3.96 \(\int (\frac{x}{\sec ^{\frac{7}{2}}(x)}-\frac{5}{21} x \sqrt{\sec (x)}) \, dx\)

Optimal. Leaf size=47 \[ \frac{20}{63 \sec ^{\frac{3}{2}}(x)}+\frac{4}{49 \sec ^{\frac{7}{2}}(x)}+\frac{2 x \sin (x)}{7 \sec ^{\frac{5}{2}}(x)}+\frac{10 x \sin (x)}{21 \sqrt{\sec (x)}} \]

[Out]

4/(49*Sec[x]^(7/2)) + 20/(63*Sec[x]^(3/2)) + (2*x*Sin[x])/(7*Sec[x]^(5/2)) + (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.

[In]

Int[x/Sec[x]^(7/2) - (5*x*Sqrt[Sec[x]])/21,x]

[Out]

4/(49*Sec[x]^(7/2)) + 20/(63*Sec[x]^(3/2)) + (2*x*Sin[x])/(7*Sec[x]^(5/2)) + (10*x*Sin[x])/(21*Sqrt[Sec[x]])

Rule 4187

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(d*(b*Csc[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)*(b*Csc[e + f*x])^(n + 2), x], x] + Simp[((c + d*x)*Cos[e + f*x]
*(b*Csc[e + f*x])^(n + 1))/(b*f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[n, -1]

Rule 4189

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*Sin[e + f*x])^n*(b*C
sc[e + f*x])^n, Int[(c + d*x)^m/(b*Sin[e + f*x])^n, x], x] /; FreeQ[{b, c, d, e, f, m, n}, x] &&  !IntegerQ[n]

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.

[In]

Integrate[x/Sec[x]^(7/2) - (5*x*Sqrt[Sec[x]])/21,x]

[Out]

Sqrt[Sec[x]]*(167/882 + (88*Cos[2*x])/441 + Cos[4*x]/98 + (13*x*Sin[2*x])/42 + (x*Sin[4*x])/28)

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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.

[In]

int(x/sec(x)^(7/2)-5/21*x*sec(x)^(1/2),x)

[Out]

int(x/sec(x)^(7/2)-5/21*x*sec(x)^(1/2),x)

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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.

[In]

integrate(x/sec(x)^(7/2)-5/21*x*sec(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(-5/21*x*sqrt(sec(x)) + x/sec(x)^(7/2), x)

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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.

[In]

integrate(x/sec(x)^(7/2)-5/21*x*sec(x)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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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.

[In]

integrate(x/sec(x)**(7/2)-5/21*x*sec(x)**(1/2),x)

[Out]

Timed out

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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.

[In]

integrate(x/sec(x)^(7/2)-5/21*x*sec(x)^(1/2),x, algorithm="giac")

[Out]

integrate(-5/21*x*sqrt(sec(x)) + x/sec(x)^(7/2), x)