3.97 \(\int (\frac {x^2}{\sec ^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\sec (x)}) \, dx\)

Optimal. Leaf size=62 \[ \frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}+\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}-\frac {16}{27} \sqrt {\cos (x)} \sqrt {\sec (x)} F\left (\left .\frac {x}{2}\right |2\right ) \]

[Out]

8/9*x/sec(x)^(3/2)-16/27*sin(x)/sec(x)^(1/2)+2/3*x^2*sin(x)/sec(x)^(1/2)-16/27*(cos(1/2*x)^2)^(1/2)/cos(1/2*x)
*EllipticF(sin(1/2*x),2^(1/2))*cos(x)^(1/2)*sec(x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.15, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4188, 4189, 3769, 3771, 2641} \[ \frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}+\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}-\frac {16}{27} \sqrt {\cos (x)} \sqrt {\sec (x)} F\left (\left .\frac {x}{2}\right |2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^2/Sec[x]^(3/2) - (x^2*Sqrt[Sec[x]])/3,x]

[Out]

(8*x)/(9*Sec[x]^(3/2)) - (16*Sqrt[Cos[x]]*EllipticF[x/2, 2]*Sqrt[Sec[x]])/27 - (16*Sin[x])/(27*Sqrt[Sec[x]]) +
(2*x^2*Sin[x])/(3*Sqrt[Sec[x]])

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 3769

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4188

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Csc[e + f*x])^n)/(f^2*n^2), x] + (Dist[(n + 1)/(b^2*n), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n + 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^n, x], x] + Simp[((c + d*x)^m*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] && GtQ[m, 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^2}{\sec ^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\sec (x)}\right ) \, dx &=-\left (\frac {1}{3} \int x^2 \sqrt {\sec (x)} \, dx\right )+\int \frac {x^2}{\sec ^{\frac {3}{2}}(x)} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}+\frac {1}{3} \int x^2 \sqrt {\sec (x)} \, dx-\frac {8}{9} \int \frac {1}{\sec ^{\frac {3}{2}}(x)} \, dx-\frac {1}{3} \left (\sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int \frac {x^2}{\sqrt {\cos (x)}} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}-\frac {8}{27} \int \sqrt {\sec (x)} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}-\frac {1}{27} \left (8 \sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int \frac {1}{\sqrt {\cos (x)}} \, dx\\ &=\frac {8 x}{9 \sec ^{\frac {3}{2}}(x)}-\frac {16}{27} \sqrt {\cos (x)} F\left (\left .\frac {x}{2}\right |2\right ) \sqrt {\sec (x)}-\frac {16 \sin (x)}{27 \sqrt {\sec (x)}}+\frac {2 x^2 \sin (x)}{3 \sqrt {\sec (x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 51, normalized size = 0.82 \[ \frac {1}{27} \sqrt {\sec (x)} \left (9 x^2 \sin (2 x)+12 x-8 \sin (2 x)+12 x \cos (2 x)-16 \sqrt {\cos (x)} F\left (\left .\frac {x}{2}\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sec[x]^(3/2) - (x^2*Sqrt[Sec[x]])/3,x]

[Out]

(Sqrt[Sec[x]]*(12*x + 12*x*Cos[2*x] - 16*Sqrt[Cos[x]]*EllipticF[x/2, 2] - 8*Sin[2*x] + 9*x^2*Sin[2*x]))/27

________________________________________________________________________________________

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^2/sec(x)^(3/2)-1/3*x^2*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 \[ \int -\frac {1}{3} \, x^{2} \sqrt {\sec \relax (x)} + \frac {x^{2}}{\sec \relax (x)^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/sec(x)^(3/2)-1/3*x^2*sec(x)^(1/2),x, algorithm="giac")

[Out]

integrate(-1/3*x^2*sqrt(sec(x)) + x^2/sec(x)^(3/2), x)

________________________________________________________________________________________

maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sec \relax (x )^{\frac {3}{2}}}-\frac {x^{2} \left (\sqrt {\sec }\relax (x )\right )}{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/sec(x)^(3/2)-1/3*x^2*sec(x)^(1/2),x)

[Out]

int(x^2/sec(x)^(3/2)-1/3*x^2*sec(x)^(1/2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{3} \, x^{2} \sqrt {\sec \relax (x)} + \frac {x^{2}}{\sec \relax (x)^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/sec(x)^(3/2)-1/3*x^2*sec(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(-1/3*x^2*sqrt(sec(x)) + x^2/sec(x)^(3/2), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ -\int \frac {x^2\,\sqrt {\frac {1}{\cos \relax (x)}}}{3}-\frac {x^2}{{\left (\frac {1}{\cos \relax (x)}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(1/cos(x))^(3/2) - (x^2*(1/cos(x))^(1/2))/3,x)

[Out]

-int((x^2*(1/cos(x))^(1/2))/3 - x^2/(1/cos(x))^(3/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \left (- \frac {3 x^{2}}{\sec ^{\frac {3}{2}}{\relax (x )}}\right )\, dx + \int x^{2} \sqrt {\sec {\relax (x )}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/sec(x)**(3/2)-1/3*x**2*sec(x)**(1/2),x)

[Out]

-(Integral(-3*x**2/sec(x)**(3/2), x) + Integral(x**2*sqrt(sec(x)), x))/3

________________________________________________________________________________________