### 3.305 $$\int \frac{1}{4-5 \sin (x)} \, dx$$

Optimal. Leaf size=43 $\frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right )$

[Out]

-Log[Cos[x/2] - 2*Sin[x/2]]/3 + Log[2*Cos[x/2] - Sin[x/2]]/3

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Rubi [A]  time = 0.0176698, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.375, Rules used = {2660, 616, 31} $\frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(4 - 5*Sin[x])^(-1),x]

[Out]

-Log[Cos[x/2] - 2*Sin[x/2]]/3 + Log[2*Cos[x/2] - Sin[x/2]]/3

Rule 2660

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[(2*e)/d, Subst[Int[1/(a + 2*b*e*x + a*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 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{4-5 \sin (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{4-10 x+4 x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-8+4 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{-2+4 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{1}{3} \log \left (1-2 \tan \left (\frac{x}{2}\right )\right )+\frac{1}{3} \log \left (2-\tan \left (\frac{x}{2}\right )\right )\\ \end{align*}

Mathematica [A]  time = 0.0118386, size = 43, normalized size = 1. $\frac{1}{3} \log \left (2 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{3} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(4 - 5*Sin[x])^(-1),x]

[Out]

-Log[Cos[x/2] - 2*Sin[x/2]]/3 + Log[2*Cos[x/2] - Sin[x/2]]/3

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Maple [A]  time = 0.012, size = 22, normalized size = 0.5 \begin{align*} -{\frac{1}{3}\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) }+{\frac{1}{3}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -2 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4-5*sin(x)),x)

[Out]

-1/3*ln(2*tan(1/2*x)-1)+1/3*ln(tan(1/2*x)-2)

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Maxima [A]  time = 0.936871, size = 41, normalized size = 0.95 \begin{align*} -\frac{1}{3} \, \log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \frac{1}{3} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-5*sin(x)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.29027, size = 105, normalized size = 2.44 \begin{align*} \frac{1}{6} \, \log \left (\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) - \frac{1}{6} \, \log \left (-\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-5*sin(x)),x, algorithm="fricas")

[Out]

1/6*log(3/2*cos(x) - 2*sin(x) + 5/2) - 1/6*log(-3/2*cos(x) - 2*sin(x) + 5/2)

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Sympy [A]  time = 0.222954, size = 20, normalized size = 0.47 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - 2 \right )}}{3} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{2} \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-5*sin(x)),x)

[Out]

log(tan(x/2) - 2)/3 - log(tan(x/2) - 1/2)/3

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Giac [A]  time = 1.08024, size = 31, normalized size = 0.72 \begin{align*} -\frac{1}{3} \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4-5*sin(x)),x, algorithm="giac")

[Out]

-1/3*log(abs(2*tan(1/2*x) - 1)) + 1/3*log(abs(tan(1/2*x) - 2))