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)+2/7*x*sin(x)/sec(x)^(5/2)+10/21*x*sin(x)/sec(x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.09, 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 = {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 = 0.11, 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)

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {5}{21} \, x \sqrt {\sec \relax (x)} + \frac {x}{\sec \relax (x)^{\frac {7}{2}}}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sec \relax (x )^{\frac {7}{2}}}-\frac {5 x \left (\sqrt {\sec }\relax (x )\right )}{21}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {5}{21} \, x \sqrt {\sec \relax (x)} + \frac {x}{\sec \relax (x)^{\frac {7}{2}}}\,{d x} \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________