Optimal. Leaf size=44 \[ -2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right )-\frac {2 a^2 \cot (x)}{\sqrt {a \csc (x)+a}} \]
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Rubi [A] time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3775, 21, 3774, 203} \[ -\frac {2 a^2 \cot (x)}{\sqrt {a \csc (x)+a}}-2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a \csc (x)+a}}\right ) \]
Antiderivative was successfully verified.
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Rule 21
Rule 203
Rule 3774
Rule 3775
Rubi steps
\begin {align*} \int (a+a \csc (x))^{3/2} \, dx &=-\frac {2 a^2 \cot (x)}{\sqrt {a+a \csc (x)}}+(2 a) \int \frac {\frac {a}{2}+\frac {1}{2} a \csc (x)}{\sqrt {a+a \csc (x)}} \, dx\\ &=-\frac {2 a^2 \cot (x)}{\sqrt {a+a \csc (x)}}+a \int \sqrt {a+a \csc (x)} \, dx\\ &=-\frac {2 a^2 \cot (x)}{\sqrt {a+a \csc (x)}}-\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\frac {a \cot (x)}{\sqrt {a+a \csc (x)}}\right )\\ &=-2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \cot (x)}{\sqrt {a+a \csc (x)}}\right )-\frac {2 a^2 \cot (x)}{\sqrt {a+a \csc (x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 69, normalized size = 1.57 \[ -\frac {2 a \sqrt {a (\csc (x)+1)} \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right ) \left (\sqrt {\csc (x)-1}+\tan ^{-1}\left (\sqrt {\csc (x)-1}\right )\right )}{\sqrt {\csc (x)-1} \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 212, normalized size = 4.82 \[ \left [\frac {{\left (a \cos \relax (x) + a \sin \relax (x) + a\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \relax (x)^{2} - 2 \, {\left (\cos \relax (x)^{2} + {\left (\cos \relax (x) + 1\right )} \sin \relax (x) - 1\right )} \sqrt {-a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} + a \cos \relax (x) - {\left (2 \, a \cos \relax (x) + a\right )} \sin \relax (x) - a}{\cos \relax (x) + \sin \relax (x) + 1}\right ) - 2 \, {\left (a \cos \relax (x) - a \sin \relax (x) + a\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}}{\cos \relax (x) + \sin \relax (x) + 1}, \frac {2 \, {\left ({\left (a \cos \relax (x) + a \sin \relax (x) + a\right )} \sqrt {a} \arctan \left (-\frac {\sqrt {a} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}} {\left (\cos \relax (x) - \sin \relax (x) + 1\right )}}{a \cos \relax (x) + a \sin \relax (x) + a}\right ) - {\left (a \cos \relax (x) - a \sin \relax (x) + a\right )} \sqrt {\frac {a \sin \relax (x) + a}{\sin \relax (x)}}\right )}}{\cos \relax (x) + \sin \relax (x) + 1}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 195, normalized size = 4.43 \[ \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} a - \frac {\sqrt {2} a^{2}}{\sqrt {a \tan \left (\frac {1}{2} \, x\right )}} + {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} + 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + {\left (a \sqrt {{\left | a \right |}} + {\left | a \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | a \right |}} - 2 \, \sqrt {a \tan \left (\frac {1}{2} \, x\right )}\right )}}{2 \, \sqrt {{\left | a \right |}}}\right ) + \frac {1}{2} \, {\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) + \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) - \frac {1}{2} \, {\left (a \sqrt {{\left | a \right |}} - {\left | a \right |}^{\frac {3}{2}}\right )} \log \left (a \tan \left (\frac {1}{2} \, x\right ) - \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )} \sqrt {{\left | a \right |}} + {\left | a \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.63, size = 273, normalized size = 6.20 \[ -\frac {\left (\sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )+\sin \relax (x )-\cos \relax (x )+1}{\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )-\sin \relax (x )+\cos \relax (x )-1}\right ) \sin \relax (x )+4 \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}+1\right ) \sin \relax (x )+4 \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}-1\right ) \sin \relax (x )+\sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \ln \left (-\frac {\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )-\sin \relax (x )+\cos \relax (x )-1}{\sqrt {2}\, \sqrt {-\frac {-1+\cos \relax (x )}{\sin \relax (x )}}\, \sin \relax (x )+\sin \relax (x )-\cos \relax (x )+1}\right ) \sin \relax (x )-2 \sin \relax (x ) \sqrt {2}-2 \cos \relax (x ) \sqrt {2}+2 \sqrt {2}\right ) \sin \relax (x ) \left (\frac {a \left (1+\sin \relax (x )\right )}{\sin \relax (x )}\right )^{\frac {3}{2}} \sqrt {2}}{2 \left (\cos \relax (x ) \sin \relax (x )+\cos ^{2}\relax (x )-2 \sin \relax (x )+\cos \relax (x )-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 200, normalized size = 4.55 \[ \sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )}\right )\right )} a^{\frac {3}{2}} - \frac {1}{5} \, \sqrt {2} {\left (a^{\frac {3}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {5}{2}} + 5 \, a^{\frac {3}{2}} \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {3}{2}} + 10 \, a^{\frac {3}{2}} \sqrt {\frac {\sin \relax (x)}{\cos \relax (x) + 1}}\right )} - \frac {\frac {5 \, \sqrt {2} a^{\frac {3}{2}} \sin \relax (x)}{\cos \relax (x) + 1} - \frac {15 \, \sqrt {2} a^{\frac {3}{2}} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {5 \, \sqrt {2} a^{\frac {3}{2}} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {\sqrt {2} a^{\frac {3}{2}} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}}}{5 \, \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+\frac {a}{\sin \relax (x)}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \csc {\relax (x )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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