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

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

[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 = 0.52, 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 = {} \begin {gather*} \int \frac {-1+4 \cos (x)+5 \cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx \end {gather*}

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 = 0.08, size = 61, normalized size = 1.42 \begin {gather*} \tan ^{-1}\left (\frac {1}{4} \sec ^3\left (\frac {x}{2}\right ) \left (\sin \left (\frac {x}{2}\right )-3 \sin \left (\frac {3 x}{2}\right )\right )\right )-\tan ^{-1}\left (\frac {1}{4} \sec ^3\left (\frac {x}{2}\right ) \left (-\sin \left (\frac {x}{2}\right )+3 \sin \left (\frac {3 x}{2}\right )\right )\right ) \end {gather*}

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

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Timed out

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Maple [A]
time = 0.09, size = 17, normalized size = 0.40

method	result	size
default	\(2 \arctan \left (\tan ^{3}\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )\right )\)	\(17\)
risch	\(-i \ln \left ({\mathrm e}^{3 i x}-2 \,{\mathrm e}^{2 i x}-{\mathrm e}^{i x}-2\right )+i \ln \left ({\mathrm e}^{3 i x}+\frac {{\mathrm e}^{2 i x}}{2}+{\mathrm e}^{i x}-\frac {1}{2}\right )\)	\(50\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.76, size = 63, normalized size = 1.47 \begin {gather*} -\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 {gather*}

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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Fricas [A]
time = 0.33, size = 31, normalized size = 0.72 \begin {gather*} \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 {gather*}

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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Sympy [F(-1)]
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+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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Giac [A]
time = 0.01, size = 20, normalized size = 0.47 \begin {gather*} -2 \arctan \left (-\tan ^{3}\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )\right ) \end {gather*}

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]

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

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Mupad [B]
time = 0.30, size = 27, normalized size = 0.63 \begin {gather*} 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 ) \end {gather*}

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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