Optimal. Leaf size=91 \[ \frac {\tan (e+f x) \sqrt {a \cos ^2(e+f x)}}{f}-\frac {\csc ^3(e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{3 f}+\frac {2 \csc (e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{f} \]
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Rubi [A] time = 0.12, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3176, 3207, 2590, 270} \[ \frac {\tan (e+f x) \sqrt {a \cos ^2(e+f x)}}{f}-\frac {\csc ^3(e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{3 f}+\frac {2 \csc (e+f x) \sec (e+f x) \sqrt {a \cos ^2(e+f x)}}{f} \]
Antiderivative was successfully verified.
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Rule 270
Rule 2590
Rule 3176
Rule 3207
Rubi steps
\begin {align*} \int \cot ^4(e+f x) \sqrt {a-a \sin ^2(e+f x)} \, dx &=\int \sqrt {a \cos ^2(e+f x)} \cot ^4(e+f x) \, dx\\ &=\left (\sqrt {a \cos ^2(e+f x)} \sec (e+f x)\right ) \int \cos (e+f x) \cot ^4(e+f x) \, dx\\ &=-\frac {\left (\sqrt {a \cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x^4} \, dx,x,-\sin (e+f x)\right )}{f}\\ &=-\frac {\left (\sqrt {a \cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}-\frac {2}{x^2}\right ) \, dx,x,-\sin (e+f x)\right )}{f}\\ &=\frac {2 \sqrt {a \cos ^2(e+f x)} \csc (e+f x) \sec (e+f x)}{f}-\frac {\sqrt {a \cos ^2(e+f x)} \csc ^3(e+f x) \sec (e+f x)}{3 f}+\frac {\sqrt {a \cos ^2(e+f x)} \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 47, normalized size = 0.52 \[ -\frac {\tan (e+f x) \left (\csc ^4(e+f x)-6 \csc ^2(e+f x)-3\right ) \sqrt {a \cos ^2(e+f x)}}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 66, normalized size = 0.73 \[ -\frac {{\left (3 \, \cos \left (f x + e\right )^{4} - 12 \, \cos \left (f x + e\right )^{2} + 8\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
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 = 0.90, size = 55, normalized size = 0.60 \[ \frac {\cos \left (f x +e \right ) a \left (3 \left (\sin ^{4}\left (f x +e \right )\right )+6 \left (\sin ^{2}\left (f x +e \right )\right )-1\right )}{3 \sin \left (f x +e \right )^{3} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 57, normalized size = 0.63 \[ \frac {8 \, \sqrt {a} \tan \left (f x + e\right )^{4} + 4 \, \sqrt {a} \tan \left (f x + e\right )^{2} - \sqrt {a}}{3 \, \sqrt {\tan \left (f x + e\right )^{2} + 1} f \tan \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 18.42, size = 364, normalized size = 4.00 \[ \frac {\left (\frac {1{}\mathrm {i}}{f}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{f}\right )\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,8{}\mathrm {i}}{f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}\,16{}\mathrm {i}}{3\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]
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 \[ \int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \cot ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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