Optimal. Leaf size=77 \[ \frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 77, 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 = {3255, 3286,
2686, 14} \begin {gather*} \frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2686
Rule 3255
Rule 3286
Rubi steps
\begin {align*} \int \frac {\cot ^6(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\cot ^6(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\cos (e+f x) \int \cot ^3(e+f x) \csc ^3(e+f x) \, dx}{a \sqrt {a \cos ^2(e+f x)}}\\ &=-\frac {\cos (e+f x) \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\csc (e+f x)\right )}{a f \sqrt {a \cos ^2(e+f x)}}\\ &=-\frac {\cos (e+f x) \text {Subst}\left (\int \left (-x^2+x^4\right ) \, 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)}}-\frac {\cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 41, normalized size = 0.53 \begin {gather*} -\frac {\cot ^3(e+f x) \left (-5+3 \csc ^2(e+f x)\right )}{15 f \left (a \cos ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.49, size = 67, normalized size = 0.87
method | result | size |
default | \(-\frac {\cos \left (f x +e \right ) \left (5 \left (\cos ^{2}\left (f x +e \right )\right )-2\right )}{15 \left (1+\cos \left (f x +e \right )\right )^{2} \left (-1+\cos \left (f x +e \right )\right )^{2} a \sin \left (f x +e \right ) \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(67\) |
risch | \(-\frac {8 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (5 \,{\mathrm e}^{6 i \left (f x +e \right )}+2 \,{\mathrm e}^{4 i \left (f x +e \right )}+5 \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{15 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{5} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(94\) |
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. 1146 vs.
\(2 (75) = 150\).
time = 0.69, size = 1146, normalized size = 14.88 \begin {gather*} \frac {8 \, {\left ({\left (5 \, \sin \left (7 \, f x + 7 \, e\right ) + 2 \, \sin \left (5 \, f x + 5 \, e\right ) + 5 \, \sin \left (3 \, f x + 3 \, e\right )\right )} \cos \left (10 \, f x + 10 \, e\right ) - 5 \, {\left (5 \, \sin \left (7 \, f x + 7 \, e\right ) + 2 \, \sin \left (5 \, f x + 5 \, e\right ) + 5 \, \sin \left (3 \, f x + 3 \, e\right )\right )} \cos \left (8 \, f x + 8 \, e\right ) - 25 \, {\left (2 \, \sin \left (6 \, f x + 6 \, e\right ) - 2 \, \sin \left (4 \, f x + 4 \, e\right ) + \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (7 \, f x + 7 \, e\right ) + 10 \, {\left (2 \, \sin \left (5 \, f x + 5 \, e\right ) + 5 \, \sin \left (3 \, f x + 3 \, e\right )\right )} \cos \left (6 \, f x + 6 \, e\right ) + 10 \, {\left (2 \, \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 ) - {\left (5 \, \cos \left (7 \, f x + 7 \, e\right ) + 2 \, \cos \left (5 \, f x + 5 \, e\right ) + 5 \, \cos \left (3 \, f x + 3 \, e\right )\right )} \sin \left (10 \, f x + 10 \, e\right ) + 5 \, {\left (5 \, \cos \left (7 \, f x + 7 \, e\right ) + 2 \, \cos \left (5 \, f x + 5 \, e\right ) + 5 \, \cos \left (3 \, f x + 3 \, e\right )\right )} \sin \left (8 \, f x + 8 \, e\right ) + 5 \, {\left (10 \, \cos \left (6 \, f x + 6 \, e\right ) - 10 \, \cos \left (4 \, f x + 4 \, e\right ) + 5 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \sin \left (7 \, f x + 7 \, e\right ) - 10 \, {\left (2 \, \cos \left (5 \, f x + 5 \, e\right ) + 5 \, \cos \left (3 \, f x + 3 \, e\right )\right )} \sin \left (6 \, f x + 6 \, e\right ) - 2 \, {\left (10 \, \cos \left (4 \, f x + 4 \, e\right ) - 5 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \sin \left (5 \, f x + 5 \, e\right ) + 50 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (4 \, f x + 4 \, e\right ) + 5 \, {\left (5 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \sin \left (3 \, f x + 3 \, e\right ) - 50 \, \cos \left (4 \, f x + 4 \, e\right ) \sin \left (3 \, f x + 3 \, e\right ) - 25 \, \cos \left (3 \, f x + 3 \, e\right ) \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a}}{15 \, {\left (a^{2} \cos \left (10 \, f x + 10 \, e\right )^{2} + 25 \, a^{2} \cos \left (8 \, f x + 8 \, e\right )^{2} + 100 \, a^{2} \cos \left (6 \, f x + 6 \, e\right )^{2} + 100 \, a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 25 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (10 \, f x + 10 \, e\right )^{2} + 25 \, a^{2} \sin \left (8 \, f x + 8 \, e\right )^{2} + 100 \, a^{2} \sin \left (6 \, f x + 6 \, e\right )^{2} + 100 \, a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 100 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 25 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 10 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} - 2 \, {\left (5 \, a^{2} \cos \left (8 \, f x + 8 \, e\right ) - 10 \, a^{2} \cos \left (6 \, f x + 6 \, e\right ) + 10 \, a^{2} \cos \left (4 \, f x + 4 \, e\right ) - 5 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (10 \, f x + 10 \, e\right ) - 10 \, {\left (10 \, a^{2} \cos \left (6 \, f x + 6 \, e\right ) - 10 \, a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 5 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2}\right )} \cos \left (8 \, f x + 8 \, e\right ) - 20 \, {\left (10 \, a^{2} \cos \left (4 \, f x + 4 \, e\right ) - 5 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (6 \, f x + 6 \, e\right ) - 20 \, {\left (5 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right ) - 10 \, {\left (a^{2} \sin \left (8 \, f x + 8 \, e\right ) - 2 \, a^{2} \sin \left (6 \, f x + 6 \, e\right ) + 2 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) - a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (10 \, f x + 10 \, e\right ) - 50 \, {\left (2 \, a^{2} \sin \left (6 \, f x + 6 \, e\right ) - 2 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) + a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \sin \left (8 \, f x + 8 \, e\right ) - 100 \, {\left (2 \, 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 75, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {a \cos \left (f x + e\right )^{2}} {\left (5 \, \cos \left (f x + e\right )^{2} - 2\right )}}{15 \, {\left (a^{2} f \cos \left (f x + e\right )^{5} - 2 \, 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 )} \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 ^{6}{\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 151 vs.
\(2 (75) = 150\).
time = 0.93, size = 151, normalized size = 1.96 \begin {gather*} -\frac {\frac {30 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 5 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, \sqrt {a}}{a^{2} \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 )^{5}} - \frac {3 \, a^{\frac {17}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 5 \, a^{\frac {17}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 30 \, a^{\frac {17}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{10} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}}{480 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 20.61, size = 393, normalized size = 5.10 \begin {gather*} -\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\,a^2\,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}\,272{}\mathrm {i}}{15\,a^2\,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 )}-\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}\,128{}\mathrm {i}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^4\,\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}\,64{}\mathrm {i}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^5\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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