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

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

[Out]

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

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Rubi [A]  time = 0.0766032, antiderivative size = 34, 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} \[ \frac{b \log (a \cos (c+d x)+b)}{a^2 d}-\frac{\cos (c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0180372, size = 30, normalized size = 0.88 \[ \frac{b \log (a \cos (c+d x)+b)-a \cos (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.027, size = 53, normalized size = 1.6 \begin{align*}{\frac{b\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{d{a}^{2}}}-{\frac{1}{ad\sec \left ( dx+c \right ) }}-{\frac{b\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02355, size = 45, normalized size = 1.32 \begin{align*} -\frac{\frac{\cos \left (d x + c\right )}{a} - \frac{b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2}}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.70301, size = 74, normalized size = 2.18 \begin{align*} -\frac{a \cos \left (d x + c\right ) - b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.30374, size = 51, normalized size = 1.5 \begin{align*} -\frac{\cos \left (d x + c\right )}{a d} + \frac{b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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