Optimal. Leaf size=86 \[ \frac {5 x \sec ^2(x)}{16 a \sqrt {a \sec ^4(x)}}+\frac {5 \tan (x)}{16 a \sqrt {a \sec ^4(x)}}+\frac {\sin (x) \cos ^3(x)}{6 a \sqrt {a \sec ^4(x)}}+\frac {5 \sin (x) \cos (x)}{24 a \sqrt {a \sec ^4(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4123, 2635, 8} \[ \frac {5 x \sec ^2(x)}{16 a \sqrt {a \sec ^4(x)}}+\frac {5 \tan (x)}{16 a \sqrt {a \sec ^4(x)}}+\frac {\sin (x) \cos ^3(x)}{6 a \sqrt {a \sec ^4(x)}}+\frac {5 \sin (x) \cos (x)}{24 a \sqrt {a \sec ^4(x)}} \]
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 )^{3/2}} \, dx &=\frac {\sec ^2(x) \int \cos ^6(x) \, dx}{a \sqrt {a \sec ^4(x)}}\\ &=\frac {\cos ^3(x) \sin (x)}{6 a \sqrt {a \sec ^4(x)}}+\frac {\left (5 \sec ^2(x)\right ) \int \cos ^4(x) \, dx}{6 a \sqrt {a \sec ^4(x)}}\\ &=\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \sec ^4(x)}}+\frac {\cos ^3(x) \sin (x)}{6 a \sqrt {a \sec ^4(x)}}+\frac {\left (5 \sec ^2(x)\right ) \int \cos ^2(x) \, dx}{8 a \sqrt {a \sec ^4(x)}}\\ &=\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \sec ^4(x)}}+\frac {\cos ^3(x) \sin (x)}{6 a \sqrt {a \sec ^4(x)}}+\frac {5 \tan (x)}{16 a \sqrt {a \sec ^4(x)}}+\frac {\left (5 \sec ^2(x)\right ) \int 1 \, dx}{16 a \sqrt {a \sec ^4(x)}}\\ &=\frac {5 x \sec ^2(x)}{16 a \sqrt {a \sec ^4(x)}}+\frac {5 \cos (x) \sin (x)}{24 a \sqrt {a \sec ^4(x)}}+\frac {\cos ^3(x) \sin (x)}{6 a \sqrt {a \sec ^4(x)}}+\frac {5 \tan (x)}{16 a \sqrt {a \sec ^4(x)}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 38, normalized size = 0.44 \[ \frac {(60 x+45 \sin (2 x)+9 \sin (4 x)+\sin (6 x)) \sec ^6(x)}{192 \left (a \sec ^4(x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 43, normalized size = 0.50 \[ \frac {{\left (15 \, x \cos \relax (x)^{2} + {\left (8 \, \cos \relax (x)^{7} + 10 \, \cos \relax (x)^{5} + 15 \, \cos \relax (x)^{3}\right )} \sin \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}}}{48 \, a^{2}} \]
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 = 0.38, size = 41, normalized size = 0.48 \[ \frac {8 \sin \relax (x ) \left (\cos ^{5}\relax (x )\right )+10 \left (\cos ^{3}\relax (x )\right ) \sin \relax (x )+15 \cos \relax (x ) \sin \relax (x )+15 x}{48 \cos \relax (x )^{6} \left (\frac {a}{\cos \relax (x )^{4}}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 58, normalized size = 0.67 \[ \frac {15 \, \tan \relax (x)^{5} + 40 \, \tan \relax (x)^{3} + 33 \, \tan \relax (x)}{48 \, {\left (a^{\frac {3}{2}} \tan \relax (x)^{6} + 3 \, a^{\frac {3}{2}} \tan \relax (x)^{4} + 3 \, a^{\frac {3}{2}} \tan \relax (x)^{2} + a^{\frac {3}{2}}\right )}} + \frac {5 \, x}{16 \, a^{\frac {3}{2}}} \]
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 \[ \int \frac {1}{{\left (\frac {a}{{\cos \relax (x)}^4}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sec ^{4}{\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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