Optimal. Leaf size=62 \[ \frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16}{27} \sqrt {\cos (x)} F\left (\left .\frac {x}{2}\right |2\right ) \sqrt {\sec (x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}} \]
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Rubi [A]
time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4273, 4274,
3854, 3856, 2720} \begin {gather*} \frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}+\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}-\frac {16}{27} \sqrt {\cos (x)} \sqrt {\sec (x)} F\left (\left .\frac {x}{2}\right |2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3854
Rule 3856
Rule 4273
Rule 4274
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\sec ^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\sec (x)}\right ) \, dx &=-\left (\frac {1}{3} \int x^2 \sqrt {\sec (x)} \, dx\right )+\int \frac {x^2}{\sec ^{\frac {3}{2}}(x)} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}+\frac {1}{3} \int x^2 \sqrt {\sec (x)} \, dx-\frac {8}{9} \int \frac {1}{\sec ^{\frac {3}{2}}(x)} \, dx-\frac {1}{3} \left (\sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int \frac {x^2}{\sqrt {\cos (x)}} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}-\frac {8}{27} \int \sqrt {\sec (x)} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}-\frac {1}{27} \left (8 \sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int \frac {1}{\sqrt {\cos (x)}} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16}{27} \sqrt {\cos (x)} F\left (\left .\frac {x}{2}\right |2\right ) \sqrt {\sec (x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.82 \begin {gather*} \frac {1}{27} \sqrt {\sec (x)} \left (12 x+12 x \cos (2 x)-16 \sqrt {\cos (x)} F\left (\left .\frac {x}{2}\right |2\right )-8 \sin (2 x)+9 x^2 \sin (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sec \left (x \right )^{\frac {3}{2}}}-\frac {x^{2} \left (\sqrt {\sec }\left (x \right )\right )}{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} - \frac {\int \left (- \frac {3 x^{2}}{\sec ^{\frac {3}{2}}{\left (x \right )}}\right )\, dx + \int x^{2} \sqrt {\sec {\left (x \right )}}\, dx}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} -\int \frac {x^2\,\sqrt {\frac {1}{\cos \left (x\right )}}}{3}-\frac {x^2}{{\left (\frac {1}{\cos \left (x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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