\(\int (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}) \, dx\) [92]

Optimal result

Integrand size = 20, antiderivative size = 47 \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]

[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] (verified)

Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396} \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=-\frac {4}{15 \cos ^{\frac {3}{2}}(x)}+\frac {12 \sqrt {\cos (x)}}{5}+\frac {2 x \sin (x)}{5 \cos ^{\frac {5}{2}}(x)}+\frac {6 x \sin (x)}{5 \sqrt {\cos (x)}} \]

[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 3396

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*} \text {integral}& = \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] (verified)

Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.70 \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\frac {46 \cos (x)+18 \cos (3 x)+21 x \sin (x)+9 x \sin (3 x)}{30 \cos ^{\frac {5}{2}}(x)} \]

[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))

Maple [F]

\[\int \left (\frac {x}{\cos \left (x \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cos }\left (x \right )\right )}{5}\right )d x\]

[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)

Fricas [F(-2)]

Exception generated. \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\text {Exception raised: TypeError} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\cos \left (x\right )} + \frac {x}{\cos \left (x\right )^{\frac {7}{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\int { \frac {3}{5} \, x \sqrt {\cos \left (x\right )} + \frac {x}{\cos \left (x\right )^{\frac {7}{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 13.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \left (\frac {x}{\cos ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cos (x)}\right ) \, dx=\frac {36\,{\cos \left (x\right )}^3+18\,x\,\sin \left (x\right )\,{\cos \left (x\right )}^2-4\,\cos \left (x\right )+6\,x\,\sin \left (x\right )}{15\,{\cos \left (x\right )}^{5/2}} \]

[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))