Optimal. Leaf size=50 \[ -\frac {2}{5} \sin ^2(x) \tan (x) \sqrt {\sin (x) \tan (x)}+\frac {16}{15} \tan (x) \sqrt {\sin (x) \tan (x)}+\frac {64}{15} \cot (x) \sqrt {\sin (x) \tan (x)} \]
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Rubi [A] time = 0.09, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4397, 4400, 2598, 2594, 2589} \[ -\frac {2}{5} \sin ^2(x) \tan (x) \sqrt {\sin (x) \tan (x)}+\frac {16}{15} \tan (x) \sqrt {\sin (x) \tan (x)}+\frac {64}{15} \cot (x) \sqrt {\sin (x) \tan (x)} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 2594
Rule 2598
Rule 4397
Rule 4400
Rubi steps
\begin {align*} \int (-\cos (x)+\sec (x))^{5/2} \, dx &=\int (\sin (x) \tan (x))^{5/2} \, dx\\ &=\frac {\sqrt {\sin (x) \tan (x)} \int \sin ^{\frac {5}{2}}(x) \tan ^{\frac {5}{2}}(x) \, dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=-\frac {2}{5} \sin ^2(x) \tan (x) \sqrt {\sin (x) \tan (x)}+\frac {\left (8 \sqrt {\sin (x) \tan (x)}\right ) \int \sqrt {\sin (x)} \tan ^{\frac {5}{2}}(x) \, dx}{5 \sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=\frac {16}{15} \tan (x) \sqrt {\sin (x) \tan (x)}-\frac {2}{5} \sin ^2(x) \tan (x) \sqrt {\sin (x) \tan (x)}-\frac {\left (32 \sqrt {\sin (x) \tan (x)}\right ) \int \sqrt {\sin (x)} \sqrt {\tan (x)} \, dx}{15 \sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=\frac {64}{15} \cot (x) \sqrt {\sin (x) \tan (x)}+\frac {16}{15} \tan (x) \sqrt {\sin (x) \tan (x)}-\frac {2}{5} \sin ^2(x) \tan (x) \sqrt {\sin (x) \tan (x)}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 29, normalized size = 0.58 \[ \frac {2}{15} \tan (x) \sqrt {\sin (x) \tan (x)} \left (3 \cos ^2(x)+32 \cot ^2(x)+5\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 38, normalized size = 0.76 \[ -\frac {2 \, {\left (3 \, \cos \relax (x)^{4} - 30 \, \cos \relax (x)^{2} - 5\right )} \sqrt {-\frac {\cos \relax (x)^{2} - 1}{\cos \relax (x)}}}{15 \, \cos \relax (x) \sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-\cos \relax (x) + \sec \relax (x)\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 321, normalized size = 6.42 \[ -\frac {\left (-1+\cos \relax (x )\right )^{2} \left (6 \left (\cos ^{4}\relax (x )\right )-15 \left (\cos ^{2}\relax (x )\right ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}\, \ln \left (-\frac {2 \left (\cos ^{2}\relax (x )\right ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-\left (\cos ^{2}\relax (x )\right )+2 \cos \relax (x )-2 \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-1}{\sin \relax (x )^{2}}\right )+15 \left (\cos ^{2}\relax (x )\right ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}\, \ln \left (-\frac {2 \left (2 \left (\cos ^{2}\relax (x )\right ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-\left (\cos ^{2}\relax (x )\right )+2 \cos \relax (x )-2 \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-1\right )}{\sin \relax (x )^{2}}\right )-15 \cos \relax (x ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}\, \ln \left (-\frac {2 \left (\cos ^{2}\relax (x )\right ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-\left (\cos ^{2}\relax (x )\right )+2 \cos \relax (x )-2 \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-1}{\sin \relax (x )^{2}}\right )+15 \cos \relax (x ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}\, \ln \left (-\frac {2 \left (2 \left (\cos ^{2}\relax (x )\right ) \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-\left (\cos ^{2}\relax (x )\right )+2 \cos \relax (x )-2 \sqrt {-\frac {\cos \relax (x )}{\left (1+\cos \relax (x )\right )^{2}}}-1\right )}{\sin \relax (x )^{2}}\right )-60 \left (\cos ^{2}\relax (x )\right )-10\right ) \cos \relax (x ) \left (1+\cos \relax (x )\right )^{2} \left (-\frac {-1+\cos ^{2}\relax (x )}{\cos \relax (x )}\right )^{\frac {5}{2}}}{15 \sin \relax (x )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 82, normalized size = 1.64 \[ -\frac {32 \, {\left (\frac {5 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {5 \, \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {2 \, \sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}} - 2\right )}}{15 \, {\left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}^{\frac {5}{2}} {\left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (\frac {1}{\cos \relax (x)}-\cos \relax (x)\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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