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

Optimal. Leaf size=24 \[ \frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}} \]

[Out]

4/9/sec(x)^(3/2)+2/3*x*sin(x)/sec(x)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4187, 4189} \[ \frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4/(9*Sec[x]^(3/2)) + (2*x*Sin[x])/(3*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 {3}{2}}(x)}-\frac {1}{3} x \sqrt {\sec (x)}\right ) \, dx &=-\left (\frac {1}{3} \int x \sqrt {\sec (x)} \, dx\right )+\int \frac {x}{\sec ^{\frac {3}{2}}(x)} \, dx\\ &=\frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}}+\frac {1}{3} \int x \sqrt {\sec (x)} \, dx-\frac {1}{3} \left (\sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int \frac {x}{\sqrt {\cos (x)}} \, dx\\ &=\frac {4}{9 \sec ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{3 \sqrt {\sec (x)}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 17, normalized size = 0.71 \[ \frac {2 (3 x \tan (x)+2)}{9 \sec ^{\frac {3}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2 + 3*x*Tan[x]))/(9*Sec[x]^(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sec \relax (x )^{\frac {3}{2}}}-\frac {x \left (\sqrt {\sec }\relax (x )\right )}{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ -\int \frac {x\,\sqrt {\frac {1}{\cos \relax (x)}}}{3}-\frac {x}{{\left (\frac {1}{\cos \relax (x)}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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