Optimal. Leaf size=141 \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {2} a^{5/2} f}-\frac {2 \cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}-\frac {\cot (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}} \]
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Rubi [A] time = 0.35, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2715, 2978, 2985, 2649, 206, 2773} \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {2} a^{5/2} f}-\frac {2 \cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}-\frac {\cot (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2715
Rule 2773
Rule 2978
Rule 2985
Rubi steps
\begin {align*} \int \frac {\cot ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\csc (e+f x) \left (-\frac {5 a}{2}+\frac {3}{2} a \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{a^2}\\ &=-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\csc (e+f x) \left (-5 a^2+2 a^2 \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^4}\\ &=-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {5 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{2 a^3}+\frac {7 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2}\\ &=-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}+\frac {5 \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^2 f}-\frac {7 \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^2 f}\\ &=\frac {5 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {2} a^{5/2} f}-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.74, size = 451, normalized size = 3.20 \[ \frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (8 \sin \left (\frac {1}{2} (e+f x)\right )+\frac {2 \sin \left (\frac {1}{4} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}{\cos \left (\frac {1}{4} (e+f x)\right )-\sin \left (\frac {1}{4} (e+f x)\right )}-\frac {2 \sin \left (\frac {1}{4} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}{\sin \left (\frac {1}{4} (e+f x)\right )+\cos \left (\frac {1}{4} (e+f x)\right )}+2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2-4 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+10 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \log \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )+1\right )-10 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )-\cos \left (\frac {1}{2} (e+f x)\right )+1\right )-\tan \left (\frac {1}{4} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2-\cot \left (\frac {1}{4} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2+(28+28 i) (-1)^{3/4} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (e+f x)\right )-1\right )\right )\right )}{4 f (a (\sin (e+f x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 539, normalized size = 3.82 \[ \frac {5 \, {\left (\cos \left (f x + e\right )^{3} + 2 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2\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 {7 \, \sqrt {2} {\left (a \cos \left (f x + e\right )^{3} + 2 \, a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) - 2 \, 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 \, {\left (2 \, \cos \left (f x + e\right )^{2} + {\left (2 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{4 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 2 \, a^{3} f \cos \left (f x + e\right )^{2} - a^{3} f \cos \left (f x + e\right ) - 2 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - a^{3} f \cos \left (f x + e\right ) - 2 \, a^{3} 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 = 0.69, size = 219, normalized size = 1.55 \[ -\frac {\left (7 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) a -10 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) a +7 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )+4 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a}\, \sin \left (f x +e \right )-10 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) a \sin \left (f x +e \right )+2 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{2 a^{\frac {7}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\cot ^{2}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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