\(\int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx\) [211]

Optimal result

Integrand size = 21, antiderivative size = 11 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (2-3 \sin (x)+\sin ^2(x)\right ) \]

[Out]

ln(2-3*sin(x)+sin(x)^2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4419, 642} \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (\sin ^2(x)-3 \sin (x)+2\right ) \]

[In]

Int[(Cos[x]*(-3 + 2*Sin[x]))/(2 - 3*Sin[x] + Sin[x]^2),x]

[Out]

Log[2 - 3*Sin[x] + Sin[x]^2]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 4419

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Sin[c*(a + b*x)], x]}, Dist[d/(b
*c), Subst[Int[SubstFor[1, Sin[c*(a + b*x)]/d, u, x], x], x, Sin[c*(a + b*x)]/d], x] /; FunctionOfQ[Sin[c*(a +
b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Cos] || EqQ[F, cos])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {-3+2 x}{2-3 x+x^2} \, dx,x,\sin (x)\right ) \\ & = \log \left (2-3 \sin (x)+\sin ^2(x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.55 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=2 (\text {arctanh}(3-2 \sin (x))+\log (1-\sin (x))) \]

[In]

Integrate[(Cos[x]*(-3 + 2*Sin[x]))/(2 - 3*Sin[x] + Sin[x]^2),x]

[Out]

2*(ArcTanh[3 - 2*Sin[x]] + Log[1 - Sin[x]])

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\ln \left (2-3 \sin \left (x \right )+\sin ^{2}\left (x \right )\right )\)	\(12\)
default	\(\ln \left (2-3 \sin \left (x \right )+\sin ^{2}\left (x \right )\right )\)	\(12\)
risch	\(-2 i x +2 \ln \left ({\mathrm e}^{i x}-i\right )+\ln \left (-4 i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )\)	\(33\)
norman	\(2 \ln \left (\tan \left (\frac {x}{2}\right )-1\right )-2 \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )+\ln \left (\tan ^{2}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )+1\right )\)	\(37\)
parallelrisch	\(2 \ln \left (-\cot \left (x \right )+\csc \left (x \right )-1\right )-2 \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+\ln \left (\frac {-\sin \left (x \right )+2}{4 \cos \left (x \right )+4}\right )\)	\(38\)

[In]

int(cos(x)*(-3+2*sin(x))/(2-3*sin(x)+sin(x)^2),x,method=_RETURNVERBOSE)

[Out]

ln(2-3*sin(x)+sin(x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (-\frac {1}{2} \, \sin \left (x\right ) + 1\right ) + \log \left (-\sin \left (x\right ) + 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log {\left (\sin {\left (x \right )} - 2 \right )} + \log {\left (\sin {\left (x \right )} - 1 \right )} \]

[In]

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

[Out]

log(sin(x) - 2) + log(sin(x) - 1)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (\sin \left (x\right )^{2} - 3 \, \sin \left (x\right ) + 2\right ) \]

[In]

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

[Out]

log(sin(x)^2 - 3*sin(x) + 2)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\log \left (-\sin \left (x\right ) + 2\right ) + \log \left (-\sin \left (x\right ) + 1\right ) \]

[In]

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

[Out]

log(-sin(x) + 2) + log(-sin(x) + 1)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x) (-3+2 \sin (x))}{2-3 \sin (x)+\sin ^2(x)} \, dx=\ln \left ({\sin \left (x\right )}^2-3\,\sin \left (x\right )+2\right ) \]

[In]

int((cos(x)*(2*sin(x) - 3))/(sin(x)^2 - 3*sin(x) + 2),x)

[Out]

log(sin(x)^2 - 3*sin(x) + 2)