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

Optimal. Leaf size=20 \[ 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \]

[Out]

2*x*sin(x)/cos(x)^(1/2)+4*cos(x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3315} \[ 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

4*Sqrt[Cos[x]] + (2*x*Sin[x])/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 {3}{2}}(x)}+x \sqrt {\cos (x)}\right ) \, dx &=\int \frac {x}{\cos ^{\frac {3}{2}}(x)} \, dx+\int x \sqrt {\cos (x)} \, dx\\ &=4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 17, normalized size = 0.85 \[ \frac {2 (x \sin (x)+2 \cos (x))}{\sqrt {\cos (x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(2*Cos[x] + x*Sin[x]))/Sqrt[Cos[x]]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cos \relax (x )^{\frac {3}{2}}}+x \left (\sqrt {\cos }\relax (x )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.33, size = 15, normalized size = 0.75 \[ \frac {4\,\cos \relax (x)+2\,x\,\sin \relax (x)}{\sqrt {\cos \relax (x)}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*cos(x) + 2*x*sin(x))/cos(x)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (\cos ^{2}{\relax (x )} + 1\right )}{\cos ^{\frac {3}{2}}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*(cos(x)**2 + 1)/cos(x)**(3/2), x)

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