3.42 \(\int \frac{1}{-3+5 \cos (c+d x)} \, dx\)

Optimal. Leaf size=63 \[ \frac{\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]

[Out]

-Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]]/(4*d) + Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]/(4*d)

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Rubi [A]  time = 0.0170166, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2659, 206} \[ \frac{\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Int[(-3 + 5*Cos[c + d*x])^(-1),x]

[Out]

-Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]]/(4*d) + Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]/(4*d)

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{-3+5 \cos (c+d x)} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{2-8 x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{d}\\ &=-\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}+\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )+2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}\\ \end{align*}

Mathematica [A]  time = 0.0208122, size = 63, normalized size = 1. \[ \frac{\log \left (2 \sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{4 d}-\frac{\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 5*Cos[c + d*x])^(-1),x]

[Out]

-Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]]/(4*d) + Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]/(4*d)

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Maple [A]  time = 0.035, size = 40, normalized size = 0.6 \begin{align*} -{\frac{1}{4\,d}\ln \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }+{\frac{1}{4\,d}\ln \left ( 1+2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d*ln(2*tan(1/2*d*x+1/2*c)-1)+1/4/d*ln(1+2*tan(1/2*d*x+1/2*c))

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Maxima [A]  time = 2.1163, size = 68, normalized size = 1.08 \begin{align*} \frac{\log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right ) - \log \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.64575, size = 139, normalized size = 2.21 \begin{align*} \frac{\log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right ) - \log \left (-\frac{3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac{5}{2}\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(log(-3/2*cos(d*x + c) + 2*sin(d*x + c) + 5/2) - log(-3/2*cos(d*x + c) - 2*sin(d*x + c) + 5/2))/d

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Sympy [A]  time = 1.20591, size = 44, normalized size = 0.7 \begin{align*} \begin{cases} - \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} - \frac{1}{2} \right )}}{4 d} + \frac{\log{\left (\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} + \frac{1}{2} \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x}{5 \cos{\left (c \right )} - 3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-log(tan(c/2 + d*x/2) - 1/2)/(4*d) + log(tan(c/2 + d*x/2) + 1/2)/(4*d), Ne(d, 0)), (x/(5*cos(c) - 3
), True))

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Giac [A]  time = 1.1589, size = 51, normalized size = 0.81 \begin{align*} \frac{\log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(log(abs(2*tan(1/2*d*x + 1/2*c) + 1)) - log(abs(2*tan(1/2*d*x + 1/2*c) - 1)))/d