Optimal. Leaf size=38 \[ -\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3176, 3207, 2606, 30} \[ -\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2606
Rule 3176
Rule 3207
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\cot ^4(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\cos (e+f x) \int \cot (e+f x) \csc ^3(e+f x) \, dx}{a \sqrt {a \cos ^2(e+f x)}}\\ &=-\frac {\cos (e+f x) \operatorname {Subst}\left (\int x^2 \, dx,x,\csc (e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}}\\ &=-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 0.76 \[ -\frac {\cot ^3(e+f x)}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 50, normalized size = 1.32 \[ \frac {\sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{3} - a^{2} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
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 = 0.63, size = 35, normalized size = 0.92 \[ -\frac {\cos \left (f x +e \right )}{3 a \sin \left (f x +e \right )^{3} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 382, normalized size = 10.05 \[ \frac {8 \, {\left (\cos \left (3 \, f x + 3 \, e\right ) \sin \left (6 \, f x + 6 \, e\right ) - 3 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (4 \, f x + 4 \, e\right ) - {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - \cos \left (6 \, f x + 6 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (4 \, f x + 4 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a}}{3 \, {\left (a^{2} \cos \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 18 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 6 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} - 2 \, {\left (3 \, a^{2} \cos \left (4 \, f x + 4 \, e\right ) - 3 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (6 \, f x + 6 \, e\right ) - 6 \, {\left (3 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 6 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) - a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right )\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.69, size = 88, normalized size = 2.32 \[ \frac {{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )} \]
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 \[ \int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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