Optimal. Leaf size=81 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{a^{3/2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3777, 3920, 3774, 203, 3795} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )}{a^{3/2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a \csc (x)+a}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\cot (x)}{2 (a \csc (x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 3774
Rule 3777
Rule 3795
Rule 3920
Rubi steps
\begin {align*} \int \frac {1}{(a+a \csc (x))^{3/2}} \, dx &=\frac {\cot (x)}{2 (a+a \csc (x))^{3/2}}-\frac {\int \frac {-2 a+\frac {1}{2} a \csc (x)}{\sqrt {a+a \csc (x)}} \, dx}{2 a^2}\\ &=\frac {\cot (x)}{2 (a+a \csc (x))^{3/2}}+\frac {\int \sqrt {a+a \csc (x)} \, dx}{a^2}-\frac {5 \int \frac {\csc (x)}{\sqrt {a+a \csc (x)}} \, dx}{4 a}\\ &=\frac {\cot (x)}{2 (a+a \csc (x))^{3/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a}+\frac {5 \operatorname {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{2 a}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )}{a^{3/2}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {2} \sqrt {a+a \csc (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {\cot (x)}{2 (a+a \csc (x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 129, normalized size = 1.59 \[ -\frac {\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \left (-2 \csc (x)+8 \sqrt {\csc (x)-1} (\csc (x)+1) \tan ^{-1}\left (\sqrt {\csc (x)-1}\right )-5 \sqrt {2} \sqrt {\csc (x)-1} \csc (x) \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2 \tan ^{-1}\left (\frac {\sqrt {\csc (x)-1}}{\sqrt {2}}\right )+2\right )}{4 a (\csc (x)-1) \sqrt {a (\csc (x)+1)} \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 427, normalized size = 5.27 \[ \left [-\frac {5 \, \sqrt {2} {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )} \sqrt {-a} \log \left (-\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} \sin \relax (x) - a \cos \relax (x)}{\sin \relax (x) + 1}\right ) + 4 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \relax (x)^{2} + 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} + a \cos \relax (x) - {\left (2 \, a \cos \relax (x) + a\right )} \sin \relax (x) - a}{\cos \relax (x) + \sin \relax (x) + 1}\right ) + 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}}{4 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2} - {\left (a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}}, \frac {5 \, \sqrt {2} {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right ) + 4 \, {\left (\cos \relax (x)^{2} - {\left (\cos \relax (x) + 2\right )} \sin \relax (x) - \cos \relax (x) - 2\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) - \sin \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right ) - {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}}{2 \, {\left (a^{2} \cos \relax (x)^{2} - a^{2} \cos \relax (x) - 2 \, a^{2} - {\left (a^{2} \cos \relax (x) + 2 \, a^{2}\right )} \sin \relax (x)\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.93, size = 304, normalized size = 3.75 \[ -\frac {1}{2} \, {\left (\frac {5 \, \sqrt {2} \arctan \left (\frac {\sqrt {a \tan \left (\frac {1}{2} \, x\right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} - \frac {2 \, {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} - \frac {2 \, {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right )}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} - \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} + \frac {{\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right )}{a^{3} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )} + \frac {\sqrt {2} {\left (\sqrt {a \tan \left (\frac {1}{2} \, x\right )} a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2} a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.81, size = 1141, normalized size = 14.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 150, normalized size = 1.85 \[ -\frac {\sqrt {2} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {3}{2}} - \sqrt {2} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}}{2 \, {\left (a^{\frac {3}{2}} + \frac {2 \, a^{\frac {3}{2}} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a^{\frac {3}{2}} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}} + \frac {\sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right )\right )}}{a^{\frac {3}{2}}} - \frac {5 \, \sqrt {2} \arctan \left (\sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}{2 \, a^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (a+\frac {a}{\sin \relax (x)}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \csc {\relax (x )} + a\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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