3.5 \(\int \frac{-1+4 \cos (x)+5 \cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx\)

Optimal. Leaf size=43 \[ x-2 \tan ^{-1}\left (\frac{3 \sin (x)+7 \sin (x) \cos (x)}{5 \cos ^2(x)+2 \cos (x)+1}\right )-2 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)+3}\right ) \]

[Out]

x - 2*ArcTan[Sin[x]/(3 + Cos[x])] - 2*ArcTan[(3*Sin[x] + 7*Cos[x]*Sin[x])/(1 + 2*Cos[x] + 5*Cos[x]^2)]

________________________________________________________________________________________

Rubi [F]  time = 0.804075, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{-1+4 \cos (x)+5 \cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(-1 + 4*Cos[x] + 5*Cos[x]^2)/(-1 - 4*Cos[x] - 3*Cos[x]^2 + 4*Cos[x]^3),x]

[Out]

Defer[Int][(1 + 4*Cos[x] + 3*Cos[x]^2 - 4*Cos[x]^3)^(-1), x] + 4*Defer[Int][Cos[x]/(-1 - 4*Cos[x] - 3*Cos[x]^2
+ 4*Cos[x]^3), x] + 5*Defer[Int][Cos[x]^2/(-1 - 4*Cos[x] - 3*Cos[x]^2 + 4*Cos[x]^3), x]

Rubi steps

\begin{align*} \int \frac{-1+4 \cos (x)+5 \cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx &=\int \left (\frac{1}{1+4 \cos (x)+3 \cos ^2(x)-4 \cos ^3(x)}+\frac{4 \cos (x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)}+\frac{5 \cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)}\right ) \, dx\\ &=4 \int \frac{\cos (x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx+5 \int \frac{\cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx+\int \frac{1}{1+4 \cos (x)+3 \cos ^2(x)-4 \cos ^3(x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.141942, size = 61, normalized size = 1.42 \[ \tan ^{-1}\left (\frac{1}{4} \left (\sin \left (\frac{x}{2}\right )-3 \sin \left (\frac{3 x}{2}\right )\right ) \sec ^3\left (\frac{x}{2}\right )\right )-\tan ^{-1}\left (\frac{1}{4} \left (3 \sin \left (\frac{3 x}{2}\right )-\sin \left (\frac{x}{2}\right )\right ) \sec ^3\left (\frac{x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 4*Cos[x] + 5*Cos[x]^2)/(-1 - 4*Cos[x] - 3*Cos[x]^2 + 4*Cos[x]^3),x]

[Out]

ArcTan[(Sec[x/2]^3*(Sin[x/2] - 3*Sin[(3*x)/2]))/4] - ArcTan[(Sec[x/2]^3*(-Sin[x/2] + 3*Sin[(3*x)/2]))/4]

________________________________________________________________________________________

Maple [A]  time = 0.055, size = 17, normalized size = 0.4 \begin{align*} 2\,\arctan \left ( \left ( \tan \left ( x/2 \right ) \right ) ^{3}-2\,\tan \left ( x/2 \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+4*cos(x)+5*cos(x)^2)/(-1-4*cos(x)-3*cos(x)^2+4*cos(x)^3),x)

[Out]

2*arctan(tan(1/2*x)^3-2*tan(1/2*x))

________________________________________________________________________________________

Maxima [A]  time = 7.82381, size = 85, normalized size = 1.98 \begin{align*} -\arctan \left (\sin \left (3 \, x\right ) + \frac{1}{2} \, \sin \left (2 \, x\right ) + \sin \left (x\right ), \cos \left (3 \, x\right ) + \frac{1}{2} \, \cos \left (2 \, x\right ) + \cos \left (x\right ) - \frac{1}{2}\right ) + \arctan \left (\sin \left (3 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - \sin \left (x\right ), \cos \left (3 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - \cos \left (x\right ) - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*cos(x)+5*cos(x)^2)/(-1-4*cos(x)-3*cos(x)^2+4*cos(x)^3),x, algorithm="maxima")

[Out]

-arctan2(sin(3*x) + 1/2*sin(2*x) + sin(x), cos(3*x) + 1/2*cos(2*x) + cos(x) - 1/2) + arctan2(sin(3*x) - 2*sin(
2*x) - sin(x), cos(3*x) - 2*cos(2*x) - cos(x) - 2)

________________________________________________________________________________________

Fricas [A]  time = 1.77823, size = 92, normalized size = 2.14 \begin{align*} \arctan \left (\frac{5 \, \cos \left (x\right )^{3} - \cos \left (x\right )}{{\left (3 \, \cos \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1\right )} \sin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*cos(x)+5*cos(x)^2)/(-1-4*cos(x)-3*cos(x)^2+4*cos(x)^3),x, algorithm="fricas")

[Out]

arctan((5*cos(x)^3 - cos(x))/((3*cos(x)^2 + 4*cos(x) + 1)*sin(x)))

________________________________________________________________________________________

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+4*cos(x)+5*cos(x)**2)/(-1-4*cos(x)-3*cos(x)**2+4*cos(x)**3),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.08918, size = 24, normalized size = 0.56 \begin{align*} -2 \, \arctan \left (-\tan \left (\frac{1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac{1}{2} \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+4*cos(x)+5*cos(x)^2)/(-1-4*cos(x)-3*cos(x)^2+4*cos(x)^3),x, algorithm="giac")

[Out]

-2*arctan(-tan(1/2*x)^3 + 2*tan(1/2*x))