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]

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

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Rubi [A]  time = 0.042698, antiderivative size = 20, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0858378, 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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Maple [F]  time = 0.22, size = 0, normalized size = 0. \begin{align*} \int{x \left ( \cos \left ( x \right ) \right ) ^{-{\frac{3}{2}}}}+x\sqrt{\cos \left ( x \right ) }\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\cos \left (x\right )} + \frac{x}{\cos \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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

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., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\cos \left (x\right )} + \frac{x}{\cos \left (x\right )^{\frac{3}{2}}}\,{d x} \end{align*}

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)