Optimal. Leaf size=113 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{a^{3/2} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{a^{3/2} f}-\frac {\cot (e+f x)}{a f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.23, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2715, 2985, 2649, 206, 2773} \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{a^{3/2} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{a^{3/2} f}-\frac {\cot (e+f x)}{a f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2715
Rule 2773
Rule 2985
Rubi steps
\begin {align*} \int \frac {\cot ^2(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx &=-\frac {\cot (e+f x)}{a f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\csc (e+f x) \left (-\frac {3 a}{2}+\frac {1}{2} a \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {\cot (e+f x)}{a f \sqrt {a+a \sin (e+f x)}}-\frac {3 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{2 a^2}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{a}\\ &=-\frac {\cot (e+f x)}{a f \sqrt {a+a \sin (e+f x)}}+\frac {3 \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 )}{a f}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a f}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a^{3/2} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{a^{3/2} f}-\frac {\cot (e+f x)}{a f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 2.11, size = 206, normalized size = 1.82 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (-\cot \left (\frac {1}{4} (e+f x)\right )+(16+16 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )+2 \left (\sec \left (\frac {1}{2} (e+f x)\right )+\sin ^2\left (\frac {1}{4} (e+f x)\right ) \csc (e+f x)-\sin \left (\frac {3}{4} (e+f x)\right ) \sin \left (\frac {1}{4} (e+f x)\right ) \csc (e+f x)+3 \log \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )+1\right )-3 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )-\cos \left (\frac {1}{2} (e+f x)\right )+1\right )\right )\right )}{4 f (a (\sin (e+f x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 421, normalized size = 3.73 \[ \frac {3 \, {\left (\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + \frac {4 \, \sqrt {2} {\left (a \cos \left (f x + e\right )^{2} - {\left (a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) - a\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {a}} + 4 \, \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{4 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.80, size = 134, normalized size = 1.19 \[ -\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin \left (f x +e \right ) a^{2} \left (2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-3 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}}{\sqrt {a}}\right )\right )+\sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {3}{2}}\right )}{a^{\frac {7}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {\cot \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cot ^{2}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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