Optimal. Leaf size=47 \[ \frac{20}{63 \sec ^{\frac{3}{2}}(x)}+\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{10 x \sin (x)}{21 \sqrt{\sec (x)}} \]
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Rubi [A] time = 0.0943382, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4187, 4189} \[ \frac{20}{63 \sec ^{\frac{3}{2}}(x)}+\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{10 x \sin (x)}{21 \sqrt{\sec (x)}} \]
Antiderivative was successfully verified.
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Rule 4187
Rule 4189
Rubi steps
\begin{align*} \int \left (\frac{x}{\sec ^{\frac{7}{2}}(x)}-\frac{5}{21} x \sqrt{\sec (x)}\right ) \, dx &=-\left (\frac{5}{21} \int x \sqrt{\sec (x)} \, dx\right )+\int \frac{x}{\sec ^{\frac{7}{2}}(x)} \, dx\\ &=\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{5}{7} \int \frac{x}{\sec ^{\frac{3}{2}}(x)} \, dx-\frac{1}{21} \left (5 \sqrt{\cos (x)} \sqrt{\sec (x)}\right ) \int \frac{x}{\sqrt{\cos (x)}} \, dx\\ &=\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{20}{63 \sec ^{\frac{3}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{10 x \sin (x)}{21 \sqrt{\sec (x)}}+\frac{5}{21} \int x \sqrt{\sec (x)} \, dx-\frac{1}{21} \left (5 \sqrt{\cos (x)} \sqrt{\sec (x)}\right ) \int \frac{x}{\sqrt{\cos (x)}} \, dx\\ &=\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{20}{63 \sec ^{\frac{3}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{10 x \sin (x)}{21 \sqrt{\sec (x)}}\\ \end{align*}
Mathematica [A] time = 0.1065, size = 45, normalized size = 0.96 \[ \sqrt{\sec (x)} \left (\frac{13}{42} x \sin (2 x)+\frac{1}{28} x \sin (4 x)+\frac{88}{441} \cos (2 x)+\frac{1}{98} \cos (4 x)+\frac{167}{882}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \sec \left ( x \right ) \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,x}{21}\sqrt{\sec \left ( x \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\sec \left (x\right )} + \frac{x}{\sec \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{5}{21} \, x \sqrt{\sec \left (x\right )} + \frac{x}{\sec \left (x\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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