3.530 \(\int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=145 \[ \frac{a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{10 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{4 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d} \]

[Out]

(-4*a^4*Csc[c + d*x])/d - (2*a^4*Csc[c + d*x]^2)/d - (a^4*Csc[c + d*x]^3)/(3*d) - (4*a^4*Log[Sin[c + d*x]])/d
- (10*a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/d +
(a^4*Sin[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.120683, antiderivative size = 145, 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 = {2836, 12, 88} \[ \frac{a^4 \sin ^5(c+d x)}{5 d}+\frac{a^4 \sin ^4(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}-\frac{2 a^4 \sin ^2(c+d x)}{d}-\frac{10 a^4 \sin (c+d x)}{d}-\frac{a^4 \csc ^3(c+d x)}{3 d}-\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{4 a^4 \csc (c+d x)}{d}-\frac{4 a^4 \log (\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^4*Csc[c + d*x])/d - (2*a^4*Csc[c + d*x]^2)/d - (a^4*Csc[c + d*x]^3)/(3*d) - (4*a^4*Log[Sin[c + d*x]])/d
- (10*a^4*Sin[c + d*x])/d - (2*a^4*Sin[c + d*x]^2)/d + (4*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^4)/d +
(a^4*Sin[c + d*x]^5)/(5*d)

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \cos (c+d x) \cot ^4(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{a^4 (a-x)^2 (a+x)^6}{x^4} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^2 (a+x)^6}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-10 a^4+\frac{a^8}{x^4}+\frac{4 a^7}{x^3}+\frac{4 a^6}{x^2}-\frac{4 a^5}{x}-4 a^3 x+4 a^2 x^2+4 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac{4 a^4 \csc (c+d x)}{d}-\frac{2 a^4 \csc ^2(c+d x)}{d}-\frac{a^4 \csc ^3(c+d x)}{3 d}-\frac{4 a^4 \log (\sin (c+d x))}{d}-\frac{10 a^4 \sin (c+d x)}{d}-\frac{2 a^4 \sin ^2(c+d x)}{d}+\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{a^4 \sin ^4(c+d x)}{d}+\frac{a^4 \sin ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.155741, size = 96, normalized size = 0.66 \[ -\frac{a^4 \left (-3 \sin ^5(c+d x)-15 \sin ^4(c+d x)-20 \sin ^3(c+d x)+30 \sin ^2(c+d x)+150 \sin (c+d x)+5 \csc ^3(c+d x)+30 \csc ^2(c+d x)+60 \csc (c+d x)+60 \log (\sin (c+d x))\right )}{15 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^4*(60*Csc[c + d*x] + 30*Csc[c + d*x]^2 + 5*Csc[c + d*x]^3 + 60*Log[Sin[c + d*x]] + 150*Sin[c + d*x] + 30*S
in[c + d*x]^2 - 20*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 3*Sin[c + d*x]^5))/(15*d)

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Maple [A]  time = 0.095, size = 179, normalized size = 1.2 \begin{align*} -{\frac{64\,{a}^{4}\sin \left ( dx+c \right ) }{5\,d}}-{\frac{24\,{a}^{4}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5\,d}}-{\frac{32\,{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) }{5\,d}}-{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{d}}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{d}}-4\,{\frac{{a}^{4}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-5\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d\sin \left ( dx+c \right ) }}-2\,{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{3\,d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x)

[Out]

-64/5*a^4*sin(d*x+c)/d-24/5/d*a^4*sin(d*x+c)*cos(d*x+c)^4-32/5/d*a^4*cos(d*x+c)^2*sin(d*x+c)-1/d*a^4*cos(d*x+c
)^4-2/d*a^4*cos(d*x+c)^2-4*a^4*ln(sin(d*x+c))/d-5/d*a^4/sin(d*x+c)*cos(d*x+c)^6-2/d*a^4/sin(d*x+c)^2*cos(d*x+c
)^6-1/3/d*a^4/sin(d*x+c)^3*cos(d*x+c)^6

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Maxima [A]  time = 1.18049, size = 161, normalized size = 1.11 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 150 \, a^{4} \sin \left (d x + c\right ) - \frac{5 \,{\left (12 \, a^{4} \sin \left (d x + c\right )^{2} + 6 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*(3*a^4*sin(d*x + c)^5 + 15*a^4*sin(d*x + c)^4 + 20*a^4*sin(d*x + c)^3 - 30*a^4*sin(d*x + c)^2 - 60*a^4*lo
g(sin(d*x + c)) - 150*a^4*sin(d*x + c) - 5*(12*a^4*sin(d*x + c)^2 + 6*a^4*sin(d*x + c) + a^4)/sin(d*x + c)^3)/
d

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Fricas [A]  time = 1.49109, size = 433, normalized size = 2.99 \begin{align*} -\frac{24 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, a^{4} \cos \left (d x + c\right )^{6} - 576 \, a^{4} \cos \left (d x + c\right )^{4} + 2304 \, a^{4} \cos \left (d x + c\right )^{2} - 1536 \, a^{4} + 480 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 15 \,{\left (8 \, a^{4} \cos \left (d x + c\right )^{6} - 8 \, a^{4} \cos \left (d x + c\right )^{4} - 3 \, a^{4} \cos \left (d x + c\right )^{2} + 19 \, a^{4}\right )} \sin \left (d x + c\right )}{120 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/120*(24*a^4*cos(d*x + c)^8 - 256*a^4*cos(d*x + c)^6 - 576*a^4*cos(d*x + c)^4 + 2304*a^4*cos(d*x + c)^2 - 15
36*a^4 + 480*(a^4*cos(d*x + c)^2 - a^4)*log(1/2*sin(d*x + c))*sin(d*x + c) - 15*(8*a^4*cos(d*x + c)^6 - 8*a^4*
cos(d*x + c)^4 - 3*a^4*cos(d*x + c)^2 + 19*a^4)*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*csc(d*x+c)**4*(a+a*sin(d*x+c))**4,x)

[Out]

Timed out

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Giac [A]  time = 1.35505, size = 182, normalized size = 1.26 \begin{align*} \frac{3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 150 \, a^{4} \sin \left (d x + c\right ) + \frac{5 \,{\left (22 \, a^{4} \sin \left (d x + c\right )^{3} - 12 \, a^{4} \sin \left (d x + c\right )^{2} - 6 \, a^{4} \sin \left (d x + c\right ) - a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*csc(d*x+c)^4*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

1/15*(3*a^4*sin(d*x + c)^5 + 15*a^4*sin(d*x + c)^4 + 20*a^4*sin(d*x + c)^3 - 30*a^4*sin(d*x + c)^2 - 60*a^4*lo
g(abs(sin(d*x + c))) - 150*a^4*sin(d*x + c) + 5*(22*a^4*sin(d*x + c)^3 - 12*a^4*sin(d*x + c)^2 - 6*a^4*sin(d*x
+ c) - a^4)/sin(d*x + c)^3)/d