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(d*x+c)/d-2*a^4*csc(d*x+c)^2/d-1/3*a^4*csc(d*x+c)^3/d-4*a^4*ln(sin(d*x+c))/d-10*a^4*sin(d*x+c)/d-2*a
^4*sin(d*x+c)^2/d+4/3*a^4*sin(d*x+c)^3/d+a^4*sin(d*x+c)^4/d+1/5*a^4*sin(d*x+c)^5/d

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Rubi [A]  time = 0.12, antiderivative size = 145, normalized size of antiderivative = 1.00, 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 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]))

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]

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*}

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Mathematica [A]  time = 0.16, 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]

-1/15*(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*Sin[c + d*x]^2 - 20*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 - 3*Sin[c + d*x]^5))/d

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fricas [A]  time = 0.68, size = 172, normalized size = 1.19 \[ -\frac {24 \, a^{4} \cos \left (d x + c\right )^{8} - 256 \, a^{4} \cos \left (d x + c\right )^{6} - 576 \, a^{4} \cos \left (d x + c\right )^{4} + 2304 \, a^{4} \cos \left (d x + c\right )^{2} - 1536 \, a^{4} + 480 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - 15 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{6} - 8 \, a^{4} \cos \left (d x + c\right )^{4} - 3 \, a^{4} \cos \left (d x + c\right )^{2} + 19 \, a^{4}\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

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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giac [A]  time = 0.41, size = 135, normalized size = 0.93 \[ \frac {3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 150 \, a^{4} \sin \left (d x + c\right ) + \frac {5 \, {\left (22 \, a^{4} \sin \left (d x + c\right )^{3} - 12 \, a^{4} \sin \left (d x + c\right )^{2} - 6 \, a^{4} \sin \left (d x + c\right ) - a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \]

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

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maple [A]  time = 0.45, size = 179, normalized size = 1.23 \[ -\frac {64 a^{4} \sin \left (d x +c \right )}{5 d}-\frac {24 a^{4} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}-\frac {32 a^{4} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}-\frac {a^{4} \left (\cos ^{4}\left (d x +c \right )\right )}{d}-\frac {2 a^{4} \left (\cos ^{2}\left (d x +c \right )\right )}{d}-\frac {4 a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {5 a^{4} \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {2 a^{4} \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{2}}-\frac {a^{4} \left (\cos ^{6}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}} \]

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 = 0.79, size = 119, normalized size = 0.82 \[ \frac {3 \, a^{4} \sin \left (d x + c\right )^{5} + 15 \, a^{4} \sin \left (d x + c\right )^{4} + 20 \, a^{4} \sin \left (d x + c\right )^{3} - 30 \, a^{4} \sin \left (d x + c\right )^{2} - 60 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) - 150 \, a^{4} \sin \left (d x + c\right ) - \frac {5 \, {\left (12 \, a^{4} \sin \left (d x + c\right )^{2} + 6 \, a^{4} \sin \left (d x + c\right ) + a^{4}\right )}}{\sin \left (d x + c\right )^{3}}}{15 \, d} \]

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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mupad [B]  time = 8.97, size = 378, normalized size = 2.61 \[ \frac {4\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {4\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {177\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+68\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+640\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+84\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {4549\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{5}+104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+728\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+104\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {745\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {56\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+4\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+80\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {17\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*a^4*log(tan(c/2 + (d*x)/2)^2 + 1))/d - (a^4*tan(c/2 + (d*x)/2)^3)/(24*d) - (4*a^4*log(tan(c/2 + (d*x)/2)))/
d - ((56*a^4*tan(c/2 + (d*x)/2)^2)/3 + 20*a^4*tan(c/2 + (d*x)/2)^3 + (745*a^4*tan(c/2 + (d*x)/2)^4)/3 + 104*a^
4*tan(c/2 + (d*x)/2)^5 + 728*a^4*tan(c/2 + (d*x)/2)^6 + 104*a^4*tan(c/2 + (d*x)/2)^7 + (4549*a^4*tan(c/2 + (d*
x)/2)^8)/5 + 84*a^4*tan(c/2 + (d*x)/2)^9 + 640*a^4*tan(c/2 + (d*x)/2)^10 + 68*a^4*tan(c/2 + (d*x)/2)^11 + 177*
a^4*tan(c/2 + (d*x)/2)^12 + a^4/3 + 4*a^4*tan(c/2 + (d*x)/2))/(d*(8*tan(c/2 + (d*x)/2)^3 + 40*tan(c/2 + (d*x)/
2)^5 + 80*tan(c/2 + (d*x)/2)^7 + 80*tan(c/2 + (d*x)/2)^9 + 40*tan(c/2 + (d*x)/2)^11 + 8*tan(c/2 + (d*x)/2)^13)
) - (17*a^4*tan(c/2 + (d*x)/2))/(8*d) - (a^4*tan(c/2 + (d*x)/2)^2)/(2*d)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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