3.1.25 \(\int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\) [25]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {3090} \begin {gather*} -\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{d} \end {gather*}

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 3090

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 = 0.07, size = 21, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{d} \end {gather*}

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(19)=38\).
time = 0.24, size = 99, normalized size = 4.71

method	result	size
default	\(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(99\)

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,method=_RETURNVERBOSE)

[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

________________________________________________________________________________________

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((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)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2 \, \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d} \end {gather*}

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

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((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)

________________________________________________________________________________________

Mupad [B]
time = 0.77, size = 19, normalized size = 0.90 \begin {gather*} -\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________