Optimal. Leaf size=99 \[ -\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}-\frac {3 \cos (x) \tan ^{-1}\left (\sqrt {-\sin (x)}\right )}{32 \sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}+\frac {3 \cos (x) \tanh ^{-1}\left (\sqrt {-\sin (x)}\right )}{32 \sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}+\frac {\tan (x) \sec ^2(x)}{4 \sqrt {\cos (x) \cot (x)}} \]
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Rubi [A] time = 0.15, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {4397, 4400, 2597, 2599, 2601, 2564, 329, 298, 203, 206} \[ -\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}-\frac {3 \cos (x) \tan ^{-1}\left (\sqrt {-\sin (x)}\right )}{32 \sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}+\frac {3 \cos (x) \tanh ^{-1}\left (\sqrt {-\sin (x)}\right )}{32 \sqrt {-\sin (x)} \sqrt {\cos (x) \cot (x)}}+\frac {\tan (x) \sec ^2(x)}{4 \sqrt {\cos (x) \cot (x)}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 298
Rule 329
Rule 2564
Rule 2597
Rule 2599
Rule 2601
Rule 4397
Rule 4400
Rubi steps
\begin {align*} \int \frac {1}{(\csc (x)-\sin (x))^{5/2}} \, dx &=\int \frac {1}{(\cos (x) \cot (x))^{5/2}} \, dx\\ &=\frac {\left (\sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(x) \cot ^{\frac {5}{2}}(x)} \, dx}{\sqrt {\cos (x) \cot (x)}}\\ &=\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}-\frac {\left (3 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\cos ^{\frac {5}{2}}(x) \sqrt {\cot (x)}} \, dx}{8 \sqrt {\cos (x) \cot (x)}}\\ &=-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}-\frac {\left (3 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\sqrt {\cos (x)} \sqrt {\cot (x)}} \, dx}{32 \sqrt {\cos (x) \cot (x)}}\\ &=-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}-\frac {(3 \cos (x)) \int \sec (x) \sqrt {-\sin (x)} \, dx}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}\\ &=-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}+\frac {(3 \cos (x)) \operatorname {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,-\sin (x)\right )}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}\\ &=-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}+\frac {(3 \cos (x)) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {-\sin (x)}\right )}{16 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}\\ &=-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}+\frac {(3 \cos (x)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-\sin (x)}\right )}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}-\frac {(3 \cos (x)) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-\sin (x)}\right )}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}\\ &=-\frac {3 \tan ^{-1}\left (\sqrt {-\sin (x)}\right ) \cos (x)}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}+\frac {3 \tanh ^{-1}\left (\sqrt {-\sin (x)}\right ) \cos (x)}{32 \sqrt {\cos (x) \cot (x)} \sqrt {-\sin (x)}}-\frac {3 \tan (x)}{16 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^2(x) \tan (x)}{4 \sqrt {\cos (x) \cot (x)}}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 69, normalized size = 0.70 \[ -\frac {\sin (x) \tan (x) \sqrt {\cos (x) \cot (x)} \left (-3 \tan ^{-1}\left (\sqrt [4]{\sin ^2(x)}\right )+3 \tanh ^{-1}\left (\sqrt [4]{\sin ^2(x)}\right )+\sin ^2(x)^{3/4} (3 \cos (2 x)-5) \sec ^4(x)\right )}{32 \sin ^2(x)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.95, size = 165, normalized size = 1.67 \[ -\frac {6 \, \arctan \left (\frac {2 \, \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}} \sin \relax (x)}{\cos \relax (x) \sin \relax (x) - \cos \relax (x)}\right ) \cos \relax (x)^{5} - 3 \, \cos \relax (x)^{5} \log \left (\frac {\cos \relax (x)^{3} - 5 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{2} + 6 \, \cos \relax (x) + 4\right )} \sin \relax (x) - 4 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}} - 2 \, \cos \relax (x) + 4}{\cos \relax (x)^{3} + 3 \, \cos \relax (x)^{2} - {\left (\cos \relax (x)^{2} - 2 \, \cos \relax (x) - 4\right )} \sin \relax (x) - 2 \, \cos \relax (x) - 4}\right ) - 8 \, {\left (3 \, \cos \relax (x)^{4} - 7 \, \cos \relax (x)^{2} + 4\right )} \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}}}{128 \, \cos \relax (x)^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (\csc \relax (x) - \sin \relax (x)\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 382, normalized size = 3.86 \[ -\frac {\left (-1+\cos \relax (x )\right ) \left (3 i \left (\cos ^{4}\relax (x )\right ) \EllipticPi \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}-3 i \left (\cos ^{4}\relax (x )\right ) \EllipticPi \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}-3 \left (\cos ^{4}\relax (x )\right ) \EllipticPi \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}-3 \left (\cos ^{4}\relax (x )\right ) \EllipticPi \left (\sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-i \cos \relax (x )+\sin \relax (x )+i}{\sin \relax (x )}}\, \sqrt {\frac {i \cos \relax (x )+\sin \relax (x )-i}{\sin \relax (x )}}\, \sqrt {-\frac {i \left (-1+\cos \relax (x )\right )}{\sin \relax (x )}}+6 \left (\cos ^{3}\relax (x )\right ) \sqrt {2}-6 \left (\cos ^{2}\relax (x )\right ) \sqrt {2}-8 \cos \relax (x ) \sqrt {2}+8 \sqrt {2}\right ) \cos \relax (x ) \left (1+\cos \relax (x )\right )^{2} \sqrt {2}}{64 \sin \relax (x )^{5} \left (\frac {\cos ^{2}\relax (x )}{\sin \relax (x )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (\csc \relax (x) - \sin \relax (x)\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {1}{\sin \relax (x)}-\sin \relax (x)\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \sin {\relax (x )} + \csc {\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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