3.24 \(\int \sqrt{\cos (c+d x)} (3-5 \cos ^2(c+d x)) \, dx\)

Optimal. Leaf size=21 \[ -\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d} \]

[Out]

(-2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/d

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Rubi [A]  time = 0.0228646, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {3011} \[ -\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Cos[c + d*x]]*(3 - 5*Cos[c + d*x]^2),x]

[Out]

(-2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/d

Rule 3011

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] /; FreeQ[{b, e, f, A, C, m}, x] && EqQ[A*(m + 2) + C*(m +
1), 0]

Rubi steps

\begin{align*} \int \sqrt{\cos (c+d x)} \left (3-5 \cos ^2(c+d x)\right ) \, dx &=-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0559298, size = 23, normalized size = 1.1 \[ -\frac{\sin (2 (c+d x)) \sqrt{\cos (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Cos[c + d*x]]*(3 - 5*Cos[c + d*x]^2),x]

[Out]

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

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Maple [B]  time = 1.698, size = 99, normalized size = 4.7 \begin{align*} -4\,{\frac{\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\cos \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}{\sin \left ( 1/2\,dx+c/2 \right ) d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="maxima")

[Out]

-integrate((5*cos(d*x + c)^2 - 3)*sqrt(cos(d*x + c)), x)

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Fricas [A]  time = 1.58204, size = 51, normalized size = 2.43 \begin{align*} -\frac{2 \, \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

-2*cos(d*x + c)^(3/2)*sin(d*x + c)/d

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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((3-5*cos(d*x+c)**2)*cos(d*x+c)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} \sqrt{\cos \left (d x + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3-5*cos(d*x+c)^2)*cos(d*x+c)^(1/2),x, algorithm="giac")

[Out]

integrate(-(5*cos(d*x + c)^2 - 3)*sqrt(cos(d*x + c)), x)