3.25 \(\int \frac{1-3 \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0227313, 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} \[ -\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 3*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]

[Out]

(-2*Sqrt[Cos[c + d*x]]*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 \frac{1-3 \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx &=-\frac{2 \sqrt{\cos (c+d x)} \sin (c+d x)}{d}\\ \end{align*}

Mathematica [A]  time = 0.0567009, size = 21, normalized size = 1. \[ -\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 3*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]

[Out]

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

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Maple [B]  time = 2.026, 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}}}{\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-integrate((3*cos(d*x + c)^2 - 1)/sqrt(cos(d*x + c)), x)

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Fricas [A]  time = 1.62265, size = 51, normalized size = 2.43 \begin{align*} -\frac{2 \, \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(cos(d*x + c))*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((1-3*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 -\frac{3 \, \cos \left (d x + c\right )^{2} - 1}{\sqrt{\cos \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(-(3*cos(d*x + c)^2 - 1)/sqrt(cos(d*x + c)), x)