3.224 \(\int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=83 \[ -\frac{b^3}{2 a^4 d (a \cos (c+d x)+b)^2}+\frac{3 b^2}{a^4 d (a \cos (c+d x)+b)}+\frac{3 b \log (a \cos (c+d x)+b)}{a^4 d}-\frac{\cos (c+d x)}{a^3 d} \]

[Out]

-(Cos[c + d*x]/(a^3*d)) - b^3/(2*a^4*d*(b + a*Cos[c + d*x])^2) + (3*b^2)/(a^4*d*(b + a*Cos[c + d*x])) + (3*b*L
og[b + a*Cos[c + d*x]])/(a^4*d)

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Rubi [A]  time = 0.133559, antiderivative size = 83, 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} \[ -\frac{b^3}{2 a^4 d (a \cos (c+d x)+b)^2}+\frac{3 b^2}{a^4 d (a \cos (c+d x)+b)}+\frac{3 b \log (a \cos (c+d x)+b)}{a^4 d}-\frac{\cos (c+d x)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*Sec[c + d*x])^3,x]

[Out]

-(Cos[c + d*x]/(a^3*d)) - b^3/(2*a^4*d*(b + a*Cos[c + d*x])^2) + (3*b^2)/(a^4*d*(b + a*Cos[c + d*x])) + (3*b*L
og[b + a*Cos[c + d*x]])/(a^4*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))^3} \, dx &=-\int \frac{\cos ^3(c+d x) \sin (c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{a^3 (-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{(-b+x)^3} \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{b^3}{(b-x)^3}+\frac{3 b^2}{(b-x)^2}-\frac{3 b}{b-x}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^4 d}\\ &=-\frac{\cos (c+d x)}{a^3 d}-\frac{b^3}{2 a^4 d (b+a \cos (c+d x))^2}+\frac{3 b^2}{a^4 d (b+a \cos (c+d x))}+\frac{3 b \log (b+a \cos (c+d x))}{a^4 d}\\ \end{align*}

Mathematica [A]  time = 0.399499, size = 111, normalized size = 1.34 \[ \frac{2 a^2 b \cos ^2(c+d x) (3 \log (a \cos (c+d x)+b)-2)-2 a^3 \cos ^3(c+d x)+b^3 (6 \log (a \cos (c+d x)+b)+5)+4 a b^2 \cos (c+d x) (3 \log (a \cos (c+d x)+b)+1)}{2 a^4 d (a \cos (c+d x)+b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(a + b*Sec[c + d*x])^3,x]

[Out]

(-2*a^3*Cos[c + d*x]^3 + 2*a^2*b*Cos[c + d*x]^2*(-2 + 3*Log[b + a*Cos[c + d*x]]) + 4*a*b^2*Cos[c + d*x]*(1 + 3
*Log[b + a*Cos[c + d*x]]) + b^3*(5 + 6*Log[b + a*Cos[c + d*x]]))/(2*a^4*d*(b + a*Cos[c + d*x])^2)

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Maple [A]  time = 0.033, size = 96, normalized size = 1.2 \begin{align*} -{\frac{b}{2\,d{a}^{2} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{2}}}+3\,{\frac{b\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{d{a}^{4}}}-2\,{\frac{b}{d{a}^{3} \left ( a+b\sec \left ( dx+c \right ) \right ) }}-{\frac{1}{d{a}^{3}\sec \left ( dx+c \right ) }}-3\,{\frac{b\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/(a+b*sec(d*x+c))^3,x)

[Out]

-1/2/d*b/a^2/(a+b*sec(d*x+c))^2+3/d/a^4*b*ln(a+b*sec(d*x+c))-2/d/a^3*b/(a+b*sec(d*x+c))-1/d/a^3/sec(d*x+c)-3/d
/a^4*b*ln(sec(d*x+c))

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Maxima [A]  time = 0.961331, size = 117, normalized size = 1.41 \begin{align*} \frac{\frac{6 \, a b^{2} \cos \left (d x + c\right ) + 5 \, b^{3}}{a^{6} \cos \left (d x + c\right )^{2} + 2 \, a^{5} b \cos \left (d x + c\right ) + a^{4} b^{2}} - \frac{2 \, \cos \left (d x + c\right )}{a^{3}} + \frac{6 \, b \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{4}}}{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))^3,x, algorithm="maxima")

[Out]

1/2*((6*a*b^2*cos(d*x + c) + 5*b^3)/(a^6*cos(d*x + c)^2 + 2*a^5*b*cos(d*x + c) + a^4*b^2) - 2*cos(d*x + c)/a^3
+ 6*b*log(a*cos(d*x + c) + b)/a^4)/d

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Fricas [A]  time = 1.88912, size = 304, normalized size = 3.66 \begin{align*} -\frac{2 \, a^{3} \cos \left (d x + c\right )^{3} + 4 \, a^{2} b \cos \left (d x + c\right )^{2} - 4 \, a b^{2} \cos \left (d x + c\right ) - 5 \, b^{3} - 6 \,{\left (a^{2} b \cos \left (d x + c\right )^{2} + 2 \, a b^{2} \cos \left (d x + c\right ) + b^{3}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{2 \,{\left (a^{6} d \cos \left (d x + c\right )^{2} + 2 \, a^{5} b d \cos \left (d x + c\right ) + a^{4} b^{2} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/(a+b*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2*(2*a^3*cos(d*x + c)^3 + 4*a^2*b*cos(d*x + c)^2 - 4*a*b^2*cos(d*x + c) - 5*b^3 - 6*(a^2*b*cos(d*x + c)^2 +
2*a*b^2*cos(d*x + c) + b^3)*log(a*cos(d*x + c) + b))/(a^6*d*cos(d*x + c)^2 + 2*a^5*b*d*cos(d*x + c) + a^4*b^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 )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{3}}\, 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))**3,x)

[Out]

Integral(sin(c + d*x)/(a + b*sec(c + d*x))**3, x)

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Giac [A]  time = 1.33973, size = 104, normalized size = 1.25 \begin{align*} -\frac{\cos \left (d x + c\right )}{a^{3} d} + \frac{3 \, b \log \left ({\left | -a \cos \left (d x + c\right ) - b \right |}\right )}{a^{4} d} + \frac{6 \, a b^{2} \cos \left (d x + c\right ) + 5 \, b^{3}}{2 \,{\left (a \cos \left (d x + c\right ) + b\right )}^{2} a^{4} 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))^3,x, algorithm="giac")

[Out]

-cos(d*x + c)/(a^3*d) + 3*b*log(abs(-a*cos(d*x + c) - b))/(a^4*d) + 1/2*(6*a*b^2*cos(d*x + c) + 5*b^3)/((a*cos
(d*x + c) + b)^2*a^4*d)