(x)^{5/2} \sin ^4(x)+\frac {1}{6} \cot ^2(x)^{7/2} \sin ^6(x)+\frac {35}{32} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\csc ^2(x)\right )\\ &=-\frac {35}{16} \sqrt {\cot ^2(x)}+\frac {35}{48} \cot ^2(x)^{3/2} \sin ^2(x)+\frac {7}{24} \cot ^2(x)^{5/2} \sin ^4(x)+\frac {1}{6} \cot ^2(x)^{7/2} \sin ^6(x)+\frac {35}{16} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\cot ^2(x)}\right )\\ &=\frac {35}{16} \tan ^{-1}\left (\sqrt {\cot ^2(x)}\right )-\frac {35}{16} \sqrt {\cot ^2(x)}+\frac {35}{48} \cot ^2(x)^{3/2} \sin ^2(x)+\frac {7}{24} \cot ^2(x)^{5/2} \sin ^4(x)+\frac {1}{6} \cot ^2(x)^{7/2} \sin ^6(x)\\ \end {align*}
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Mathematica [A] time = 0.09, size = 40, normalized size = 0.53 \[ \frac {1}{384} \sqrt {\cot ^2(x)} \sec (x) (-840 x \sin (x)-525 \cos (x)+126 \cos (3 x)+14 \cos (5 x)+\cos (7 x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.92, size = 34, normalized size = 0.45 \[ -\frac {8 \, \cos \relax (x)^{7} + 14 \, \cos \relax (x)^{5} + 35 \, \cos \relax (x)^{3} - 105 \, x \sin \relax (x) - 105 \, \cos \relax (x)}{48 \, \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 97, normalized size = 1.28 \[ -\frac {1}{48} \, {\left ({\left (2 \, {\left (4 \, \sin \relax (x)^{2} - 19\right )} \sin \relax (x)^{2} + 87\right )} \sqrt {-\sin \relax (x)^{2} + 1} \sin \relax (x) - 105 \, {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor - x\right )} \left (-1\right )^{\left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor } + \frac {24 \, {\left (\sqrt {-\sin \relax (x)^{2} + 1} - 1\right )}}{\sin \relax (x)} - \frac {24 \, \sin \relax (x)}{\sqrt {-\sin \relax (x)^{2} + 1} - 1}\right )} \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 54, normalized size = 0.71 \[ -\frac {\left (-8 \left (\cos ^{7}\relax (x )\right )-14 \left (\cos ^{5}\relax (x )\right )-35 \left (\cos ^{3}\relax (x )\right )+105 x \sin \relax (x )+105 \cos \relax (x )\right ) \sqrt {-\frac {\cos ^{2}\relax (x )}{-1+\cos ^{2}\relax (x )}}\, \sqrt {4}}{96 \cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 136, normalized size = 1.79 \[ -\frac {3}{2} \, \sqrt {\frac {1}{\sin \relax (x)^{2}} - 1} \sin \relax (x)^{2} - \sqrt {\frac {1}{\sin \relax (x)^{2}} - 1} + \frac {3 \, {\left (\frac {1}{\sin \relax (x)^{2}} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (\frac {1}{\sin \relax (x)^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\frac {1}{\sin \relax (x)^{2}} - 1}}{48 \, {\left ({\left (\frac {1}{\sin \relax (x)^{2}} - 1\right )}^{3} + 3 \, {\left (\frac {1}{\sin \relax (x)^{2}} - 1\right )}^{2} + \frac {3}{\sin \relax (x)^{2}} - 2\right )}} - \frac {3 \, {\left ({\left (\frac {1}{\sin \relax (x)^{2}} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {1}{\sin \relax (x)^{2}} - 1}\right )}}{8 \, {\left ({\left (\frac {1}{\sin \relax (x)^{2}} - 1\right )}^{2} + \frac {2}{\sin \relax (x)^{2}} - 1\right )}} + \frac {35}{16} \, \arctan \left (\sqrt {\frac {1}{\sin \relax (x)^{2}} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int -\mathrm {cot}\relax (x)\,\sqrt {\frac {1}{{\sin \relax (x)}^2}-1}\,{\left ({\sin \relax (x)}^2-1\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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