Optimal. Leaf size=73 \[ -\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {8}{7} \sin (x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {256}{35} \csc (x) \sqrt {\sin (x) \tan (x)}+\frac {64}{35} \tan (x) \sec (x) \sqrt {\sin (x) \tan (x)} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {8}{7} \sin (x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {256}{35} \csc (x) \sqrt {\sin (x) \tan (x)}+\frac {64}{35} \tan (x) \sec (x) \sqrt {\sin (x) \tan (x)} \]
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))^{7/2} \, dx &=\int (\sin (x) \tan (x))^{7/2} \, dx\\ &=\frac {\sqrt {\sin (x) \tan (x)} \int \sin ^{\frac {7}{2}}(x) \tan ^{\frac {7}{2}}(x) \, dx}{\sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=-\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}+\frac {\left (12 \sqrt {\sin (x) \tan (x)}\right ) \int \sin ^{\frac {3}{2}}(x) \tan ^{\frac {7}{2}}(x) \, dx}{7 \sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=-\frac {8}{7} \sin (x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}+\frac {\left (32 \sqrt {\sin (x) \tan (x)}\right ) \int \frac {\tan ^{\frac {7}{2}}(x)}{\sqrt {\sin (x)}} \, dx}{7 \sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=\frac {64}{35} \sec (x) \tan (x) \sqrt {\sin (x) \tan (x)}-\frac {8}{7} \sin (x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {\left (128 \sqrt {\sin (x) \tan (x)}\right ) \int \frac {\tan ^{\frac {3}{2}}(x)}{\sqrt {\sin (x)}} \, dx}{35 \sqrt {\sin (x)} \sqrt {\tan (x)}}\\ &=-\frac {256}{35} \csc (x) \sqrt {\sin (x) \tan (x)}+\frac {64}{35} \sec (x) \tan (x) \sqrt {\sin (x) \tan (x)}-\frac {8}{7} \sin (x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}-\frac {2}{7} \sin ^3(x) \tan ^2(x) \sqrt {\sin (x) \tan (x)}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 37, normalized size = 0.51 \[ \frac {1}{70} \sec (x) \sqrt {\sin (x) \tan (x)} (28 \tan (x)-512 \cot (x)-5 (\sin (3 x)-23 \sin (x)) \cos (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 44, normalized size = 0.60 \[ \frac {2 \, {\left (5 \, \cos \relax (x)^{6} - 35 \, \cos \relax (x)^{4} - 105 \, \cos \relax (x)^{2} + 7\right )} \sqrt {-\frac {\cos \relax (x)^{2} - 1}{\cos \relax (x)}}}{35 \, \cos \relax (x)^{2} \sin \relax (x)} \]
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 \[ \int {\left (-\cos \relax (x) + \sec \relax (x)\right )}^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 603, normalized size = 8.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 82, normalized size = 1.12 \[ \frac {128 \, {\left (\frac {7 \, \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} - \frac {7 \, \sin \relax (x)^{10}}{{\left (\cos \relax (x) + 1\right )}^{10}} + \frac {2 \, \sin \relax (x)^{14}}{{\left (\cos \relax (x) + 1\right )}^{14}} - 2\right )}}{35 \, {\left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right )}^{\frac {7}{2}} {\left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\cos \relax (x)}-\cos \relax (x)\right )}^{7/2} \,d x \]
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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