3.1.96 \(\int (\frac {x}{\sec ^{\frac {7}{2}}(x)}-\frac {5}{21} x \sqrt {\sec (x)}) \, dx\) [96]

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

[Out]

4/49/sec(x)^(7/2)+20/63/sec(x)^(3/2)+2/7*x*sin(x)/sec(x)^(5/2)+10/21*x*sin(x)/sec(x)^(1/2)

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Rubi [A]
time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \begin {gather*} \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)}} \end {gather*}

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 4272

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 4274

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*}

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Mathematica [A]
time = 0.08, size = 45, normalized size = 0.96 \begin {gather*} \sqrt {\sec (x)} \left (\frac {167}{882}+\frac {88}{441} \cos (2 x)+\frac {1}{98} \cos (4 x)+\frac {13}{42} x \sin (2 x)+\frac {1}{28} x \sin (4 x)\right ) \end {gather*}

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 = 0.14, size = 0, normalized size = 0.00 \[\int \frac {x}{\sec \left (x \right )^{\frac {7}{2}}}-\frac {5 x \left (\sqrt {\sec }\left (x \right )\right )}{21}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \left (- \frac {21 x}{\sec ^{\frac {7}{2}}{\left (x \right )}}\right )\, dx + \int 5 x \sqrt {\sec {\left (x \right )}}\, dx}{21} \end {gather*}

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]

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

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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.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int \frac {5\,x\,\sqrt {\frac {1}{\cos \left (x\right )}}}{21}-\frac {x}{{\left (\frac {1}{\cos \left (x\right )}\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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