3.57 \(\int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=265 \[ \frac {a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^3 x}{128}-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}-\frac {3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^2 x+\frac {b^3 \cos ^8(c+d x)}{8 d}-\frac {b^3 \cos ^6(c+d x)}{6 d} \]

[Out]

35/128*a^3*x+15/128*a*b^2*x-1/6*b^3*cos(d*x+c)^6/d-3/8*a^2*b*cos(d*x+c)^8/d+1/8*b^3*cos(d*x+c)^8/d+35/128*a^3*
cos(d*x+c)*sin(d*x+c)/d+15/128*a*b^2*cos(d*x+c)*sin(d*x+c)/d+35/192*a^3*cos(d*x+c)^3*sin(d*x+c)/d+5/64*a*b^2*c
os(d*x+c)^3*sin(d*x+c)/d+7/48*a^3*cos(d*x+c)^5*sin(d*x+c)/d+1/16*a*b^2*cos(d*x+c)^5*sin(d*x+c)/d+1/8*a^3*cos(d
*x+c)^7*sin(d*x+c)/d-3/8*a*b^2*cos(d*x+c)^7*sin(d*x+c)/d

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Rubi [A]  time = 0.25, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3090, 2635, 8, 2565, 30, 2568, 14} \[ -\frac {3 a^2 b \cos ^8(c+d x)}{8 d}+\frac {a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {35 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {35 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {35 a^3 x}{128}-\frac {3 a b^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac {a b^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac {5 a b^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac {15 a b^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac {15}{128} a b^2 x+\frac {b^3 \cos ^8(c+d x)}{8 d}-\frac {b^3 \cos ^6(c+d x)}{6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*a^3*x)/128 + (15*a*b^2*x)/128 - (b^3*Cos[c + d*x]^6)/(6*d) - (3*a^2*b*Cos[c + d*x]^8)/(8*d) + (b^3*Cos[c +
d*x]^8)/(8*d) + (35*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a*b^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (
35*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (5*a*b^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (7*a^3*Cos[c + d*
x]^5*Sin[c + d*x])/(48*d) + (a*b^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) + (a^3*Cos[c + d*x]^7*Sin[c + d*x])/(8*
d) - (3*a*b^2*Cos[c + d*x]^7*Sin[c + d*x])/(8*d)

Rule 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 2568

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

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 3090

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Sym
bol] :> Int[ExpandTrig[cos[c + d*x]^m*(a*cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] &&
IntegerQ[m] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos ^8(c+d x)+3 a^2 b \cos ^7(c+d x) \sin (c+d x)+3 a b^2 \cos ^6(c+d x) \sin ^2(c+d x)+b^3 \cos ^5(c+d x) \sin ^3(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^8(c+d x) \, dx+\left (3 a^2 b\right ) \int \cos ^7(c+d x) \sin (c+d x) \, dx+\left (3 a b^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+b^3 \int \cos ^5(c+d x) \sin ^3(c+d x) \, dx\\ &=\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{8} \left (7 a^3\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} \left (3 a b^2\right ) \int \cos ^6(c+d x) \, dx-\frac {\left (3 a^2 b\right ) \operatorname {Subst}\left (\int x^7 \, dx,x,\cos (c+d x)\right )}{d}-\frac {b^3 \operatorname {Subst}\left (\int x^5 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}+\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{48} \left (35 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (5 a b^2\right ) \int \cos ^4(c+d x) \, dx-\frac {b^3 \operatorname {Subst}\left (\int \left (x^5-x^7\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {b^3 \cos ^6(c+d x)}{6 d}-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}+\frac {b^3 \cos ^8(c+d x)}{8 d}+\frac {35 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{64} \left (35 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{64} \left (15 a b^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac {b^3 \cos ^6(c+d x)}{6 d}-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}+\frac {b^3 \cos ^8(c+d x)}{8 d}+\frac {35 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac {1}{128} \left (35 a^3\right ) \int 1 \, dx+\frac {1}{128} \left (15 a b^2\right ) \int 1 \, dx\\ &=\frac {35 a^3 x}{128}+\frac {15}{128} a b^2 x-\frac {b^3 \cos ^6(c+d x)}{6 d}-\frac {3 a^2 b \cos ^8(c+d x)}{8 d}+\frac {b^3 \cos ^8(c+d x)}{8 d}+\frac {35 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {15 a b^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {35 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {5 a b^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}+\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac {a b^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}+\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b^2 \cos ^7(c+d x) \sin (c+d x)}{8 d}\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 235, normalized size = 0.89 \[ \frac {5 a \left (7 a^2+3 b^2\right ) (c+d x)}{128 d}+\frac {a \left (14 a^2+3 b^2\right ) \sin (2 (c+d x))}{64 d}+\frac {a \left (7 a^2-3 b^2\right ) \sin (4 (c+d x))}{128 d}+\frac {a \left (2 a^2-3 b^2\right ) \sin (6 (c+d x))}{192 d}+\frac {a \left (a^2-3 b^2\right ) \sin (8 (c+d x))}{1024 d}-\frac {3 b \left (7 a^2+b^2\right ) \cos (2 (c+d x))}{128 d}-\frac {b \left (21 a^2+b^2\right ) \cos (4 (c+d x))}{256 d}-\frac {b \left (9 a^2-b^2\right ) \cos (6 (c+d x))}{384 d}-\frac {b \left (3 a^2-b^2\right ) \cos (8 (c+d x))}{1024 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*(7*a^2 + 3*b^2)*(c + d*x))/(128*d) - (3*b*(7*a^2 + b^2)*Cos[2*(c + d*x)])/(128*d) - (b*(21*a^2 + b^2)*Cos
[4*(c + d*x)])/(256*d) - (b*(9*a^2 - b^2)*Cos[6*(c + d*x)])/(384*d) - (b*(3*a^2 - b^2)*Cos[8*(c + d*x)])/(1024
*d) + (a*(14*a^2 + 3*b^2)*Sin[2*(c + d*x)])/(64*d) + (a*(7*a^2 - 3*b^2)*Sin[4*(c + d*x)])/(128*d) + (a*(2*a^2
- 3*b^2)*Sin[6*(c + d*x)])/(192*d) + (a*(a^2 - 3*b^2)*Sin[8*(c + d*x)])/(1024*d)

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fricas [A]  time = 0.72, size = 150, normalized size = 0.57 \[ -\frac {64 \, b^{3} \cos \left (d x + c\right )^{6} + 48 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{8} - 15 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} d x - {\left (48 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{7} + 8 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/384*(64*b^3*cos(d*x + c)^6 + 48*(3*a^2*b - b^3)*cos(d*x + c)^8 - 15*(7*a^3 + 3*a*b^2)*d*x - (48*(a^3 - 3*a*
b^2)*cos(d*x + c)^7 + 8*(7*a^3 + 3*a*b^2)*cos(d*x + c)^5 + 10*(7*a^3 + 3*a*b^2)*cos(d*x + c)^3 + 15*(7*a^3 + 3
*a*b^2)*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.35, size = 218, normalized size = 0.82 \[ \frac {5}{128} \, {\left (7 \, a^{3} + 3 \, a b^{2}\right )} x - \frac {{\left (3 \, a^{2} b - b^{3}\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (9 \, a^{2} b - b^{3}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (21 \, a^{2} b + b^{3}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {3 \, {\left (7 \, a^{2} b + b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (7 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (14 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/128*(7*a^3 + 3*a*b^2)*x - 1/1024*(3*a^2*b - b^3)*cos(8*d*x + 8*c)/d - 1/384*(9*a^2*b - b^3)*cos(6*d*x + 6*c)
/d - 1/256*(21*a^2*b + b^3)*cos(4*d*x + 4*c)/d - 3/128*(7*a^2*b + b^3)*cos(2*d*x + 2*c)/d + 1/1024*(a^3 - 3*a*
b^2)*sin(8*d*x + 8*c)/d + 1/192*(2*a^3 - 3*a*b^2)*sin(6*d*x + 6*c)/d + 1/128*(7*a^3 - 3*a*b^2)*sin(4*d*x + 4*c
)/d + 1/64*(14*a^3 + 3*a*b^2)*sin(2*d*x + 2*c)/d

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maple [A]  time = 11.01, size = 175, normalized size = 0.66 \[ \frac {b^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{8}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{24}\right )+3 b^{2} a \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )-\frac {3 a^{2} b \left (\cos ^{8}\left (d x +c \right )\right )}{8}+a^{3} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+3*b^2*a*(-1/8*sin(d*x+c)*cos(d*x+c)^7+1/48*(cos(d*
x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d*x+5/128*c)-3/8*a^2*b*cos(d*x+c)^8+a^3*(1/8*(cos(d*
x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+35/128*d*x+35/128*c))

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maxima [A]  time = 0.34, size = 163, normalized size = 0.62 \[ -\frac {1152 \, a^{2} b \cos \left (d x + c\right )^{8} + {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 3 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 128 \, {\left (3 \, \sin \left (d x + c\right )^{8} - 8 \, \sin \left (d x + c\right )^{6} + 6 \, \sin \left (d x + c\right )^{4}\right )} b^{3}}{3072 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3072*(1152*a^2*b*cos(d*x + c)^8 + (128*sin(2*d*x + 2*c)^3 - 840*d*x - 840*c - 3*sin(8*d*x + 8*c) - 168*sin(
4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*a^3 - 3*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 2
4*sin(4*d*x + 4*c))*a*b^2 - 128*(3*sin(d*x + c)^8 - 8*sin(d*x + c)^6 + 6*sin(d*x + c)^4)*b^3)/d

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mupad [B]  time = 2.29, size = 523, normalized size = 1.97 \[ \frac {4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15\,a\,b^2}{64}-\frac {93\,a^3}{64}\right )+\frac {40\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+4\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {15\,a\,b^2}{64}-\frac {93\,a^3}{64}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {91\,a^3}{192}+\frac {397\,a\,b^2}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {91\,a^3}{192}+\frac {397\,a\,b^2}{64}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {895\,a\,b^2}{64}-\frac {1799\,a^3}{192}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {895\,a\,b^2}{64}-\frac {1799\,a^3}{192}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {1765\,a\,b^2}{64}-\frac {1085\,a^3}{192}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {1765\,a\,b^2}{64}-\frac {1085\,a^3}{192}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (42\,a^2\,b-\frac {16\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (42\,a^2\,b-\frac {16\,b^3}{3}\right )+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {5\,a\,\mathrm {atan}\left (\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,a^2+3\,b^2\right )}{64\,\left (\frac {35\,a^3}{64}+\frac {15\,a\,b^2}{64}\right )}\right )\,\left (7\,a^2+3\,b^2\right )}{64\,d}-\frac {5\,a\,\left (7\,a^2+3\,b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^3*tan(c/2 + (d*x)/2)^4 - tan(c/2 + (d*x)/2)*((15*a*b^2)/64 - (93*a^3)/64) + (40*b^3*tan(c/2 + (d*x)/2)^8)
/3 + 4*b^3*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^15*((15*a*b^2)/64 - (93*a^3)/64) + tan(c/2 + (d*x)/2)^3*
((397*a*b^2)/64 + (91*a^3)/192) - tan(c/2 + (d*x)/2)^13*((397*a*b^2)/64 + (91*a^3)/192) - tan(c/2 + (d*x)/2)^5
*((895*a*b^2)/64 - (1799*a^3)/192) + tan(c/2 + (d*x)/2)^11*((895*a*b^2)/64 - (1799*a^3)/192) + tan(c/2 + (d*x)
/2)^7*((1765*a*b^2)/64 - (1085*a^3)/192) - tan(c/2 + (d*x)/2)^9*((1765*a*b^2)/64 - (1085*a^3)/192) + tan(c/2 +
(d*x)/2)^6*(42*a^2*b - (16*b^3)/3) + tan(c/2 + (d*x)/2)^10*(42*a^2*b - (16*b^3)/3) + 6*a^2*b*tan(c/2 + (d*x)/
2)^2 + 6*a^2*b*tan(c/2 + (d*x)/2)^14)/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x
)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan(c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^
14 + tan(c/2 + (d*x)/2)^16 + 1)) + (5*a*atan((5*a*tan(c/2 + (d*x)/2)*(7*a^2 + 3*b^2))/(64*((15*a*b^2)/64 + (35
*a^3)/64)))*(7*a^2 + 3*b^2))/(64*d) - (5*a*(7*a^2 + 3*b^2)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64*d)

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sympy [A]  time = 10.33, size = 532, normalized size = 2.01 \[ \begin {cases} \frac {35 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {105 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {35 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {35 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {35 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {385 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {511 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {93 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {3 a^{2} b \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {15 a b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {15 a b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {15 a b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {73 a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {15 a b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {b^{3} \sin ^{8}{\left (c + d x \right )}}{24 d} + \frac {b^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{6 d} + \frac {b^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{4 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\relax (c )} + b \sin {\relax (c )}\right )^{3} \cos ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((35*a**3*x*sin(c + d*x)**8/128 + 35*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 105*a**3*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 35*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*a**3*x*cos(c + d*x)**8/128 + 35*
a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 511*a**3*sin(c
+ d*x)**3*cos(c + d*x)**5/(384*d) + 93*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 3*a**2*b*cos(c + d*x)**8/(8
*d) + 15*a*b**2*x*sin(c + d*x)**8/128 + 15*a*b**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a*b**2*x*sin(c + d
*x)**4*cos(c + d*x)**4/64 + 15*a*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 15*a*b**2*x*cos(c + d*x)**8/128 +
15*a*b**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*a*b**2*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) + 73*a*b**
2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 15*a*b**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) + b**3*sin(c + d*x)
**8/(24*d) + b**3*sin(c + d*x)**6*cos(c + d*x)**2/(6*d) + b**3*sin(c + d*x)**4*cos(c + d*x)**4/(4*d), Ne(d, 0)
), (x*(a*cos(c) + b*sin(c))**3*cos(c)**5, True))

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