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 = 0.0508577, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 4123
Rule 2635
Rule 8
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*}
Mathematica [A] time = 0.0824578, 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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Maple [A] time = 0.221, size = 57, normalized size = 0.4 \begin{align*}{\frac{128\, \left ( \cos \left ( x \right ) \right ) ^{9}\sin \left ( x \right ) +144\, \left ( \cos \left ( x \right ) \right ) ^{7}\sin \left ( x \right ) +168\, \left ( \cos \left ( x \right ) \right ) ^{5}\sin \left ( x \right ) +210\, \left ( \cos \left ( x \right ) \right ) ^{3}\sin \left ( x \right ) +315\,\cos \left ( x \right ) \sin \left ( x \right ) +315\,x}{1280\, \left ( \cos \left ( x \right ) \right ) ^{10}} \left ({\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{4}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.71142, size = 119, normalized size = 0.9 \begin{align*} \frac{315 \, \tan \left (x\right )^{9} + 1470 \, \tan \left (x\right )^{7} + 2688 \, \tan \left (x\right )^{5} + 2370 \, \tan \left (x\right )^{3} + 965 \, \tan \left (x\right )}{1280 \,{\left (a^{\frac{5}{2}} \tan \left (x\right )^{10} + 5 \, a^{\frac{5}{2}} \tan \left (x\right )^{8} + 10 \, a^{\frac{5}{2}} \tan \left (x\right )^{6} + 10 \, a^{\frac{5}{2}} \tan \left (x\right )^{4} + 5 \, a^{\frac{5}{2}} \tan \left (x\right )^{2} + a^{\frac{5}{2}}\right )}} + \frac{63 \, x}{256 \, a^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45152, size = 177, normalized size = 1.34 \begin{align*} \frac{{\left (315 \, x \cos \left (x\right )^{2} +{\left (128 \, \cos \left (x\right )^{11} + 144 \, \cos \left (x\right )^{9} + 168 \, \cos \left (x\right )^{7} + 210 \, \cos \left (x\right )^{5} + 315 \, \cos \left (x\right )^{3}\right )} \sin \left (x\right )\right )} \sqrt{\frac{a}{\cos \left (x\right )^{4}}}}{1280 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sec ^{4}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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