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

Optimal result

Integrand size = 23, antiderivative size = 21 \[ \int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {3090} \[ \int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{d} \]

[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*} \text {integral}& = -\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \sqrt {\cos (c+d x)} \sin (c+d x)}{d} \]

[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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(98\) vs. \(2(19)=38\).

Time = 4.30 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.71

method	result	size
default	\(-\frac {4 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(99\)
parts	\(\frac {2 \,\operatorname {am}^{-1}\left (\frac {d x}{2}+\frac {c}{2}| \sqrt {2}\right )}{d}+\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\)	\(197\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \, \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d} \]

[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. \[ \int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

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

[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)

Giac [F]

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

[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] (verification not implemented)

Time = 0.76 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {1-3 \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{d} \]

[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