3.61 \(\int \frac{\sin (c+d x)}{a+a \sec (c+d x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.0714901, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3872, 2833, 12, 43} \[ \frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

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

Mathematica [A]  time = 0.0800689, size = 28, normalized size = 0.9 \[ -\frac{\cos (c+d x)-2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.023, size = 49, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{da}}-{\frac{1}{da\sec \left ( dx+c \right ) }}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{da}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00837, size = 41, normalized size = 1.32 \begin{align*} -\frac{\frac{\cos \left (d x + c\right )}{a} - \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.70961, size = 72, normalized size = 2.32 \begin{align*} -\frac{\cos \left (d x + c\right ) - \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(cos(d*x + c) - log(1/2*cos(d*x + c) + 1/2))/(a*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sin{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(c + d*x)/(sec(c + d*x) + 1), x)/a

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Giac [A]  time = 1.25922, size = 46, normalized size = 1.48 \begin{align*} -\frac{\cos \left (d x + c\right )}{a d} + \frac{\log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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