Optimal. Leaf size=132 \[ \frac {63 x \sec ^2(x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {63 \tan (x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {\sin (x) \cos ^7(x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \sin (x) \cos ^5(x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \sin (x) \cos ^3(x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \sin (x) \cos (x)}{128 a^2 \sqrt {a \sec ^4(x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4123, 2635, 8} \[ \frac {63 x \sec ^2(x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {63 \tan (x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {\sin (x) \cos ^7(x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \sin (x) \cos ^5(x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \sin (x) \cos ^3(x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \sin (x) \cos (x)}{128 a^2 \sqrt {a \sec ^4(x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 4123
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sec ^4(x)\right )^{5/2}} \, dx &=\frac {\sec ^2(x) \int \cos ^{10}(x) \, dx}{a^2 \sqrt {a \sec ^4(x)}}\\ &=\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {\left (9 \sec ^2(x)\right ) \int \cos ^8(x) \, dx}{10 a^2 \sqrt {a \sec ^4(x)}}\\ &=\frac {9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {\left (63 \sec ^2(x)\right ) \int \cos ^6(x) \, dx}{80 a^2 \sqrt {a \sec ^4(x)}}\\ &=\frac {21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {\left (21 \sec ^2(x)\right ) \int \cos ^4(x) \, dx}{32 a^2 \sqrt {a \sec ^4(x)}}\\ &=\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {\left (63 \sec ^2(x)\right ) \int \cos ^2(x) \, dx}{128 a^2 \sqrt {a \sec ^4(x)}}\\ &=\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {63 \tan (x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {\left (63 \sec ^2(x)\right ) \int 1 \, dx}{256 a^2 \sqrt {a \sec ^4(x)}}\\ &=\frac {63 x \sec ^2(x)}{256 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \cos (x) \sin (x)}{128 a^2 \sqrt {a \sec ^4(x)}}+\frac {21 \cos ^3(x) \sin (x)}{160 a^2 \sqrt {a \sec ^4(x)}}+\frac {9 \cos ^5(x) \sin (x)}{80 a^2 \sqrt {a \sec ^4(x)}}+\frac {\cos ^7(x) \sin (x)}{10 a^2 \sqrt {a \sec ^4(x)}}+\frac {63 \tan (x)}{256 a^2 \sqrt {a \sec ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 55, normalized size = 0.42 \[ \frac {(2520 x+2100 \sin (2 x)+600 \sin (4 x)+150 \sin (6 x)+25 \sin (8 x)+2 \sin (10 x)) \cos ^2(x) \sqrt {a \sec ^4(x)}}{10240 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 55, normalized size = 0.42 \[ \frac {{\left (315 \, x \cos \relax (x)^{2} + {\left (128 \, \cos \relax (x)^{11} + 144 \, \cos \relax (x)^{9} + 168 \, \cos \relax (x)^{7} + 210 \, \cos \relax (x)^{5} + 315 \, \cos \relax (x)^{3}\right )} \sin \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}}}{1280 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 57, normalized size = 0.43 \[ \frac {128 \sin \relax (x ) \left (\cos ^{9}\relax (x )\right )+144 \sin \relax (x ) \left (\cos ^{7}\relax (x )\right )+168 \sin \relax (x ) \left (\cos ^{5}\relax (x )\right )+210 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x )+315 \cos \relax (x ) \sin \relax (x )+315 x}{1280 \cos \relax (x )^{10} \left (\frac {a}{\cos \relax (x )^{4}}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 88, normalized size = 0.67 \[ \frac {315 \, \tan \relax (x)^{9} + 1470 \, \tan \relax (x)^{7} + 2688 \, \tan \relax (x)^{5} + 2370 \, \tan \relax (x)^{3} + 965 \, \tan \relax (x)}{1280 \, {\left (a^{\frac {5}{2}} \tan \relax (x)^{10} + 5 \, a^{\frac {5}{2}} \tan \relax (x)^{8} + 10 \, a^{\frac {5}{2}} \tan \relax (x)^{6} + 10 \, a^{\frac {5}{2}} \tan \relax (x)^{4} + 5 \, a^{\frac {5}{2}} \tan \relax (x)^{2} + a^{\frac {5}{2}}\right )}} + \frac {63 \, x}{256 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (\frac {a}{{\cos \relax (x)}^4}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sec ^{4}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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