Optimal. Leaf size=163 \[ a^3 \sin (x) \cos (x) \sqrt {a \sec ^4(x)}+\frac {1}{13} a^3 \sin ^2(x) \tan ^{11}(x) \sqrt {a \sec ^4(x)}+\frac {6}{11} a^3 \sin ^2(x) \tan ^9(x) \sqrt {a \sec ^4(x)}+\frac {5}{3} a^3 \sin ^2(x) \tan ^7(x) \sqrt {a \sec ^4(x)}+\frac {20}{7} a^3 \sin ^2(x) \tan ^5(x) \sqrt {a \sec ^4(x)}+3 a^3 \sin ^2(x) \tan ^3(x) \sqrt {a \sec ^4(x)}+2 a^3 \sin ^2(x) \tan (x) \sqrt {a \sec ^4(x)} \]
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Rubi [A] time = 0.04, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4123, 3767} \[ a^3 \sin (x) \cos (x) \sqrt {a \sec ^4(x)}+\frac {1}{13} a^3 \sin ^2(x) \tan ^{11}(x) \sqrt {a \sec ^4(x)}+\frac {6}{11} a^3 \sin ^2(x) \tan ^9(x) \sqrt {a \sec ^4(x)}+\frac {5}{3} a^3 \sin ^2(x) \tan ^7(x) \sqrt {a \sec ^4(x)}+\frac {20}{7} a^3 \sin ^2(x) \tan ^5(x) \sqrt {a \sec ^4(x)}+3 a^3 \sin ^2(x) \tan ^3(x) \sqrt {a \sec ^4(x)}+2 a^3 \sin ^2(x) \tan (x) \sqrt {a \sec ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3767
Rule 4123
Rubi steps
\begin {align*} \int \left (a \sec ^4(x)\right )^{7/2} \, dx &=\left (a^3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \int \sec ^{14}(x) \, dx\\ &=-\left (\left (a^3 \cos ^2(x) \sqrt {a \sec ^4(x)}\right ) \operatorname {Subst}\left (\int \left (1+6 x^2+15 x^4+20 x^6+15 x^8+6 x^{10}+x^{12}\right ) \, dx,x,-\tan (x)\right )\right )\\ &=a^3 \cos (x) \sqrt {a \sec ^4(x)} \sin (x)+2 a^3 \sqrt {a \sec ^4(x)} \sin ^2(x) \tan (x)+3 a^3 \sqrt {a \sec ^4(x)} \sin ^2(x) \tan ^3(x)+\frac {20}{7} a^3 \sqrt {a \sec ^4(x)} \sin ^2(x) \tan ^5(x)+\frac {5}{3} a^3 \sqrt {a \sec ^4(x)} \sin ^2(x) \tan ^7(x)+\frac {6}{11} a^3 \sqrt {a \sec ^4(x)} \sin ^2(x) \tan ^9(x)+\frac {1}{13} a^3 \sqrt {a \sec ^4(x)} \sin ^2(x) \tan ^{11}(x)\\ \end {align*}
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Mathematica [A] time = 0.18, size = 54, normalized size = 0.33 \[ \frac {\sin (x) \cos (x) (2380 \cos (2 x)+1093 \cos (4 x)+378 \cos (6 x)+92 \cos (8 x)+14 \cos (10 x)+\cos (12 x)+2048) \left (a \sec ^4(x)\right )^{7/2}}{6006} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 76, normalized size = 0.47 \[ \frac {{\left (1024 \, a^{3} \cos \relax (x)^{12} + 512 \, a^{3} \cos \relax (x)^{10} + 384 \, a^{3} \cos \relax (x)^{8} + 320 \, a^{3} \cos \relax (x)^{6} + 280 \, a^{3} \cos \relax (x)^{4} + 252 \, a^{3} \cos \relax (x)^{2} + 231 \, a^{3}\right )} \sqrt {\frac {a}{\cos \relax (x)^{4}}} \sin \relax (x)}{3003 \, \cos \relax (x)^{11}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 67, normalized size = 0.41 \[ \frac {1}{3003} \, {\left (231 \, a^{3} \tan \relax (x)^{13} + 1638 \, a^{3} \tan \relax (x)^{11} + 5005 \, a^{3} \tan \relax (x)^{9} + 8580 \, a^{3} \tan \relax (x)^{7} + 9009 \, a^{3} \tan \relax (x)^{5} + 6006 \, a^{3} \tan \relax (x)^{3} + 3003 \, a^{3} \tan \relax (x)\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 53, normalized size = 0.33 \[ \frac {\left (1024 \left (\cos ^{12}\relax (x )\right )+512 \left (\cos ^{10}\relax (x )\right )+384 \left (\cos ^{8}\relax (x )\right )+320 \left (\cos ^{6}\relax (x )\right )+280 \left (\cos ^{4}\relax (x )\right )+252 \left (\cos ^{2}\relax (x )\right )+231\right ) \left (\frac {a}{\cos \relax (x )^{4}}\right )^{\frac {7}{2}} \sin \relax (x ) \cos \relax (x )}{3003} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 61, normalized size = 0.37 \[ \frac {1}{13} \, a^{\frac {7}{2}} \tan \relax (x)^{13} + \frac {6}{11} \, a^{\frac {7}{2}} \tan \relax (x)^{11} + \frac {5}{3} \, a^{\frac {7}{2}} \tan \relax (x)^{9} + \frac {20}{7} \, a^{\frac {7}{2}} \tan \relax (x)^{7} + 3 \, a^{\frac {7}{2}} \tan \relax (x)^{5} + 2 \, a^{\frac {7}{2}} \tan \relax (x)^{3} + a^{\frac {7}{2}} \tan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 589, normalized size = 3.61 \[ \text {result too large to display} \]
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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