Optimal. Leaf size=121 \[ -\frac {3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}+\frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}+\frac {5 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f} \]
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Rubi [A] time = 0.32, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2716, 2976, 2981, 2773, 206} \[ \frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{f}+\frac {5 a \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f}-\frac {\cot (e+f x) (a \sin (e+f x)+a)^{3/2}}{f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2716
Rule 2773
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \cot ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx &=-\frac {\cot (e+f x) (a+a \sin (e+f x))^{3/2}}{f}+\frac {\int \csc (e+f x) \left (\frac {3 a}{2}-\frac {5}{2} a \sin (e+f x)\right ) (a+a \sin (e+f x))^{3/2} \, dx}{a}\\ &=\frac {5 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {\cot (e+f x) (a+a \sin (e+f x))^{3/2}}{f}+\frac {2 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \left (\frac {9 a^2}{4}-\frac {11}{4} a^2 \sin (e+f x)\right ) \, dx}{3 a}\\ &=\frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {5 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {\cot (e+f x) (a+a \sin (e+f x))^{3/2}}{f}+\frac {1}{2} (3 a) \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx\\ &=\frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {5 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {\cot (e+f x) (a+a \sin (e+f x))^{3/2}}{f}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac {3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f}+\frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {5 a \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}-\frac {\cot (e+f x) (a+a \sin (e+f x))^{3/2}}{f}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 233, normalized size = 1.93 \[ -\frac {a \csc ^4\left (\frac {1}{2} (e+f x)\right ) \sqrt {a (\sin (e+f x)+1)} \left (-14 \sin \left (\frac {1}{2} (e+f x)\right )-9 \sin \left (\frac {3}{2} (e+f x)\right )-\sin \left (\frac {5}{2} (e+f x)\right )+14 \cos \left (\frac {1}{2} (e+f x)\right )-9 \cos \left (\frac {3}{2} (e+f x)\right )+\cos \left (\frac {5}{2} (e+f x)\right )+9 \sin (e+f x) \log \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )+1\right )-9 \sin (e+f x) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )-\cos \left (\frac {1}{2} (e+f x)\right )+1\right )\right )}{3 f \left (\cot \left (\frac {1}{2} (e+f x)\right )+1\right ) \left (\csc \left (\frac {1}{4} (e+f x)\right )-\sec \left (\frac {1}{4} (e+f x)\right )\right ) \left (\csc \left (\frac {1}{4} (e+f x)\right )+\sec \left (\frac {1}{4} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 315, normalized size = 2.60 \[ \frac {9 \, {\left (a \cos \left (f x + e\right )^{2} - {\left (a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, {\left (2 \, a \cos \left (f x + e\right )^{3} - 8 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - {\left (2 \, a \cos \left (f x + e\right )^{2} + 10 \, a \cos \left (f x + e\right ) + 11 \, a\right )} \sin \left (f x + e\right ) + 11 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{12 \, {\left (f \cos \left (f x + e\right )^{2} - {\left (f \cos \left (f x + e\right ) + f\right )} \sin \left (f x + e\right ) - f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.90, size = 144, normalized size = 1.19 \[ \frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin \left (f x +e \right ) \left (12 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}-2 \sqrt {a}\, \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}}-9 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}}{\sqrt {a}}\right ) a^{2}\right )-3 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}\right )}{3 \sin \left (f x +e \right ) \sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \cot \left (f x + e\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \cot ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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