Optimal. Leaf size=46 \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{2} a \tan (x) \sqrt {a \sec ^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 206} \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{2} a \tan (x) \sqrt {a \sec ^2(x)} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 4122
Rubi steps
\begin {align*} \int \left (a \sec ^2(x)\right )^{3/2} \, dx &=a \operatorname {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x)+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tan (x)}{\sqrt {a \sec ^2(x)}}\right )\\ &=\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (x)}{\sqrt {a \sec ^2(x)}}\right )+\frac {1}{2} a \sqrt {a \sec ^2(x)} \tan (x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 55, normalized size = 1.20 \[ \frac {1}{2} a \cos (x) \sqrt {a \sec ^2(x)} \left (\tan (x) \sec (x)-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 39, normalized size = 0.85 \[ -\frac {{\left (a \cos \relax (x)^{2} \log \left (-\frac {\sin \relax (x) - 1}{\sin \relax (x) + 1}\right ) - 2 \, a \sin \relax (x)\right )} \sqrt {\frac {a}{\cos \relax (x)^{2}}}}{4 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.60, size = 42, normalized size = 0.91 \[ \frac {1}{4} \, {\left (\log \left (\sin \relax (x) + 1\right ) \mathrm {sgn}\left (\cos \relax (x)\right ) - \log \left (-\sin \relax (x) + 1\right ) \mathrm {sgn}\left (\cos \relax (x)\right ) - \frac {2 \, \mathrm {sgn}\left (\cos \relax (x)\right ) \sin \relax (x)}{\sin \relax (x)^{2} - 1}\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 57, normalized size = 1.24 \[ -\frac {\left (\left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-1+\cos \relax (x )+\sin \relax (x )}{\sin \relax (x )}\right )-\left (\cos ^{2}\relax (x )\right ) \ln \left (-\frac {-\sin \relax (x )-1+\cos \relax (x )}{\sin \relax (x )}\right )-\sin \relax (x )\right ) \cos \relax (x ) \left (\frac {a}{\cos \relax (x )^{2}}\right )^{\frac {3}{2}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 324, normalized size = 7.04 \[ -\frac {{\left (8 \, a \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, a \cos \relax (x) \sin \left (2 \, x\right ) + 8 \, a \cos \left (2 \, x\right ) \sin \relax (x) - 4 \, {\left (a \sin \left (3 \, x\right ) - a \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + {\left (a \cos \left (4 \, x\right )^{2} + 4 \, a \cos \left (2 \, x\right )^{2} + a \sin \left (4 \, x\right )^{2} + 4 \, a \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, a \sin \left (2 \, x\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \cos \left (4 \, x\right ) + 4 \, a \cos \left (2 \, x\right ) + a\right )} \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) + 4 \, {\left (a \cos \left (3 \, x\right ) - a \cos \relax (x)\right )} \sin \left (4 \, x\right ) - 4 \, {\left (2 \, a \cos \left (2 \, x\right ) + a\right )} \sin \left (3 \, x\right ) + 4 \, a \sin \relax (x)\right )} \sqrt {a}}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) + 1\right )} \cos \left (4 \, x\right ) + \cos \left (4 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right )^{2} + \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) + 1\right )}} \]
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 \[ \int {\left (\frac {a}{{\cos \relax (x)}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \sec ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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