\(\int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx\) [227]

Optimal result

Integrand size = 25, antiderivative size = 31 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\log (1+\sin (c+d x))}{a d}+\frac {\sin (c+d x)}{a d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2912, 12, 45} \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\sin (c+d x)}{a d}-\frac {\log (\sin (c+d x)+1)}{a d} \]

[In]

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

[Out]

-(Log[1 + Sin[c + d*x]]/(a*d)) + Sin[c + d*x]/(a*d)

Rule 12

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

Rule 45

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])

Rule 2912

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/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {1}{2} \left (-\frac {2 \log (1+\sin (c+d x))}{a d}+\frac {2 \sin (c+d x)}{a d}\right ) \]

[In]

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

[Out]

((-2*Log[1 + Sin[c + d*x]])/(a*d) + (2*Sin[c + d*x])/(a*d))/2

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\)	\(26\)
default	\(\frac {\sin \left (d x +c \right )-\ln \left (1+\sin \left (d x +c \right )\right )}{d a}\)	\(26\)
parallelrisch	\(\frac {-2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin \left (d x +c \right )}{a d}\)	\(41\)
risch	\(\frac {i x}{a}-\frac {i {\mathrm e}^{i \left (d x +c \right )}}{2 d a}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{2 d a}+\frac {2 i c}{a d}-\frac {2 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a d}\)	\(76\)
norman	\(\frac {-\frac {2}{a d}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}-\frac {2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a d}\)	\(138\)

[In]

int(cos(d*x+c)*sin(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\log \left (\sin \left (d x + c\right ) + 1\right ) - \sin \left (d x + c\right )}{a d} \]

[In]

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

[Out]

-(log(sin(d*x + c) + 1) - sin(d*x + c))/(a*d)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} - \frac {\log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{a d} + \frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: d \neq 0 \\\frac {x \sin {\left (c \right )} \cos {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-log(sin(c + d*x) + 1)/(a*d) + sin(c + d*x)/(a*d), Ne(d, 0)), (x*sin(c)*cos(c)/(a*sin(c) + a), True
))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]

[In]

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

[Out]

-(log(sin(d*x + c) + 1)/a - sin(d*x + c)/a)/d

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {\sin \left (d x + c\right )}{a}}{d} \]

[In]

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

[Out]

-(log(abs(sin(d*x + c) + 1))/a - sin(d*x + c)/a)/d

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {\cos (c+d x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )-\sin \left (c+d\,x\right )}{a\,d} \]

[In]

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

[Out]

-(log(sin(c + d*x) + 1) - sin(c + d*x))/(a*d)