Optimal. Leaf size=60 \[ \frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3255, 3286,
2686} \begin {gather*} \frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2686
Rule 3255
Rule 3286
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{\sqrt {a-a \sin ^2(e+f x)}} \, dx &=\int \frac {\cot ^4(e+f x)}{\sqrt {a \cos ^2(e+f x)}} \, dx\\ &=\frac {\cos (e+f x) \int \cot ^3(e+f x) \csc (e+f x) \, dx}{\sqrt {a \cos ^2(e+f x)}}\\ &=-\frac {\cos (e+f x) \text {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{f \sqrt {a \cos ^2(e+f x)}}\\ &=\frac {\cot (e+f x)}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 f \sqrt {a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 37, normalized size = 0.62 \begin {gather*} -\frac {\cot (e+f x) \left (-3+\csc ^2(e+f x)\right )}{3 f \sqrt {a \cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.82, size = 44, normalized size = 0.73
method | result | size |
default | \(\frac {\cos \left (f x +e \right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )-1\right )}{3 \sin \left (f x +e \right )^{3} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(44\) |
risch | \(\frac {2 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (3 \,{\mathrm e}^{4 i \left (f x +e \right )}-2 \,{\mathrm e}^{2 i \left (f x +e \right )}+3\right )}{3 \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 571 vs.
\(2 (59) = 118\).
time = 0.63, size = 571, normalized size = 9.52 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, \sin \left (5 \, f x + 5 \, e\right ) - 2 \, \sin \left (3 \, f x + 3 \, e\right ) + 3 \, \sin \left (f x + e\right )\right )} \cos \left (6 \, f x + 6 \, e\right ) + 9 \, {\left (\sin \left (4 \, f x + 4 \, e\right ) - \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (5 \, f x + 5 \, e\right ) + 3 \, {\left (2 \, \sin \left (3 \, f x + 3 \, e\right ) - 3 \, \sin \left (f x + e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - {\left (3 \, \cos \left (5 \, f x + 5 \, e\right ) - 2 \, \cos \left (3 \, f x + 3 \, e\right ) + 3 \, \cos \left (f x + e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (3 \, \cos \left (4 \, f x + 4 \, e\right ) - 3 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sin \left (5 \, f x + 5 \, e\right ) - 3 \, {\left (2 \, \cos \left (3 \, f x + 3 \, e\right ) - 3 \, \cos \left (f x + e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) - 2 \, {\left (3 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \sin \left (3 \, f x + 3 \, e\right ) + 6 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 9 \, \cos \left (f x + e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, \cos \left (2 \, f x + 2 \, e\right ) \sin \left (f x + e\right ) - 3 \, \sin \left (f x + e\right )\right )} \sqrt {a}}{3 \, {\left (a \cos \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a \cos \left (4 \, f x + 4 \, e\right )^{2} + 9 \, a \cos \left (2 \, f x + 2 \, e\right )^{2} + a \sin \left (6 \, f x + 6 \, e\right )^{2} + 9 \, a \sin \left (4 \, f x + 4 \, e\right )^{2} - 18 \, a \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a \sin \left (2 \, f x + 2 \, e\right )^{2} - 2 \, {\left (3 \, a \cos \left (4 \, f x + 4 \, e\right ) - 3 \, a \cos \left (2 \, f x + 2 \, e\right ) + a\right )} \cos \left (6 \, f x + 6 \, e\right ) - 6 \, {\left (3 \, a \cos \left (2 \, f x + 2 \, e\right ) - a\right )} \cos \left (4 \, f x + 4 \, e\right ) - 6 \, a \cos \left (2 \, f x + 2 \, e\right ) - 6 \, {\left (a \sin \left (4 \, f x + 4 \, e\right ) - a \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) + a\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 58, normalized size = 0.97 \begin {gather*} \frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (3 \, \cos \left (f x + e\right )^{2} - 2\right )}}{3 \, {\left (a f \cos \left (f x + e\right )^{3} - a f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \end {gather*}
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 \begin {gather*} \int \frac {\cot ^{4}{\left (e + f x \right )}}{\sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 99, normalized size = 1.65 \begin {gather*} \frac {\frac {\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )} - \frac {9 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, \sqrt {a} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 19.19, size = 118, normalized size = 1.97 \begin {gather*} \frac {4\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\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}\,\left (-{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,2{}\mathrm {i}+{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,3{}\mathrm {i}+3{}\mathrm {i}\right )}{3\,a\,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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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