Optimal. Leaf size=144 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^{3/2} f}-\frac {\cot (e+f x)}{8 a f \sqrt {a+a \sin (e+f x)}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f} \]
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Rubi [A]
time = 0.36, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2796, 2851,
2852, 212, 3123, 3059} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{8 a^{3/2} f}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a \sin (e+f x)+a}}{3 a^2 f}-\frac {\cot (e+f x)}{8 a f \sqrt {a \sin (e+f x)+a}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2796
Rule 2851
Rule 2852
Rule 3059
Rule 3123
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac {\int \csc ^4(e+f x) \sqrt {a+a \sin (e+f x)} \left (1+\sin ^2(e+f x)\right ) \, dx}{a^2}-\frac {2 \int \csc ^3(e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{a^2}\\ &=\frac {\cot (e+f x) \csc (e+f x)}{a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f}+\frac {\int \csc ^3(e+f x) \sqrt {a+a \sin (e+f x)} \left (\frac {a}{2}+\frac {9}{2} a \sin (e+f x)\right ) \, dx}{3 a^3}-\frac {3 \int \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{2 a^2}\\ &=\frac {3 \cot (e+f x)}{2 a f \sqrt {a+a \sin (e+f x)}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f}-\frac {3 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{4 a^2}+\frac {13 \int \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{8 a^2}\\ &=-\frac {\cot (e+f x)}{8 a f \sqrt {a+a \sin (e+f x)}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f}+\frac {13 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{16 a^2}+\frac {3 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a f}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a^{3/2} f}-\frac {\cot (e+f x)}{8 a f \sqrt {a+a \sin (e+f x)}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f}-\frac {13 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a f}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^{3/2} f}-\frac {\cot (e+f x)}{8 a f \sqrt {a+a \sin (e+f x)}}+\frac {11 \cot (e+f x) \csc (e+f x)}{12 a f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x) \sqrt {a+a \sin (e+f x)}}{3 a^2 f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(294\) vs. \(2(144)=288\).
time = 0.52, size = 294, normalized size = 2.04 \begin {gather*} \frac {\csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (-132 \cos \left (\frac {1}{2} (e+f x)\right )+62 \cos \left (\frac {3}{2} (e+f x)\right )+6 \cos \left (\frac {5}{2} (e+f x)\right )+132 \sin \left (\frac {1}{2} (e+f x)\right )-9 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+9 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+62 \sin \left (\frac {3}{2} (e+f x)\right )-6 \sin \left (\frac {5}{2} (e+f x)\right )+3 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-3 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{24 f \left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3 (a (1+\sin (e+f x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.24, size = 144, normalized size = 1.00
method | result | size |
default | \(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (3 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+3 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (f x +e \right )\right )+8 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-3 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 a^{\frac {11}{2}} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 418 vs.
\(2 (134) = 268\).
time = 0.37, size = 418, normalized size = 2.90 \begin {gather*} \frac {3 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, {\left (3 \, \cos \left (f x + e\right )^{3} + 17 \, \cos \left (f x + e\right )^{2} - {\left (3 \, \cos \left (f x + e\right )^{2} - 14 \, \cos \left (f x + e\right ) - 25\right )} \sin \left (f x + e\right ) - 11 \, \cos \left (f x + e\right ) - 25\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{96 \, {\left (a^{2} f \cos \left (f x + e\right )^{4} - 2 \, a^{2} f \cos \left (f x + e\right )^{2} + a^{2} f - {\left (a^{2} f \cos \left (f x + e\right )^{3} + a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - a^{2} f\right )} \sin \left (f x + e\right )\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 )}}{\left (a \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 [A]
time = 9.80, size = 176, normalized size = 1.22 \begin {gather*} -\frac {\sqrt {2} \sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {4 \, {\left (12 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 16 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{96 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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