Optimal. Leaf size=24 \[ \frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4187, 4189} \[ \frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin {align*} \int \left (\frac {x}{\sec ^{\frac {3}{2}}(x)}-\frac {1}{3} x \sqrt {\sec (x)}\right ) \, dx &=-\left (\frac {1}{3} \int x \sqrt {\sec (x)} \, dx\right )+\int \frac {x}{\sec ^{\frac {3}{2}}(x)} \, dx\\ &=\frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}}+\frac {1}{3} \int x \sqrt {\sec (x)} \, dx-\frac {1}{3} \left (\sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int \frac {x}{\sqrt {\cos (x)}} \, dx\\ &=\frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 17, normalized size = 0.71 \[ \frac {2 (3 x \tan (x)+2)}{9 \sec ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int -\frac {1}{3} \, x \sqrt {\sec \relax (x)} + \frac {x}{\sec \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sec \relax (x )^{\frac {3}{2}}}-\frac {x \left (\sqrt {\sec }\relax (x )\right )}{3}\, dx \]
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 \[ \int -\frac {1}{3} \, x \sqrt {\sec \relax (x)} + \frac {x}{\sec \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {x\,\sqrt {\frac {1}{\cos \relax (x)}}}{3}-\frac {x}{{\left (\frac {1}{\cos \relax (x)}\right )}^{3/2}} \,d x \]
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 \[ - \frac {\int \left (- \frac {3 x}{\sec ^{\frac {3}{2}}{\relax (x )}}\right )\, dx + \int x \sqrt {\sec {\relax (x )}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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