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))

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Rubi [F]  time = 0.80, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.13, 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]

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fricas [A]  time = 0.45, size = 31, normalized size = 0.72 \[ \arctan \left (\frac {5 \, \cos \relax (x)^{3} - \cos \relax (x)}{{\left (3 \, \cos \relax (x)^{2} + 4 \, \cos \relax (x) + 1\right )} \sin \relax (x)}\right ) \]

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)))

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giac [A]  time = 0.96, size = 18, normalized size = 0.42 \[ -2 \, \arctan \left (-\tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, \tan \left (\frac {1}{2} \, x\right )\right ) \]

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))

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maple [A]  time = 0.10, size = 17, normalized size = 0.40 \[ 2 \arctan \left (\tan ^{3}\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )\right ) \]

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))

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maxima [A]  time = 11.09, size = 63, normalized size = 1.47 \[ -\arctan \left (\sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (2 \, x\right ) + \sin \relax (x), \cos \left (3 \, x\right ) + \frac {1}{2} \, \cos \left (2 \, x\right ) + \cos \relax (x) - \frac {1}{2}\right ) + \arctan \left (\sin \left (3 \, x\right ) - 2 \, \sin \left (2 \, x\right ) - \sin \relax (x), \cos \left (3 \, x\right ) - 2 \, \cos \left (2 \, x\right ) - \cos \relax (x) - 2\right ) \]

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)

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mupad [B]  time = 0.30, size = 27, normalized size = 0.63 \[ x-2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (\frac {x}{2}\right )-{\mathrm {tan}\left (\frac {x}{2}\right )}^3\right )-2\,\mathrm {atan}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 2*atan(2*tan(x/2) - tan(x/2)^3) - 2*atan(tan(x/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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