3.1.16 \(\int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx\) [16]

Optimal. Leaf size=18 \[ -\frac {\log \left (a^2+b^2 \cos ^2(x)\right )}{b^2} \]

[Out]

-ln(a^2+b^2*cos(x)^2)/b^2

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Rubi [A]
time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 266} \begin {gather*} -\frac {\log \left (a^2-b^2 \sin ^2(x)+b^2\right )}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[2*x]/(a^2 + b^2*Cos[x]^2),x]

[Out]

-(Log[a^2 + b^2 - b^2*Sin[x]^2]/b^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

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

Rubi steps

\begin {align*} \int \frac {\sin (2 x)}{a^2+b^2 \cos ^2(x)} \, dx &=\text {Subst}\left (\int \frac {2 x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int \frac {x}{a^2+b^2-b^2 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {\log \left (a^2+b^2-b^2 \sin ^2(x)\right )}{b^2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.22 \begin {gather*} -\frac {\log \left (a^2+b^2-b^2 \sin ^2(x)\right )}{b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[2*x]/(a^2 + b^2*Cos[x]^2),x]

[Out]

-(Log[a^2 + b^2 - b^2*Sin[x]^2]/b^2)

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Maple [A]
time = 0.09, size = 19, normalized size = 1.06

method	result	size
derivativedivides	\(-\frac {\ln \left (a^{2}+b^{2} \left (\cos ^{2}\left (x \right )\right )\right )}{b^{2}}\)	\(19\)
default	\(-\frac {\ln \left (a^{2}+b^{2} \left (\cos ^{2}\left (x \right )\right )\right )}{b^{2}}\)	\(19\)
risch	\(\frac {2 i x}{b^{2}}-\frac {\ln \left ({\mathrm e}^{4 i x}+\frac {2 \left (2 a^{2}+b^{2}\right ) {\mathrm e}^{2 i x}}{b^{2}}+1\right )}{b^{2}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(a^2+b^2*cos(x)^2),x,method=_RETURNVERBOSE)

[Out]

-ln(a^2+b^2*cos(x)^2)/b^2

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Maxima [A]
time = 3.56, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(b^2*cos(x)^2 + a^2)/b^2

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Fricas [A]
time = 0.99, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(b^2*cos(x)^2 + a^2)/b^2

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Sympy [A]
time = 1.39, size = 34, normalized size = 1.89 \begin {gather*} 2 \left (\begin {cases} - \frac {\cos ^{2}{\left (x \right )}}{2 a^{2}} & \text {for}\: b^{2} = 0 \\- \frac {\log {\left (a^{2} + b^{2} \cos ^{2}{\left (x \right )} \right )}}{2 b^{2}} & \text {otherwise} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)/(a**2+b**2*cos(x)**2),x)

[Out]

2*Piecewise((-cos(x)**2/(2*a**2), Eq(b**2, 0)), (-log(a**2 + b**2*cos(x)**2)/(2*b**2), True))

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Giac [A]
time = 0.95, size = 18, normalized size = 1.00 \begin {gather*} -\frac {\log \left (b^{2} \cos \left (x\right )^{2} + a^{2}\right )}{b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(b^2*cos(x)^2 + a^2)/b^2

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Mupad [B]
time = 0.41, size = 58, normalized size = 3.22 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {b^2}{2\,a^2+b^2\,{\cos \left (x\right )}^2+b^2}-\frac {b^2\,{\cos \left (x\right )}^2}{2\,a^2+b^2\,{\cos \left (x\right )}^2+b^2}\right )}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)/(b^2*cos(x)^2 + a^2),x)

[Out]

(2*atanh(b^2/(b^2*cos(x)^2 + 2*a^2 + b^2) - (b^2*cos(x)^2)/(b^2*cos(x)^2 + 2*a^2 + b^2)))/b^2

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