3.501 \(\int \frac{a+b \sin (d+e x)}{b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)} \]

[Out]

-(Cos[d + e*x]/(e*(b + a*Sin[d + e*x])))

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Rubi [A]  time = 0.0893867, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3288, 2754, 8} \[ -\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[d + e*x])/(b^2 + 2*a*b*Sin[d + e*x] + a^2*Sin[d + e*x]^2),x]

[Out]

-(Cos[d + e*x]/(e*(b + a*Sin[d + e*x])))

Rule 3288

Int[((A_) + (B_.)*sin[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*sin[(d_.) + (e_.)*(x_)] + (c_.)*sin[(d_.) + (e_.)*(x_
)]^2)^(n_), x_Symbol] :> Dist[1/(4^n*c^n), Int[(A + B*Sin[d + e*x])*(b + 2*c*Sin[d + e*x])^(2*n), x], x] /; Fr
eeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[n]

Rule 2754

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(a^2 - b^2
)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{a+b \sin (d+e x)}{b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \, dx &=\left (4 a^2\right ) \int \frac{a+b \sin (d+e x)}{\left (2 a b+2 a^2 \sin (d+e x)\right )^2} \, dx\\ &=-\frac{\cos (d+e x)}{e (b+a \sin (d+e x))}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=-\frac{\cos (d+e x)}{e (b+a \sin (d+e x))}\\ \end{align*}

Mathematica [A]  time = 0.0623663, size = 23, normalized size = 1. \[ -\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[d + e*x]/(e*(b + a*Sin[d + e*x])))

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Maple [B]  time = 0.092, size = 52, normalized size = 2.3 \begin{align*} 2\,{\frac{1}{e \left ( b \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}+2\,a\tan \left ( d/2+1/2\,ex \right ) +b \right ) } \left ( -{\frac{a\tan \left ( d/2+1/2\,ex \right ) }{b}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(e*x+d))/(b^2+2*a*b*sin(e*x+d)+a^2*sin(e*x+d)^2),x)

[Out]

2/e*(-a*tan(1/2*d+1/2*e*x)/b-1)/(b*tan(1/2*d+1/2*e*x)^2+2*a*tan(1/2*d+1/2*e*x)+b)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67657, size = 54, normalized size = 2.35 \begin{align*} -\frac{\cos \left (e x + d\right )}{a e \sin \left (e x + d\right ) + b e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(e*x + d)/(a*e*sin(e*x + d) + b*e)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.19627, size = 70, normalized size = 3.04 \begin{align*} -\frac{2 \,{\left (a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + b\right )} e^{\left (-1\right )}}{{\left (b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + b\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(a*tan(1/2*x*e + 1/2*d) + b)*e^(-1)/((b*tan(1/2*x*e + 1/2*d)^2 + 2*a*tan(1/2*x*e + 1/2*d) + b)*b)