3.212 \(\int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d} \]

[Out]

-(Cos[c + d*x]/(a^2*d)) + b^2/(a^3*d*(b + a*Cos[c + d*x])) + (2*b*Log[b + a*Cos[c + d*x]])/(a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.112259, antiderivative size = 57, 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} \[ \frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[c + d*x]/(a^2*d)) + b^2/(a^3*d*(b + a*Cos[c + d*x])) + (2*b*Log[b + a*Cos[c + d*x]])/(a^3*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))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin (c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{a^2 (-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^2}{(-b+x)^2} \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1+\frac{b^2}{(b-x)^2}-\frac{2 b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac{\cos (c+d x)}{a^2 d}+\frac{b^2}{a^3 d (b+a \cos (c+d x))}+\frac{2 b \log (b+a \cos (c+d x))}{a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.132602, size = 76, normalized size = 1.33 \[ \frac{-a^2 \cos ^2(c+d x)+b^2 (2 \log (a \cos (c+d x)+b)+1)+a b \cos (c+d x) (2 \log (a \cos (c+d x)+b)-1)}{a^3 d (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(a^2*Cos[c + d*x]^2) + a*b*Cos[c + d*x]*(-1 + 2*Log[b + a*Cos[c + d*x]]) + b^2*(1 + 2*Log[b + a*Cos[c + d*x]
]))/(a^3*d*(b + a*Cos[c + d*x]))

________________________________________________________________________________________

Maple [A]  time = 0.036, size = 75, normalized size = 1.3 \begin{align*} -{\frac{b}{d{a}^{2} \left ( a+b\sec \left ( dx+c \right ) \right ) }}+2\,{\frac{b\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{d{a}^{2}\sec \left ( dx+c \right ) }}-2\,{\frac{b\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*b/a^2/(a+b*sec(d*x+c))+2/d/a^3*b*ln(a+b*sec(d*x+c))-1/d/a^2/sec(d*x+c)-2/d/a^3*b*ln(sec(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.00148, size = 74, normalized size = 1.3 \begin{align*} \frac{\frac{b^{2}}{a^{4} \cos \left (d x + c\right ) + a^{3} b} - \frac{\cos \left (d x + c\right )}{a^{2}} + \frac{2 \, b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{3}}}{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))^2,x, algorithm="maxima")

[Out]

(b^2/(a^4*cos(d*x + c) + a^3*b) - cos(d*x + c)/a^2 + 2*b*log(a*cos(d*x + c) + b)/a^3)/d

________________________________________________________________________________________

Fricas [A]  time = 1.75957, size = 178, normalized size = 3.12 \begin{align*} -\frac{a^{2} \cos \left (d x + c\right )^{2} + a b \cos \left (d x + c\right ) - b^{2} - 2 \,{\left (a b \cos \left (d x + c\right ) + b^{2}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4} d \cos \left (d x + c\right ) + a^{3} b 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))^2,x, algorithm="fricas")

[Out]

-(a^2*cos(d*x + c)^2 + a*b*cos(d*x + c) - b^2 - 2*(a*b*cos(d*x + c) + b^2)*log(a*cos(d*x + c) + b))/(a^4*d*cos
(d*x + c) + a^3*b*d)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}\, 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))**2,x)

[Out]

Integral(sin(c + d*x)/(a + b*sec(c + d*x))**2, x)

________________________________________________________________________________________

Giac [A]  time = 1.2496, size = 82, normalized size = 1.44 \begin{align*} -\frac{\cos \left (d x + c\right )}{a^{2} d} + \frac{2 \, b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{3} d} + \frac{b^{2}}{{\left (a \cos \left (d x + c\right ) + b\right )} a^{3} 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))^2,x, algorithm="giac")

[Out]

-cos(d*x + c)/(a^2*d) + 2*b*log(abs(-a*cos(d*x + c) - b))/(a^3*d) + b^2/((a*cos(d*x + c) + b)*a^3*d)