3.407 \(\int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx\)

Optimal. Leaf size=187 \[ -\frac {11 a^4 \cos ^7(c+d x)}{112 d}-\frac {11 \cos ^7(c+d x) \left (a^4 \sin (c+d x)+a^4\right )}{144 d}+\frac {11 a^4 \sin (c+d x) \cos ^5(c+d x)}{96 d}+\frac {55 a^4 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {55 a^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {55 a^4 x}{256}-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^2}{18 d}-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d} \]

[Out]

55/256*a^4*x-11/112*a^4*cos(d*x+c)^7/d+55/256*a^4*cos(d*x+c)*sin(d*x+c)/d+55/384*a^4*cos(d*x+c)^3*sin(d*x+c)/d
+11/96*a^4*cos(d*x+c)^5*sin(d*x+c)/d-1/10*cos(d*x+c)^5*(a+a*sin(d*x+c))^5/a/d-1/18*cos(d*x+c)^7*(a^2+a^2*sin(d
*x+c))^2/d-11/144*cos(d*x+c)^7*(a^4+a^4*sin(d*x+c))/d

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Rubi [A]  time = 0.23, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2870, 2678, 2669, 2635, 8} \[ -\frac {11 a^4 \cos ^7(c+d x)}{112 d}-\frac {\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4 \sin (c+d x)+a^4\right )}{144 d}+\frac {11 a^4 \sin (c+d x) \cos ^5(c+d x)}{96 d}+\frac {55 a^4 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {55 a^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {55 a^4 x}{256}-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]

[Out]

(55*a^4*x)/256 - (11*a^4*Cos[c + d*x]^7)/(112*d) + (55*a^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (55*a^4*Cos[c
+ d*x]^3*Sin[c + d*x])/(384*d) + (11*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(96*d) - (Cos[c + d*x]^5*(a + a*Sin[c +
d*x])^5)/(10*a*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x])^2)/(18*d) - (11*Cos[c + d*x]^7*(a^4 + a^4*Sin[c +
d*x]))/(144*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2678

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2870

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(2*b*f*g*(m + 1)), x] + Dist[a/(2
*g^2), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&
EqQ[a^2 - b^2, 0] && EqQ[m - p, 0]

Rubi steps

\begin {align*} \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx &=-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}+\frac {1}{2} a \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}+\frac {1}{18} \left (11 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac {1}{16} \left (11 a^3\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac {11 a^4 \cos ^7(c+d x)}{112 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac {1}{16} \left (11 a^4\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac {11 a^4 \cos ^7(c+d x)}{112 d}+\frac {11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac {1}{96} \left (55 a^4\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac {11 a^4 \cos ^7(c+d x)}{112 d}+\frac {55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac {1}{128} \left (55 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {11 a^4 \cos ^7(c+d x)}{112 d}+\frac {55 a^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac {1}{256} \left (55 a^4\right ) \int 1 \, dx\\ &=\frac {55 a^4 x}{256}-\frac {11 a^4 \cos ^7(c+d x)}{112 d}+\frac {55 a^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac {\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac {11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}\\ \end {align*}

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Mathematica [A]  time = 1.19, size = 116, normalized size = 0.62 \[ \frac {a^4 (8820 \sin (2 (c+d x))-42840 \sin (4 (c+d x))-2730 \sin (6 (c+d x))+4095 \sin (8 (c+d x))-126 \sin (10 (c+d x))-181440 \cos (c+d x)-53760 \cos (3 (c+d x))+16128 \cos (5 (c+d x))+7200 \cos (7 (c+d x))-1120 \cos (9 (c+d x))+136080 c+138600 d x)}{645120 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^4,x]

[Out]

(a^4*(136080*c + 138600*d*x - 181440*Cos[c + d*x] - 53760*Cos[3*(c + d*x)] + 16128*Cos[5*(c + d*x)] + 7200*Cos
[7*(c + d*x)] - 1120*Cos[9*(c + d*x)] + 8820*Sin[2*(c + d*x)] - 42840*Sin[4*(c + d*x)] - 2730*Sin[6*(c + d*x)]
+ 4095*Sin[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)

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fricas [A]  time = 0.49, size = 124, normalized size = 0.66 \[ -\frac {35840 \, a^{4} \cos \left (d x + c\right )^{9} - 138240 \, a^{4} \cos \left (d x + c\right )^{7} + 129024 \, a^{4} \cos \left (d x + c\right )^{5} - 17325 \, a^{4} d x + 21 \, {\left (384 \, a^{4} \cos \left (d x + c\right )^{9} - 3888 \, a^{4} \cos \left (d x + c\right )^{7} + 5704 \, a^{4} \cos \left (d x + c\right )^{5} - 550 \, a^{4} \cos \left (d x + c\right )^{3} - 825 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/80640*(35840*a^4*cos(d*x + c)^9 - 138240*a^4*cos(d*x + c)^7 + 129024*a^4*cos(d*x + c)^5 - 17325*a^4*d*x + 2
1*(384*a^4*cos(d*x + c)^9 - 3888*a^4*cos(d*x + c)^7 + 5704*a^4*cos(d*x + c)^5 - 550*a^4*cos(d*x + c)^3 - 825*a
^4*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.42, size = 174, normalized size = 0.93 \[ \frac {55}{256} \, a^{4} x - \frac {a^{4} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} + \frac {5 \, a^{4} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {a^{4} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac {a^{4} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac {9 \, a^{4} \cos \left (d x + c\right )}{32 \, d} - \frac {a^{4} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {13 \, a^{4} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {13 \, a^{4} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {17 \, a^{4} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {7 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

55/256*a^4*x - 1/576*a^4*cos(9*d*x + 9*c)/d + 5/448*a^4*cos(7*d*x + 7*c)/d + 1/40*a^4*cos(5*d*x + 5*c)/d - 1/1
2*a^4*cos(3*d*x + 3*c)/d - 9/32*a^4*cos(d*x + c)/d - 1/5120*a^4*sin(10*d*x + 10*c)/d + 13/2048*a^4*sin(8*d*x +
8*c)/d - 13/3072*a^4*sin(6*d*x + 6*c)/d - 17/256*a^4*sin(4*d*x + 4*c)/d + 7/512*a^4*sin(2*d*x + 2*c)/d

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maple [A]  time = 0.28, size = 306, normalized size = 1.64 \[ \frac {a^{4} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+4 a^{4} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+6 a^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+4 a^{4} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+a^{4} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x)

[Out]

1/d*(a^4*(-1/10*sin(d*x+c)^5*cos(d*x+c)^5-1/16*sin(d*x+c)^3*cos(d*x+c)^5-1/32*sin(d*x+c)*cos(d*x+c)^5+1/128*(c
os(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+4*a^4*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*sin(d*x+c
)^2*cos(d*x+c)^5-8/315*cos(d*x+c)^5)+6*a^4*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos(d*x+c)^5+1/64*(
cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+4*a^4*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+
c)^5)+a^4*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c))

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maxima [A]  time = 0.35, size = 186, normalized size = 0.99 \[ -\frac {8192 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{4} - 73728 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{4} + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 3360 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 3780 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4}}{645120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^2*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/645120*(8192*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^4 - 73728*(5*cos(d*x + c)^7 - 7*
cos(d*x + c)^5)*a^4 + 63*(32*sin(2*d*x + 2*c)^5 - 120*d*x - 120*c - 5*sin(8*d*x + 8*c) + 40*sin(4*d*x + 4*c))*
a^4 - 3360*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^4 - 3780*(24*d*x + 24*c + sin(8*d*x +
8*c) - 8*sin(4*d*x + 4*c))*a^4)/d

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mupad [B]  time = 10.84, size = 572, normalized size = 3.06 \[ \frac {55\,a^4\,x}{256}-\frac {\frac {571\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{384}-\frac {14149\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{480}-\frac {469\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {4293\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {4293\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {469\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {14149\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{480}-\frac {571\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{384}-\frac {55\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{80640}-\frac {a^4\,\left (17325\,c+17325\,d\,x-53248\right )}{80640}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{8064}-\frac {a^4\,\left (173250\,c+173250\,d\,x\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{8064}-\frac {a^4\,\left (173250\,c+173250\,d\,x-532480\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{1792}-\frac {a^4\,\left (779625\,c+779625\,d\,x-1105920\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{1792}-\frac {a^4\,\left (779625\,c+779625\,d\,x-1290240\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{672}-\frac {a^4\,\left (2079000\,c+2079000\,d\,x-368640\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{384}-\frac {a^4\,\left (3638250\,c+3638250\,d\,x-860160\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{672}-\frac {a^4\,\left (2079000\,c+2079000\,d\,x-6021120\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{320}-\frac {a^4\,\left (4365900\,c+4365900\,d\,x-6709248\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^4\,\left (17325\,c+17325\,d\,x\right )}{384}-\frac {a^4\,\left (3638250\,c+3638250\,d\,x-10321920\right )}{80640}\right )+\frac {55\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4*sin(c + d*x)^2*(a + a*sin(c + d*x))^4,x)

[Out]

(55*a^4*x)/256 - ((571*a^4*tan(c/2 + (d*x)/2)^3)/384 - (14149*a^4*tan(c/2 + (d*x)/2)^5)/480 - (469*a^4*tan(c/2
+ (d*x)/2)^7)/32 + (4293*a^4*tan(c/2 + (d*x)/2)^9)/64 - (4293*a^4*tan(c/2 + (d*x)/2)^11)/64 + (469*a^4*tan(c/
2 + (d*x)/2)^13)/32 + (14149*a^4*tan(c/2 + (d*x)/2)^15)/480 - (571*a^4*tan(c/2 + (d*x)/2)^17)/384 - (55*a^4*ta
n(c/2 + (d*x)/2)^19)/128 + (a^4*(17325*c + 17325*d*x))/80640 - (a^4*(17325*c + 17325*d*x - 53248))/80640 + tan
(c/2 + (d*x)/2)^18*((a^4*(17325*c + 17325*d*x))/8064 - (a^4*(173250*c + 173250*d*x))/80640) + tan(c/2 + (d*x)/
2)^2*((a^4*(17325*c + 17325*d*x))/8064 - (a^4*(173250*c + 173250*d*x - 532480))/80640) + tan(c/2 + (d*x)/2)^4*
((a^4*(17325*c + 17325*d*x))/1792 - (a^4*(779625*c + 779625*d*x - 1105920))/80640) + tan(c/2 + (d*x)/2)^16*((a
^4*(17325*c + 17325*d*x))/1792 - (a^4*(779625*c + 779625*d*x - 1290240))/80640) + tan(c/2 + (d*x)/2)^6*((a^4*(
17325*c + 17325*d*x))/672 - (a^4*(2079000*c + 2079000*d*x - 368640))/80640) + tan(c/2 + (d*x)/2)^12*((a^4*(173
25*c + 17325*d*x))/384 - (a^4*(3638250*c + 3638250*d*x - 860160))/80640) + tan(c/2 + (d*x)/2)^14*((a^4*(17325*
c + 17325*d*x))/672 - (a^4*(2079000*c + 2079000*d*x - 6021120))/80640) + tan(c/2 + (d*x)/2)^10*((a^4*(17325*c
+ 17325*d*x))/320 - (a^4*(4365900*c + 4365900*d*x - 6709248))/80640) + tan(c/2 + (d*x)/2)^8*((a^4*(17325*c + 1
7325*d*x))/384 - (a^4*(3638250*c + 3638250*d*x - 10321920))/80640) + (55*a^4*tan(c/2 + (d*x)/2))/128)/(d*(tan(
c/2 + (d*x)/2)^2 + 1)^10)

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sympy [A]  time = 34.28, size = 746, normalized size = 3.99 \[ \begin {cases} \frac {3 a^{4} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{4} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {9 a^{4} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {15 a^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {9 a^{4} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {a^{4} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {27 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {3 a^{4} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {9 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{4} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {9 a^{4} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {a^{4} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{4} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {9 a^{4} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} - \frac {a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {33 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{64 d} + \frac {a^{4} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {4 a^{4} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {33 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{64 d} + \frac {a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {16 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {4 a^{4} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{4} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {9 a^{4} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {32 a^{4} \cos ^{9}{\left (c + d x \right )}}{315 d} - \frac {8 a^{4} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right )^{4} \sin ^{2}{\relax (c )} \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**2*(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((3*a**4*x*sin(c + d*x)**10/256 + 15*a**4*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 9*a**4*x*sin(c + d*
x)**8/64 + 15*a**4*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 9*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + a**4*
x*sin(c + d*x)**6/16 + 15*a**4*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 27*a**4*x*sin(c + d*x)**4*cos(c + d*x)*
*4/32 + 3*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**4*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 9*a**4*x
*sin(c + d*x)**2*cos(c + d*x)**6/16 + 3*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*a**4*x*cos(c + d*x)**10/
256 + 9*a**4*x*cos(c + d*x)**8/64 + a**4*x*cos(c + d*x)**6/16 + 3*a**4*sin(c + d*x)**9*cos(c + d*x)/(256*d) +
7*a**4*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + 9*a**4*sin(c + d*x)**7*cos(c + d*x)/(64*d) - a**4*sin(c + d*x
)**5*cos(c + d*x)**5/(10*d) + 33*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(64*d) + a**4*sin(c + d*x)**5*cos(c + d*
x)/(16*d) - 4*a**4*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 7*a**4*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 33
*a**4*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) + a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 16*a**4*sin(c + d*
x)**2*cos(c + d*x)**7/(35*d) - 4*a**4*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**4*sin(c + d*x)*cos(c + d*x)
**9/(256*d) - 9*a**4*sin(c + d*x)*cos(c + d*x)**7/(64*d) - a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 32*a**4*
cos(c + d*x)**9/(315*d) - 8*a**4*cos(c + d*x)**7/(35*d), Ne(d, 0)), (x*(a*sin(c) + a)**4*sin(c)**2*cos(c)**4,
True))

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