Optimal. Leaf size=227 \[ \frac {55 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}-\frac {9 a^3 \cos (e+f x)}{40 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a+a \sin (e+f x)}}{24 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f} \]
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Rubi [A]
time = 0.42, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2797, 2726,
2725, 3123, 3054, 3060, 2852, 212} \begin {gather*} \frac {55 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{8 f}-\frac {9 a^3 \cos (e+f x)}{40 f \sqrt {a \sin (e+f x)+a}}-\frac {16 a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a \sin (e+f x)+a}}{24 f}-\frac {2 a \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a \sin (e+f x)+a)^{5/2}}{3 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a \sin (e+f x)+a)^{3/2}}{12 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2725
Rule 2726
Rule 2797
Rule 2852
Rule 3054
Rule 3060
Rule 3123
Rubi steps
\begin {align*} \int \cot ^4(e+f x) (a+a \sin (e+f x))^{5/2} \, dx &=\int (a+a \sin (e+f x))^{5/2} \, dx+\int \csc ^4(e+f x) (a+a \sin (e+f x))^{5/2} \left (1-2 \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac {\int \csc ^3(e+f x) \left (\frac {5 a}{2}-\frac {13}{2} a \sin (e+f x)\right ) (a+a \sin (e+f x))^{5/2} \, dx}{3 a}+\frac {1}{5} (8 a) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac {\int \csc ^2(e+f x) (a+a \sin (e+f x))^{3/2} \left (-\frac {17 a^2}{4}-\frac {57}{4} a^2 \sin (e+f x)\right ) \, dx}{6 a}+\frac {1}{15} \left (32 a^2\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {64 a^3 \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a+a \sin (e+f x)}}{24 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac {\int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \left (-\frac {165 a^3}{8}-\frac {97}{8} a^3 \sin (e+f x)\right ) \, dx}{6 a}\\ &=-\frac {9 a^3 \cos (e+f x)}{40 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a+a \sin (e+f x)}}{24 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}-\frac {1}{16} \left (55 a^2\right ) \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {9 a^3 \cos (e+f x)}{40 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a+a \sin (e+f x)}}{24 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}+\frac {\left (55 a^3\right ) \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 f}\\ &=\frac {55 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 f}-\frac {9 a^3 \cos (e+f x)}{40 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}+\frac {17 a^2 \cot (e+f x) \sqrt {a+a \sin (e+f x)}}{24 f}-\frac {2 a \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 f}-\frac {5 a \cot (e+f x) \csc (e+f x) (a+a \sin (e+f x))^{3/2}}{12 f}-\frac {\cot (e+f x) \csc ^2(e+f x) (a+a \sin (e+f x))^{5/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 1.16, size = 360, normalized size = 1.59 \begin {gather*} -\frac {a^2 \csc ^{10}\left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sin (e+f x))} \left (108 \cos \left (\frac {1}{2} (e+f x)\right )+706 \cos \left (\frac {3}{2} (e+f x)\right )-450 \cos \left (\frac {5}{2} (e+f x)\right )-156 \cos \left (\frac {7}{2} (e+f x)\right )+100 \cos \left (\frac {9}{2} (e+f x)\right )+12 \cos \left (\frac {11}{2} (e+f x)\right )-108 \sin \left (\frac {1}{2} (e+f x)\right )-2475 \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)+2475 \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)+706 \sin \left (\frac {3}{2} (e+f x)\right )+450 \sin \left (\frac {5}{2} (e+f x)\right )+825 \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))-825 \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))-156 \sin \left (\frac {7}{2} (e+f x)\right )-100 \sin \left (\frac {9}{2} (e+f x)\right )+12 \sin \left (\frac {11}{2} (e+f x)\right )\right )}{120 f \left (1+\cot \left (\frac {1}{2} (e+f x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.59, size = 222, normalized size = 0.98
method | result | size |
default | \(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (48 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} \left (\sin ^{3}\left (f x +e \right )\right ) \sqrt {a}-320 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \left (\sin ^{3}\left (f x +e \right )\right ) a^{\frac {3}{2}}+480 a^{\frac {5}{2}} \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin ^{3}\left (f x +e \right )\right )-825 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) a^{3}+135 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} \sqrt {a}-440 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+345 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {5}{2}}\right )}{120 \sin \left (f x +e \right )^{3} \sqrt {a}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 526 vs.
\(2 (211) = 422\).
time = 0.38, size = 526, normalized size = 2.32 \begin {gather*} \frac {825 \, {\left (a^{2} \cos \left (f x + e\right )^{4} - 2 \, a^{2} \cos \left (f x + e\right )^{2} + a^{2} - {\left (a^{2} \cos \left (f x + e\right )^{3} + a^{2} \cos \left (f x + e\right )^{2} - a^{2} \cos \left (f x + e\right ) - a^{2}\right )} \sin \left (f x + e\right )\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 (48 \, a^{2} \cos \left (f x + e\right )^{6} + 224 \, a^{2} \cos \left (f x + e\right )^{5} - 128 \, a^{2} \cos \left (f x + e\right )^{4} - 583 \, a^{2} \cos \left (f x + e\right )^{3} + 147 \, a^{2} \cos \left (f x + e\right )^{2} + 399 \, a^{2} \cos \left (f x + e\right ) - 27 \, a^{2} + {\left (48 \, a^{2} \cos \left (f x + e\right )^{5} - 176 \, a^{2} \cos \left (f x + e\right )^{4} - 304 \, a^{2} \cos \left (f x + e\right )^{3} + 279 \, a^{2} \cos \left (f x + e\right )^{2} + 426 \, a^{2} \cos \left (f x + e\right ) + 27 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{480 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} - {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2} - f \cos \left (f x + e\right ) - f\right )} \sin \left (f x + e\right ) + f\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 = 5.16, size = 306, normalized size = 1.35 \begin {gather*} \frac {\sqrt {2} {\left (768 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 2560 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 825 \, \sqrt {2} a^{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 ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 1920 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {20 \, {\left (108 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 69 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \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}}\right )} \sqrt {a}}{480 \, 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 = -0.00 \begin {gather*} \int {\mathrm {cot}\left (e+f\,x\right )}^4\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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