3.81 \(\int (-\frac{x}{3 \sqrt{\cos (a+b x)}}+x \cos ^{\frac{3}{2}}(a+b x)) \, dx\)

Optimal. Leaf size=42 \[ \frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b} \]

[Out]

(4*Cos[a + b*x]^(3/2))/(9*b^2) + (2*x*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)

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Rubi [A]  time = 0.0586445, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3310} \[ \frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sin (a+b x) \sqrt{\cos (a+b x)}}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Cos[a + b*x]^(3/2))/(9*b^2) + (2*x*Sqrt[Cos[a + b*x]]*Sin[a + b*x])/(3*b)

Rule 3310

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

Rubi steps

\begin{align*} \int \left (-\frac{x}{3 \sqrt{\cos (a+b x)}}+x \cos ^{\frac{3}{2}}(a+b x)\right ) \, dx &=-\left (\frac{1}{3} \int \frac{x}{\sqrt{\cos (a+b x)}} \, dx\right )+\int x \cos ^{\frac{3}{2}}(a+b x) \, dx\\ &=\frac{4 \cos ^{\frac{3}{2}}(a+b x)}{9 b^2}+\frac{2 x \sqrt{\cos (a+b x)} \sin (a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.4149, size = 40, normalized size = 0.95 \[ \frac{\sqrt{\cos (a+b x)} \left (4 x \sin (a+b x)+\frac{8 \cos (a+b x)}{3 b}\right )}{6 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Cos[a + b*x]]*((8*Cos[a + b*x])/(3*b) + 4*x*Sin[a + b*x]))/(6*b)

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Maple [F]  time = 0.275, size = 0, normalized size = 0. \begin{align*} \int x \left ( \cos \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}-{\frac{x}{3}{\frac{1}{\sqrt{\cos \left ( bx+a \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right )^{\frac{3}{2}} - \frac{x}{3 \, \sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*cos(b*x + a)^(3/2) - 1/3*x/sqrt(cos(b*x + a)), 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(b*x+a)^(3/2)-1/3*x/cos(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right )^{\frac{3}{2}} - \frac{x}{3 \, \sqrt{\cos \left (b x + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*cos(b*x + a)^(3/2) - 1/3*x/sqrt(cos(b*x + a)), x)