Optimal. Leaf size=55 \[ \frac {8 \tan (x)}{15 a^2 \sqrt {a \sec ^2(x)}}+\frac {4 \tan (x)}{15 a \left (a \sec ^2(x)\right )^{3/2}}+\frac {\tan (x)}{5 \left (a \sec ^2(x)\right )^{5/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4122, 192, 191} \[ \frac {8 \tan (x)}{15 a^2 \sqrt {a \sec ^2(x)}}+\frac {4 \tan (x)}{15 a \left (a \sec ^2(x)\right )^{3/2}}+\frac {\tan (x)}{5 \left (a \sec ^2(x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 4122
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sec ^2(x)\right )^{5/2}} \, dx &=a \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{7/2}} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{5 \left (a \sec ^2(x)\right )^{5/2}}+\frac {4}{5} \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\tan (x)\right )\\ &=\frac {\tan (x)}{5 \left (a \sec ^2(x)\right )^{5/2}}+\frac {4 \tan (x)}{15 a \left (a \sec ^2(x)\right )^{3/2}}+\frac {8 \operatorname {Subst}\left (\int \frac {1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (x)\right )}{15 a}\\ &=\frac {\tan (x)}{5 \left (a \sec ^2(x)\right )^{5/2}}+\frac {4 \tan (x)}{15 a \left (a \sec ^2(x)\right )^{3/2}}+\frac {8 \tan (x)}{15 a^2 \sqrt {a \sec ^2(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 36, normalized size = 0.65 \[ \frac {(150 \sin (x)+25 \sin (3 x)+3 \sin (5 x)) \cos (x) \sqrt {a \sec ^2(x)}}{240 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 32, normalized size = 0.58 \[ \frac {{\left (3 \, \cos \relax (x)^{5} + 4 \, \cos \relax (x)^{3} + 8 \, \cos \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}} \sin \relax (x)}{15 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 84, normalized size = 1.53 \[ \frac {2 \, {\left (15 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{4} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) - 40 \, {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{2} \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 48 \, \mathrm {sgn}\left (-\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )\right )}}{15 \, a^{\frac {5}{2}} {\left (\frac {1}{\tan \left (\frac {1}{2} \, x\right )} + \tan \left (\frac {1}{2} \, x\right )\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 31, normalized size = 0.56 \[ \frac {\sin \relax (x ) \left (3 \left (\cos ^{4}\relax (x )\right )+4 \left (\cos ^{2}\relax (x )\right )+8\right )}{15 \cos \relax (x )^{5} \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 22, normalized size = 0.40 \[ \frac {3 \, \sin \left (5 \, x\right ) + 25 \, \sin \left (3 \, x\right ) + 150 \, \sin \relax (x)}{240 \, a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (\frac {a}{{\cos \relax (x)}^2}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.57, size = 60, normalized size = 1.09 \[ \frac {8 \tan ^{5}{\relax (x )}}{15 a^{\frac {5}{2}} \left (\sec ^{2}{\relax (x )}\right )^{\frac {5}{2}}} + \frac {4 \tan ^{3}{\relax (x )}}{3 a^{\frac {5}{2}} \left (\sec ^{2}{\relax (x )}\right )^{\frac {5}{2}}} + \frac {\tan {\relax (x )}}{a^{\frac {5}{2}} \left (\sec ^{2}{\relax (x )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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