3.240 \(\int \frac{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=111 \[ \frac{\sin ^3(c+d x)}{3 a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{6 \sin (c+d x)}{a^3 d}-\frac{5}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{10 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

[Out]

(-10*Log[1 + Sin[c + d*x]])/(a^3*d) + (6*Sin[c + d*x])/(a^3*d) - (3*Sin[c + d*x]^2)/(2*a^3*d) + Sin[c + d*x]^3
/(3*a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a^3 + a^3*Sin[c + d*x]))

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Rubi [A]  time = 0.10643, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2833, 12, 43} \[ \frac{\sin ^3(c+d x)}{3 a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{6 \sin (c+d x)}{a^3 d}-\frac{5}{d \left (a^3 \sin (c+d x)+a^3\right )}-\frac{10 \log (\sin (c+d x)+1)}{a^3 d}+\frac{1}{2 a d (a \sin (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-10*Log[1 + Sin[c + d*x]])/(a^3*d) + (6*Sin[c + d*x])/(a^3*d) - (3*Sin[c + d*x]^2)/(2*a^3*d) + Sin[c + d*x]^3
/(3*a^3*d) + 1/(2*a*d*(a + a*Sin[c + d*x])^2) - 5/(d*(a^3 + a^3*Sin[c + d*x]))

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{\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{a^5 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{(a+x)^3} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (6 a^2-3 a x+x^2-\frac{a^5}{(a+x)^3}+\frac{5 a^4}{(a+x)^2}-\frac{10 a^3}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=-\frac{10 \log (1+\sin (c+d x))}{a^3 d}+\frac{6 \sin (c+d x)}{a^3 d}-\frac{3 \sin ^2(c+d x)}{2 a^3 d}+\frac{\sin ^3(c+d x)}{3 a^3 d}+\frac{1}{2 a d (a+a \sin (c+d x))^2}-\frac{5}{d \left (a^3+a^3 \sin (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.700328, size = 106, normalized size = 0.95 \[ \frac{32 \sin ^5(c+d x)-80 \sin ^4(c+d x)+320 \sin ^3(c+d x)+\sin ^2(c+d x) (1023-960 \log (\sin (c+d x)+1))-6 \sin (c+d x) (320 \log (\sin (c+d x)+1)-21)-960 \log (\sin (c+d x)+1)-417}{96 a^3 d (\sin (c+d x)+1)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]

[Out]

(-417 - 960*Log[1 + Sin[c + d*x]] - 6*(-21 + 320*Log[1 + Sin[c + d*x]])*Sin[c + d*x] + (1023 - 960*Log[1 + Sin
[c + d*x]])*Sin[c + d*x]^2 + 320*Sin[c + d*x]^3 - 80*Sin[c + d*x]^4 + 32*Sin[c + d*x]^5)/(96*a^3*d*(1 + Sin[c
+ d*x])^2)

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Maple [A]  time = 0.039, size = 101, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,{a}^{3}d}}-{\frac{3\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{a}^{3}d}}+6\,{\frac{\sin \left ( dx+c \right ) }{{a}^{3}d}}+{\frac{1}{2\,{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-5\,{\frac{1}{{a}^{3}d \left ( 1+\sin \left ( dx+c \right ) \right ) }}-10\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x)

[Out]

1/3*sin(d*x+c)^3/a^3/d-3/2*sin(d*x+c)^2/a^3/d+6*sin(d*x+c)/a^3/d+1/2/d/a^3/(1+sin(d*x+c))^2-5/d/a^3/(1+sin(d*x
+c))-10*ln(1+sin(d*x+c))/a^3/d

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Maxima [A]  time = 1.108, size = 128, normalized size = 1.15 \begin{align*} -\frac{\frac{3 \,{\left (10 \, \sin \left (d x + c\right ) + 9\right )}}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac{2 \, \sin \left (d x + c\right )^{3} - 9 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right )}{a^{3}} + \frac{60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/6*(3*(10*sin(d*x + c) + 9)/(a^3*sin(d*x + c)^2 + 2*a^3*sin(d*x + c) + a^3) - (2*sin(d*x + c)^3 - 9*sin(d*x
+ c)^2 + 36*sin(d*x + c))/a^3 + 60*log(sin(d*x + c) + 1)/a^3)/d

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Fricas [A]  time = 1.50639, size = 317, normalized size = 2.86 \begin{align*} \frac{10 \, \cos \left (d x + c\right )^{4} + 115 \, \cos \left (d x + c\right )^{2} - 120 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (2 \, \cos \left (d x + c\right )^{4} - 24 \, \cos \left (d x + c\right )^{2} + 37\right )} \sin \left (d x + c\right ) - 80}{12 \,{\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/12*(10*cos(d*x + c)^4 + 115*cos(d*x + c)^2 - 120*(cos(d*x + c)^2 - 2*sin(d*x + c) - 2)*log(sin(d*x + c) + 1)
- 2*(2*cos(d*x + c)^4 - 24*cos(d*x + c)^2 + 37)*sin(d*x + c) - 80)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x +
c) - 2*a^3*d)

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Sympy [A]  time = 12.0587, size = 760, normalized size = 6.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)**5/(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((-60*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a*
*3*d) - 120*log(sin(c + d*x) + 1)*sin(c + d*x)/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d)
- 60*log(sin(c + d*x) + 1)/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - sin(c + d*x)**6/(6
*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 2*sin(c + d*x)**4*cos(c + d*x)**2/(6*a**3*d*sin
(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 16*sin(c + d*x)**4/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*s
in(c + d*x) + 6*a**3*d) - 4*sin(c + d*x)**3*cos(c + d*x)**2/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x)
+ 6*a**3*d) - sin(c + d*x)**2*cos(c + d*x)**4/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d)
- 12*sin(c + d*x)**2*cos(c + d*x)**2/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 2*sin(c
+ d*x)*cos(c + d*x)**4/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 20*sin(c + d*x)*cos(c
+ d*x)**2/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 100*sin(c + d*x)/(6*a**3*d*sin(c +
d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - cos(c + d*x)**4/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d
*x) + 6*a**3*d) - 10*cos(c + d*x)**2/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 80/(6*a*
*3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d), Ne(d, 0)), (x*sin(c)**5*cos(c)/(a*sin(c) + a)**3, T
rue))

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Giac [A]  time = 1.17292, size = 120, normalized size = 1.08 \begin{align*} -\frac{\frac{60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3}} + \frac{3 \,{\left (10 \, \sin \left (d x + c\right ) + 9\right )}}{a^{3}{\left (\sin \left (d x + c\right ) + 1\right )}^{2}} - \frac{2 \, a^{6} \sin \left (d x + c\right )^{3} - 9 \, a^{6} \sin \left (d x + c\right )^{2} + 36 \, a^{6} \sin \left (d x + c\right )}{a^{9}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-1/6*(60*log(abs(sin(d*x + c) + 1))/a^3 + 3*(10*sin(d*x + c) + 9)/(a^3*(sin(d*x + c) + 1)^2) - (2*a^6*sin(d*x
+ c)^3 - 9*a^6*sin(d*x + c)^2 + 36*a^6*sin(d*x + c))/a^9)/d