Optimal. Leaf size=115 \[ -\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 115, 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, 276} \begin {gather*} -\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}}+\frac {2 \cot (e+f x) \csc ^4(e+f x)}{5 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 {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2686
Rule 3255
Rule 3286
Rubi steps
\begin {align*} \int \frac {\cot ^8(e+f x)}{\left (a-a \sin ^2(e+f x)\right )^{3/2}} \, dx &=\int \frac {\cot ^8(e+f x)}{\left (a \cos ^2(e+f x)\right )^{3/2}} \, dx\\ &=\frac {\cos (e+f x) \int \cot ^5(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 )^2 \, 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^2-2 x^4+x^6\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 {2 \cot (e+f x) \csc ^4(e+f x)}{5 a f \sqrt {a \cos ^2(e+f x)}}-\frac {\cot (e+f x) \csc ^6(e+f x)}{7 a f \sqrt {a \cos ^2(e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 51, normalized size = 0.44 \begin {gather*} -\frac {\cot ^3(e+f x) \left (35-42 \csc ^2(e+f x)+15 \csc ^4(e+f x)\right )}{105 f \left (a \cos ^2(e+f x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.52, size = 57, normalized size = 0.50
method | result | size |
default | \(-\frac {\cos \left (f x +e \right ) \left (35 \left (\cos ^{4}\left (f x +e \right )\right )-28 \left (\cos ^{2}\left (f x +e \right )\right )+8\right )}{105 a \sin \left (f x +e \right )^{7} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(57\) |
risch | \(\frac {8 i \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) \left (35 \,{\mathrm e}^{10 i \left (f x +e \right )}+28 \,{\mathrm e}^{8 i \left (f x +e \right )}+114 \,{\mathrm e}^{6 i \left (f x +e \right )}+28 \,{\mathrm e}^{4 i \left (f x +e \right )}+35 \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{105 \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{7} f \sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, a}\) | \(116\) |
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. 2187 vs.
\(2 (112) = 224\).
time = 0.59, size = 2187, normalized size = 19.02 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 100, normalized size = 0.87 \begin {gather*} \frac {{\left (35 \, \cos \left (f x + e\right )^{4} - 28 \, \cos \left (f x + e\right )^{2} + 8\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{105 \, {\left (a^{2} f \cos \left (f x + e\right )^{7} - 3 \, a^{2} f \cos \left (f x + e\right )^{5} + 3 \, 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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 184, normalized size = 1.60 \begin {gather*} \frac {\frac {525 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 63 \, \sqrt {a} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 15 \, \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 )^{7}} + \frac {15 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 63 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 35 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 525 \, a^{\frac {25}{2}} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{14} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}}{13440 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 33.01, size = 589, normalized size = 5.12 \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}\,464{}\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}\,3072{}\mathrm {i}}{35\,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}\,4736{}\mathrm {i}}{35\,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 )}+\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}\,768{}\mathrm {i}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^6\,\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}\,256{}\mathrm {i}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}-1\right )}^7\,\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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