\(\int (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}) \, dx\) [95]

Optimal result

Integrand size = 20, antiderivative size = 24 \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)} \]

[Out]

4/25/sec(x)^(5/2)+2/5*x*sin(x)/sec(x)^(3/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274} \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)} \]

[In]

Int[x/Sec[x]^(5/2) - (3*x)/(5*Sqrt[Sec[x]]),x]

[Out]

4/(25*Sec[x]^(5/2)) + (2*x*Sin[x])/(5*Sec[x]^(3/2))

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*} \text {integral}& = -\left (\frac {3}{5} \int \frac {x}{\sqrt {\sec (x)}} \, dx\right )+\int \frac {x}{\sec ^{\frac {5}{2}}(x)} \, dx \\ & = \frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)}+\frac {3}{5} \int \frac {x}{\sqrt {\sec (x)}} \, dx-\frac {1}{5} \left (3 \sqrt {\cos (x)} \sqrt {\sec (x)}\right ) \int x \sqrt {\cos (x)} \, dx \\ & = \frac {4}{25 \sec ^{\frac {5}{2}}(x)}+\frac {2 x \sin (x)}{5 \sec ^{\frac {3}{2}}(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\frac {2 (2+5 x \tan (x))}{25 \sec ^{\frac {5}{2}}(x)} \]

[In]

Integrate[x/Sec[x]^(5/2) - (3*x)/(5*Sqrt[Sec[x]]),x]

[Out]

(2*(2 + 5*x*Tan[x]))/(25*Sec[x]^(5/2))

Maple [F]

\[\int \left (\frac {x}{\sec \left (x \right )^{\frac {5}{2}}}-\frac {3 x}{5 \sqrt {\sec \left (x \right )}}\right )d x\]

[In]

int(x/sec(x)^(5/2)-3/5*x/sec(x)^(1/2),x)

[Out]

int(x/sec(x)^(5/2)-3/5*x/sec(x)^(1/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x/sec(x)^(5/2)-3/5*x/sec(x)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=- \frac {\int \left (- \frac {5 x}{\sec ^{\frac {5}{2}}{\left (x \right )}}\right )\, dx + \int \frac {3 x}{\sqrt {\sec {\left (x \right )}}}\, dx}{5} \]

[In]

integrate(x/sec(x)**(5/2)-3/5*x/sec(x)**(1/2),x)

[Out]

-(Integral(-5*x/sec(x)**(5/2), x) + Integral(3*x/sqrt(sec(x)), x))/5

Maxima [F]

\[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\int { -\frac {3 \, x}{5 \, \sqrt {\sec \left (x\right )}} + \frac {x}{\sec \left (x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x/sec(x)^(5/2)-3/5*x/sec(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(-3/5*x/sqrt(sec(x)) + x/sec(x)^(5/2), x)

Giac [F]

\[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=\int { -\frac {3 \, x}{5 \, \sqrt {\sec \left (x\right )}} + \frac {x}{\sec \left (x\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x/sec(x)^(5/2)-3/5*x/sec(x)^(1/2),x, algorithm="giac")

[Out]

integrate(-3/5*x/sqrt(sec(x)) + x/sec(x)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \left (\frac {x}{\sec ^{\frac {5}{2}}(x)}-\frac {3 x}{5 \sqrt {\sec (x)}}\right ) \, dx=-\int \frac {3\,x}{5\,\sqrt {\frac {1}{\cos \left (x\right )}}}-\frac {x}{{\left (\frac {1}{\cos \left (x\right )}\right )}^{5/2}} \,d x \]

[In]

int(x/(1/cos(x))^(5/2) - (3*x)/(5*(1/cos(x))^(1/2)),x)

[Out]

-int((3*x)/(5*(1/cos(x))^(1/2)) - x/(1/cos(x))^(5/2), x)