3.221 \(\int \frac {1}{-4 \cos (x)+3 \sin (x)} \, dx\)

Optimal. Leaf size=18 \[ -\frac {1}{5} \tanh ^{-1}\left (\frac {1}{5} (4 \sin (x)+3 \cos (x))\right ) \]

[Out]

-1/5*arctanh(3/5*cos(x)+4/5*sin(x))

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3074, 206} \[ -\frac {1}{5} \tanh ^{-1}\left (\frac {1}{5} (4 \sin (x)+3 \cos (x))\right ) \]

Antiderivative was successfully verified.

[In]

Int[(-4*Cos[x] + 3*Sin[x])^(-1),x]

[Out]

-ArcTanh[(3*Cos[x] + 4*Sin[x])/5]/5

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])

Rule 3074

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

Rubi steps

\begin {align*} \int \frac {1}{-4 \cos (x)+3 \sin (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{25-x^2} \, dx,x,3 \cos (x)+4 \sin (x)\right )\\ &=-\frac {1}{5} \tanh ^{-1}\left (\frac {1}{5} (3 \cos (x)+4 \sin (x))\right )\\ \end {align*}

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Mathematica [B]  time = 0.02, size = 41, normalized size = 2.28 \[ \frac {1}{5} \log \left (\cos \left (\frac {x}{2}\right )-2 \sin \left (\frac {x}{2}\right )\right )-\frac {1}{5} \log \left (\sin \left (\frac {x}{2}\right )+2 \cos \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(-4*Cos[x] + 3*Sin[x])^(-1),x]

[Out]

Log[Cos[x/2] - 2*Sin[x/2]]/5 - Log[2*Cos[x/2] + Sin[x/2]]/5

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fricas [B]  time = 0.44, size = 27, normalized size = 1.50 \[ -\frac {1}{10} \, \log \left (\frac {3}{2} \, \cos \relax (x) + 2 \, \sin \relax (x) + \frac {5}{2}\right ) + \frac {1}{10} \, \log \left (-\frac {3}{2} \, \cos \relax (x) - 2 \, \sin \relax (x) + \frac {5}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*log(3/2*cos(x) + 2*sin(x) + 5/2) + 1/10*log(-3/2*cos(x) - 2*sin(x) + 5/2)

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giac [A]  time = 0.96, size = 23, normalized size = 1.28 \[ \frac {1}{5} \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) - \frac {1}{5} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*log(abs(2*tan(1/2*x) - 1)) - 1/5*log(abs(tan(1/2*x) + 2))

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maple [A]  time = 0.06, size = 22, normalized size = 1.22 \[ -\frac {\ln \left (\tan \left (\frac {x}{2}\right )+2\right )}{5}+\frac {\ln \left (2 \tan \left (\frac {x}{2}\right )-1\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*cos(x)+3*sin(x)),x)

[Out]

-1/5*ln(tan(1/2*x)+2)+1/5*ln(2*tan(1/2*x)-1)

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maxima [B]  time = 0.55, size = 30, normalized size = 1.67 \[ \frac {1}{5} \, \log \left (\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) - \frac {1}{5} \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*log(2*sin(x)/(cos(x) + 1) - 1) - 1/5*log(sin(x)/(cos(x) + 1) + 2)

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mupad [B]  time = 0.49, size = 11, normalized size = 0.61 \[ -\frac {2\,\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {x}{2}\right )}{5}+\frac {3}{5}\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(4*cos(x) - 3*sin(x)),x)

[Out]

-(2*atanh((4*tan(x/2))/5 + 3/5))/5

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sympy [A]  time = 0.29, size = 20, normalized size = 1.11 \[ \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {1}{2} \right )}}{5} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 2 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*cos(x)+3*sin(x)),x)

[Out]

log(tan(x/2) - 1/2)/5 - log(tan(x/2) + 2)/5

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