3.92 \(\int (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}) \, dx\)

Optimal. Leaf size=47 \[ -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]

[Out]

-4/15/cos(x)^(3/2)+2/5*x*sin(x)/cos(x)^(5/2)+6/5*x*sin(x)/cos(x)^(1/2)+12/5*cos(x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3315} \[ -\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]

Antiderivative was successfully verified.

[In]

Int[x/Cos[x]^(7/2) + (3*x*Sqrt[Cos[x]])/5,x]

[Out]

-4/(15*Cos[x]^(3/2)) + (12*Sqrt[Cos[x]])/5 + (2*x*Sin[x])/(5*Cos[x]^(5/2)) + (6*x*Sin[x])/(5*Sqrt[Cos[x]])

Rule 3315

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)*Cos[e + f*x]*(b*Si
n[e + f*x])^(n + 1))/(b*f*(n + 1)), x] + (Dist[(n + 2)/(b^2*(n + 1)), Int[(c + d*x)*(b*Sin[e + f*x])^(n + 2),
x], x] - Simp[(d*(b*Sin[e + f*x])^(n + 2))/(b^2*f^2*(n + 1)*(n + 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && LtQ[
n, -1] && NeQ[n, -2]

Rubi steps

\begin {align*} \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx &=\frac {3}{5} \int x \sqrt {\cos (x)} \, dx+\int \frac {x}{\cos ^{\frac {7}{2}}(x)} \, dx\\ &=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {3}{5} \int \frac {x}{\cos ^{\frac {3}{2}}(x)} \, dx+\frac {3}{5} \int x \sqrt {\cos (x)} \, dx\\ &=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 33, normalized size = 0.70 \[ \frac {21 x \sin (x)+9 x \sin (3 x)+46 \cos (x)+18 \cos (3 x)}{30 \cos ^{\frac {5}{2}}(x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Cos[x]^(7/2) + (3*x*Sqrt[Cos[x]])/5,x]

[Out]

(46*Cos[x] + 18*Cos[3*x] + 21*x*Sin[x] + 9*x*Sin[3*x])/(30*Cos[x]^(5/2))

________________________________________________________________________________________

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/cos(x)^(7/2)+3/5*x*cos(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 \[ \int \frac {3}{5} \, x \sqrt {\cos \relax (x)} + \frac {x}{\cos \relax (x)^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(x)^(7/2)+3/5*x*cos(x)^(1/2),x, algorithm="giac")

[Out]

integrate(3/5*x*sqrt(cos(x)) + x/cos(x)^(7/2), x)

________________________________________________________________________________________

maple [F]  time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cos \relax (x )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cos }\relax (x )\right )}{5}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/cos(x)^(7/2)+3/5*x*cos(x)^(1/2),x)

[Out]

int(x/cos(x)^(7/2)+3/5*x*cos(x)^(1/2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {3}{5} \, x \sqrt {\cos \relax (x)} + \frac {x}{\cos \relax (x)^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(x)^(7/2)+3/5*x*cos(x)^(1/2),x, algorithm="maxima")

[Out]

integrate(3/5*x*sqrt(cos(x)) + x/cos(x)^(7/2), x)

________________________________________________________________________________________

mupad [B]  time = 0.53, size = 31, normalized size = 0.66 \[ \frac {36\,{\cos \relax (x)}^3+18\,x\,\sin \relax (x)\,{\cos \relax (x)}^2-4\,\cos \relax (x)+6\,x\,\sin \relax (x)}{15\,{\cos \relax (x)}^{5/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x*cos(x)^(1/2))/5 + x/cos(x)^(7/2),x)

[Out]

(36*cos(x)^3 - 4*cos(x) + 6*x*sin(x) + 18*x*cos(x)^2*sin(x))/(15*cos(x)^(5/2))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/cos(x)**(7/2)+3/5*x*cos(x)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________