3.1281 \(\int \frac{\cos (c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx\)

Optimal. Leaf size=34 \[ \frac{\sin (c+d x)}{b d}-\frac{a \log (a+b \sin (c+d x))}{b^2 d} \]

[Out]

-((a*Log[a + b*Sin[c + d*x]])/(b^2*d)) + Sin[c + d*x]/(b*d)

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Rubi [A]  time = 0.050181, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{\sin (c+d x)}{b d}-\frac{a \log (a+b \sin (c+d x))}{b^2 d} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]

[Out]

-((a*Log[a + b*Sin[c + d*x]])/(b^2*d)) + Sin[c + d*x]/(b*d)

Rule 2833

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\cos (c+d x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{b (a+x)} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{a}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=-\frac{a \log (a+b \sin (c+d x))}{b^2 d}+\frac{\sin (c+d x)}{b d}\\ \end{align*}

Mathematica [A]  time = 0.0229036, size = 33, normalized size = 0.97 \[ -\frac{\frac{a \log (a+b \sin (c+d x))}{b^2}-\frac{\sin (c+d x)}{b}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Sin[c + d*x])/(a + b*Sin[c + d*x]),x]

[Out]

-(((a*Log[a + b*Sin[c + d*x]])/b^2 - Sin[c + d*x]/b)/d)

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Maple [A]  time = 0.026, size = 35, normalized size = 1. \begin{align*} -{\frac{a\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d{b}^{2}}}+{\frac{\sin \left ( dx+c \right ) }{bd}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)/(a+b*sin(d*x+c)),x)

[Out]

-1/d/b^2*a*ln(a+b*sin(d*x+c))+sin(d*x+c)/b/d

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Maxima [A]  time = 0.988223, size = 45, normalized size = 1.32 \begin{align*} -\frac{\frac{a \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{2}} - \frac{\sin \left (d x + c\right )}{b}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*log(b*sin(d*x + c) + a)/b^2 - sin(d*x + c)/b)/d

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Fricas [A]  time = 1.52505, size = 74, normalized size = 2.18 \begin{align*} -\frac{a \log \left (b \sin \left (d x + c\right ) + a\right ) - b \sin \left (d x + c\right )}{b^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*log(b*sin(d*x + c) + a) - b*sin(d*x + c))/(b^2*d)

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Sympy [A]  time = 0.869154, size = 66, normalized size = 1.94 \begin{align*} \begin{cases} \frac{x \sin{\left (c \right )} \cos{\left (c \right )}}{a} & \text{for}\: b = 0 \wedge d = 0 \\- \frac{\cos ^{2}{\left (c + d x \right )}}{2 a d} & \text{for}\: b = 0 \\\frac{x \sin{\left (c \right )} \cos{\left (c \right )}}{a + b \sin{\left (c \right )}} & \text{for}\: d = 0 \\- \frac{a \log{\left (\frac{a}{b} + \sin{\left (c + d x \right )} \right )}}{b^{2} d} + \frac{\sin{\left (c + d x \right )}}{b d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.19328, size = 46, normalized size = 1.35 \begin{align*} -\frac{\frac{a \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{2}} - \frac{\sin \left (d x + c\right )}{b}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*log(abs(b*sin(d*x + c) + a))/b^2 - sin(d*x + c)/b)/d