3.251 \(\int \frac{\cos (c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^4} \, dx\)

Optimal. Leaf size=83 \[ \frac{3}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]

[Out]

Log[1 + Sin[c + d*x]]/(a^4*d) + 1/(3*a*d*(a + a*Sin[c + d*x])^3) - 3/(2*d*(a^2 + a^2*Sin[c + d*x])^2) + 3/(d*(
a^4 + a^4*Sin[c + d*x]))

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Rubi [A]  time = 0.0928894, antiderivative size = 83, 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} \[ \frac{3}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac{3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac{\log (\sin (c+d x)+1)}{a^4 d}+\frac{1}{3 a d (a \sin (c+d x)+a)^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]

[Out]

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

Mathematica [A]  time = 0.357063, size = 61, normalized size = 0.73 \[ \frac{18 \sin ^2(c+d x)+27 \sin (c+d x)+6 (\sin (c+d x)+1)^3 \log (\sin (c+d x)+1)+11}{6 a^4 d (\sin (c+d x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]*Sin[c + d*x]^3)/(a + a*Sin[c + d*x])^4,x]

[Out]

(11 + 27*Sin[c + d*x] + 18*Sin[c + d*x]^2 + 6*Log[1 + Sin[c + d*x]]*(1 + Sin[c + d*x])^3)/(6*a^4*d*(1 + Sin[c
+ d*x])^3)

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Maple [A]  time = 0.037, size = 72, normalized size = 0.9 \begin{align*} -{\frac{3}{2\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{3\,d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}+3\,{\frac{1}{d{a}^{4} \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{d{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x)

[Out]

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

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Maxima [A]  time = 1.11051, size = 112, normalized size = 1.35 \begin{align*} \frac{\frac{18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4} \sin \left (d x + c\right )^{3} + 3 \, a^{4} \sin \left (d x + c\right )^{2} + 3 \, a^{4} \sin \left (d x + c\right ) + a^{4}} + \frac{6 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/6*((18*sin(d*x + c)^2 + 27*sin(d*x + c) + 11)/(a^4*sin(d*x + c)^3 + 3*a^4*sin(d*x + c)^2 + 3*a^4*sin(d*x + c
) + a^4) + 6*log(sin(d*x + c) + 1)/a^4)/d

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Fricas [A]  time = 1.53932, size = 292, normalized size = 3.52 \begin{align*} \frac{18 \, \cos \left (d x + c\right )^{2} + 6 \,{\left (3 \, \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 27 \, \sin \left (d x + c\right ) - 29}{6 \,{\left (3 \, a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d +{\left (a^{4} d \cos \left (d x + c\right )^{2} - 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)^3/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

1/6*(18*cos(d*x + c)^2 + 6*(3*cos(d*x + c)^2 + (cos(d*x + c)^2 - 4)*sin(d*x + c) - 4)*log(sin(d*x + c) + 1) -
27*sin(d*x + c) - 29)/(3*a^4*d*cos(d*x + c)^2 - 4*a^4*d + (a^4*d*cos(d*x + c)^2 - 4*a^4*d)*sin(d*x + c))

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Sympy [A]  time = 4.08603, size = 466, normalized size = 5.61 \begin{align*} \begin{cases} \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{18 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{18 \log{\left (\sin{\left (c + d x \right )} + 1 \right )} \sin{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{6 \log{\left (\sin{\left (c + d x \right )} + 1 \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} - \frac{6 \sin ^{3}{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{9 \sin{\left (c + d x \right )}}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} + \frac{5}{6 a^{4} d \sin ^{3}{\left (c + d x \right )} + 18 a^{4} d \sin ^{2}{\left (c + d x \right )} + 18 a^{4} d \sin{\left (c + d x \right )} + 6 a^{4} d} & \text{for}\: d \neq 0 \\\frac{x \sin ^{3}{\left (c \right )} \cos{\left (c \right )}}{\left (a \sin{\left (c \right )} + a\right )^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*sin(d*x+c)**3/(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((6*log(sin(c + d*x) + 1)*sin(c + d*x)**3/(6*a**4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*
a**4*d*sin(c + d*x) + 6*a**4*d) + 18*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(6*a**4*d*sin(c + d*x)**3 + 18*a**4
*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 18*log(sin(c + d*x) + 1)*sin(c + d*x)/(6*a**4*d*sin(
c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 6*log(sin(c + d*x) + 1)/(6*a**4
*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) - 6*sin(c + d*x)**3/(6*a**
4*d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 9*sin(c + d*x)/(6*a**4*
d*sin(c + d*x)**3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d) + 5/(6*a**4*d*sin(c + d*x)*
*3 + 18*a**4*d*sin(c + d*x)**2 + 18*a**4*d*sin(c + d*x) + 6*a**4*d), Ne(d, 0)), (x*sin(c)**3*cos(c)/(a*sin(c)
+ a)**4, True))

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Giac [A]  time = 1.23416, size = 74, normalized size = 0.89 \begin{align*} \frac{\frac{6 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} + \frac{18 \, \sin \left (d x + c\right )^{2} + 27 \, \sin \left (d x + c\right ) + 11}{a^{4}{\left (\sin \left (d x + c\right ) + 1\right )}^{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)^3/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(6*log(abs(sin(d*x + c) + 1))/a^4 + (18*sin(d*x + c)^2 + 27*sin(d*x + c) + 11)/(a^4*(sin(d*x + c) + 1)^3))
/d