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

Optimal. Leaf size=75 \[ -\frac{\cos (c+d x)}{a^3 d}+\frac{3}{d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{3 \log (\cos (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \cos (c+d x)+a)^2} \]

[Out]

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

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Rubi [A]  time = 0.116747, antiderivative size = 75, 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} \[ -\frac{\cos (c+d x)}{a^3 d}+\frac{3}{d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{3 \log (\cos (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \cos (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.322926, size = 103, normalized size = 1.37 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \left (-2 \cos (3 (c+d x))+72 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\cos (2 (c+d x)) \left (24 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-5\right )+\cos (c+d x) \left (96 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+22\right )+21\right )}{4 a^3 d (\cos (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(c + d*x)/2]^2*(21 - 2*Cos[3*(c + d*x)] + 72*Log[Cos[(c + d*x)/2]] + Cos[2*(c + d*x)]*(-5 + 24*Log[Cos[(c
+ d*x)/2]]) + Cos[c + d*x]*(22 + 96*Log[Cos[(c + d*x)/2]])))/(4*a^3*d*(1 + Cos[c + d*x])^3)

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Maple [A]  time = 0.029, size = 86, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,d{a}^{3} \left ( 1+\sec \left ( dx+c \right ) \right ) ^{2}}}-2\,{\frac{1}{d{a}^{3} \left ( 1+\sec \left ( dx+c \right ) \right ) }}+3\,{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{d{a}^{3}}}-{\frac{1}{d{a}^{3}\sec \left ( dx+c \right ) }}-3\,{\frac{\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+a*sec(d*x+c))^3,x)

[Out]

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

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Maxima [A]  time = 1.0238, size = 96, normalized size = 1.28 \begin{align*} \frac{\frac{6 \, \cos \left (d x + c\right ) + 5}{a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \cos \left (d x + c\right ) + a^{3}} - \frac{2 \, \cos \left (d x + c\right )}{a^{3}} + \frac{6 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}}}{2 \, 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))^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73379, size = 255, normalized size = 3.4 \begin{align*} -\frac{2 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} - 6 \,{\left (\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - 4 \, \cos \left (d x + c\right ) - 5}{2 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} + 2 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(c + d*x)/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x)/a**3

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Giac [A]  time = 1.34489, size = 85, normalized size = 1.13 \begin{align*} -\frac{\cos \left (d x + c\right )}{a^{3} d} + \frac{3 \, \log \left ({\left | -\cos \left (d x + c\right ) - 1 \right |}\right )}{a^{3} d} + \frac{6 \, \cos \left (d x + c\right ) + 5}{2 \, a^{3} d{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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