3.1.90 \(\int (\frac {x}{\cos ^{\frac {3}{2}}(x)}+x \sqrt {\cos (x)}) \, dx\) [90]

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 = 0.03, 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 = {3396} \begin {gather*} 4 \sqrt {\cos (x)}+\frac {2 x \sin (x)}{\sqrt {\cos (x)}} \end {gather*}

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 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*} \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 = 0.06, size = 17, normalized size = 0.85 \begin {gather*} \frac {2 (2 \cos (x)+x \sin (x))}{\sqrt {\cos (x)}} \end {gather*}

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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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {x}{\cos \left (x \right )^{\frac {3}{2}}}+x \left (\sqrt {\cos }\left (x \right )\right )\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (\cos ^{2}{\left (x \right )} + 1\right )}{\cos ^{\frac {3}{2}}{\left (x \right )}}\, dx \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [B]
time = 0.33, size = 15, normalized size = 0.75 \begin {gather*} \frac {4\,\cos \left (x\right )+2\,x\,\sin \left (x\right )}{\sqrt {\cos \left (x\right )}} \end {gather*}

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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