\(\int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx\) [413]

Optimal result

Integrand size = 21, antiderivative size = 144 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}-\frac {2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac {a (a+b \sin (c+d x))^{12}}{3 b^5 d}+\frac {(a+b \sin (c+d x))^{13}}{13 b^5 d} \]

[Out]

1/9*(a^2-b^2)^2*(a+b*sin(d*x+c))^9/b^5/d-2/5*a*(a^2-b^2)*(a+b*sin(d*x+c))^10/b^5/d+2/11*(3*a^2-b^2)*(a+b*sin(d
*x+c))^11/b^5/d-1/3*a*(a+b*sin(d*x+c))^12/b^5/d+1/13*(a+b*sin(d*x+c))^13/b^5/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2747, 711} \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac {2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac {(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac {a (a+b \sin (c+d x))^{12}}{3 b^5 d} \]

[In]

Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]

[Out]

((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/(9*b^5*d) - (2*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^10)/(5*b^5*d) + (2*(3
*a^2 - b^2)*(a + b*Sin[c + d*x])^11)/(11*b^5*d) - (a*(a + b*Sin[c + d*x])^12)/(3*b^5*d) + (a + b*Sin[c + d*x])
^13/(13*b^5*d)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 2747

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^8 \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \left (\left (a^2-b^2\right )^2 (a+x)^8-4 \left (a^3-a b^2\right ) (a+x)^9+2 \left (3 a^2-b^2\right ) (a+x)^{10}-4 a (a+x)^{11}+(a+x)^{12}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}-\frac {2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac {a (a+b \sin (c+d x))^{12}}{3 b^5 d}+\frac {(a+b \sin (c+d x))^{13}}{13 b^5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.83 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\frac {1}{9} \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9-\frac {2}{5} a (a-b) (a+b) (a+b \sin (c+d x))^{10}+\frac {2}{11} \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}-\frac {1}{3} a (a+b \sin (c+d x))^{12}+\frac {1}{13} (a+b \sin (c+d x))^{13}}{b^5 d} \]

[In]

Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]

[Out]

(((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/9 - (2*a*(a - b)*(a + b)*(a + b*Sin[c + d*x])^10)/5 + (2*(3*a^2 - b^2)
*(a + b*Sin[c + d*x])^11)/11 - (a*(a + b*Sin[c + d*x])^12)/3 + (a + b*Sin[c + d*x])^13/13)/(b^5*d)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(315\) vs. \(2(134)=268\).

Time = 4.96 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.19

method	result	size
derivativedivides	\(\frac {\frac {b^{8} \left (\sin ^{13}\left (d x +c \right )\right )}{13}+\frac {2 a \,b^{7} \left (\sin ^{12}\left (d x +c \right )\right )}{3}+\frac {\left (28 a^{2} b^{6}-2 b^{8}\right ) \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (56 a^{3} b^{5}-16 a \,b^{7}\right ) \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (70 a^{4} b^{4}-56 a^{2} b^{6}+b^{8}\right ) \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (56 a^{5} b^{3}-112 a^{3} b^{5}+8 a \,b^{7}\right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (28 a^{6} b^{2}-140 a^{4} b^{4}+28 a^{2} b^{6}\right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (8 a^{7} b -112 a^{5} b^{3}+56 a^{3} b^{5}\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (a^{8}-56 a^{6} b^{2}+70 a^{4} b^{4}\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (-16 a^{7} b +56 a^{5} b^{3}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (-2 a^{8}+28 a^{6} b^{2}\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+4 \left (\sin ^{2}\left (d x +c \right )\right ) a^{7} b +a^{8} \sin \left (d x +c \right )}{d}\)	\(316\)
default	\(\frac {\frac {b^{8} \left (\sin ^{13}\left (d x +c \right )\right )}{13}+\frac {2 a \,b^{7} \left (\sin ^{12}\left (d x +c \right )\right )}{3}+\frac {\left (28 a^{2} b^{6}-2 b^{8}\right ) \left (\sin ^{11}\left (d x +c \right )\right )}{11}+\frac {\left (56 a^{3} b^{5}-16 a \,b^{7}\right ) \left (\sin ^{10}\left (d x +c \right )\right )}{10}+\frac {\left (70 a^{4} b^{4}-56 a^{2} b^{6}+b^{8}\right ) \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (56 a^{5} b^{3}-112 a^{3} b^{5}+8 a \,b^{7}\right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (28 a^{6} b^{2}-140 a^{4} b^{4}+28 a^{2} b^{6}\right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (8 a^{7} b -112 a^{5} b^{3}+56 a^{3} b^{5}\right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (a^{8}-56 a^{6} b^{2}+70 a^{4} b^{4}\right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (-16 a^{7} b +56 a^{5} b^{3}\right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (-2 a^{8}+28 a^{6} b^{2}\right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+4 \left (\sin ^{2}\left (d x +c \right )\right ) a^{7} b +a^{8} \sin \left (d x +c \right )}{d}\)	\(316\)
parallelrisch	\(\frac {\left (2745600 a^{8}-3843840 a^{6} b^{2}-9609600 a^{4} b^{4}-2402400 a^{2} b^{6}-53625 b^{8}\right ) \sin \left (3 d x +3 c \right )+\left (329472 a^{8}-6918912 a^{6} b^{2}-5765760 a^{4} b^{4}-720720 a^{2} b^{6}-6435 b^{8}\right ) \sin \left (5 d x +5 c \right )-16473600 a \left (a^{6}+\frac {21}{10} a^{4} b^{2}+\frac {7}{8} a^{2} b^{4}+\frac {1}{16} b^{6}\right ) b \cos \left (2 d x +2 c \right )-1098240 \left (a^{6}-\frac {7}{2} a^{4} b^{2}-\frac {35}{16} a^{2} b^{4}-\frac {5}{32} b^{6}\right ) a b \cos \left (6 d x +6 c \right )+\left (-1647360 a^{6} b^{2}+1029600 a^{4} b^{4}+514800 a^{2} b^{6}+12870 b^{8}\right ) \sin \left (7 d x +7 c \right )+\left (-6589440 a^{7} b -5765760 a^{5} b^{3}+128700 a \,b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (800800 a^{4} b^{4}+80080 a^{2} b^{6}-1430 b^{8}\right ) \sin \left (9 d x +9 c \right )+\left (-288288 a^{3} b^{5}-20592 a \,b^{7}\right ) \cos \left (10 d x +10 c \right )+\left (-65520 a^{2} b^{6}-1755 b^{8}\right ) \sin \left (11 d x +11 c \right )+\left (1441440 a^{5} b^{3}-51480 a \,b^{7}\right ) \cos \left (8 d x +8 c \right )+8580 a \,b^{7} \cos \left (12 d x +12 c \right )+495 b^{8} \sin \left (13 d x +13 c \right )+\left (16473600 a^{8}[phone] a^{6} b^{2}[phone] a^{4} b^{4}+7207200 a^{2} b^{6}+128700 b^{8}\right ) \sin \left (d x +c \right )[phone] a^{7} b [phone] a^{5} b^{3}[phone] a^{3} b^{5}+792792 a \,b^{7}}{26357760 d}\)	\(449\)
risch	\(\frac {5 a^{8} \sin \left (d x +c \right )}{8 d}+\frac {b^{8} \sin \left (13 d x +13 c \right )}{53248 d}-\frac {3 \sin \left (11 d x +11 c \right ) b^{8}}{45056 d}+\frac {5 \sin \left (d x +c \right ) b^{8}}{1024 d}-\frac {\sin \left (9 d x +9 c \right ) b^{8}}{18432 d}-\frac {7 \sin \left (11 d x +11 c \right ) a^{2} b^{6}}{2816 d}-\frac {7 a^{3} b^{5} \cos \left (10 d x +10 c \right )}{640 d}-\frac {5 a^{7} b \cos \left (2 d x +2 c \right )}{8 d}-\frac {21 a^{5} b^{3} \cos \left (2 d x +2 c \right )}{16 d}+\frac {35 \sin \left (d x +c \right ) a^{6} b^{2}}{16 d}-\frac {a^{7} b \cos \left (6 d x +6 c \right )}{24 d}+\frac {7 a^{5} b^{3} \cos \left (6 d x +6 c \right )}{48 d}+\frac {35 a^{3} b^{5} \cos \left (6 d x +6 c \right )}{384 d}+\frac {5 a \,b^{7} \cos \left (6 d x +6 c \right )}{768 d}+\frac {a \,b^{7} \cos \left (12 d x +12 c \right )}{3072 d}-\frac {35 a^{3} b^{5} \cos \left (2 d x +2 c \right )}{64 d}-\frac {5 a \,b^{7} \cos \left (2 d x +2 c \right )}{128 d}-\frac {a \,b^{7} \cos \left (10 d x +10 c \right )}{1280 d}+\frac {5 \sin \left (7 d x +7 c \right ) a^{4} b^{4}}{128 d}+\frac {5 \sin \left (7 d x +7 c \right ) a^{2} b^{6}}{256 d}+\frac {\sin \left (7 d x +7 c \right ) b^{8}}{2048 d}+\frac {\sin \left (5 d x +5 c \right ) a^{8}}{80 d}-\frac {\sin \left (5 d x +5 c \right ) b^{8}}{4096 d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{8}}{48 d}-\frac {25 \sin \left (3 d x +3 c \right ) b^{8}}{12288 d}-\frac {a^{7} b \cos \left (4 d x +4 c \right )}{4 d}-\frac {7 a^{5} b^{3} \cos \left (4 d x +4 c \right )}{32 d}-\frac {\sin \left (7 d x +7 c \right ) a^{6} b^{2}}{16 d}+\frac {5 a \,b^{7} \cos \left (4 d x +4 c \right )}{1024 d}+\frac {105 \sin \left (d x +c \right ) a^{4} b^{4}}{64 d}+\frac {35 \sin \left (d x +c \right ) a^{2} b^{6}}{128 d}-\frac {7 \sin \left (3 d x +3 c \right ) a^{6} b^{2}}{48 d}-\frac {35 \sin \left (3 d x +3 c \right ) a^{4} b^{4}}{96 d}-\frac {35 \sin \left (3 d x +3 c \right ) a^{2} b^{6}}{384 d}-\frac {21 \sin \left (5 d x +5 c \right ) a^{6} b^{2}}{80 d}-\frac {7 \sin \left (5 d x +5 c \right ) a^{4} b^{4}}{32 d}-\frac {7 \sin \left (5 d x +5 c \right ) a^{2} b^{6}}{256 d}+\frac {35 \sin \left (9 d x +9 c \right ) a^{4} b^{4}}{1152 d}+\frac {7 \sin \left (9 d x +9 c \right ) a^{2} b^{6}}{2304 d}+\frac {7 a^{5} b^{3} \cos \left (8 d x +8 c \right )}{128 d}-\frac {a \,b^{7} \cos \left (8 d x +8 c \right )}{512 d}\)	\(759\)

[In]

int(cos(d*x+c)^5*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/13*b^8*sin(d*x+c)^13+2/3*a*b^7*sin(d*x+c)^12+1/11*(28*a^2*b^6-2*b^8)*sin(d*x+c)^11+1/10*(56*a^3*b^5-16*
a*b^7)*sin(d*x+c)^10+1/9*(70*a^4*b^4-56*a^2*b^6+b^8)*sin(d*x+c)^9+1/8*(56*a^5*b^3-112*a^3*b^5+8*a*b^7)*sin(d*x
+c)^8+1/7*(28*a^6*b^2-140*a^4*b^4+28*a^2*b^6)*sin(d*x+c)^7+1/6*(8*a^7*b-112*a^5*b^3+56*a^3*b^5)*sin(d*x+c)^6+1
/5*(a^8-56*a^6*b^2+70*a^4*b^4)*sin(d*x+c)^5+1/4*(-16*a^7*b+56*a^5*b^3)*sin(d*x+c)^4+1/3*(-2*a^8+28*a^6*b^2)*si
n(d*x+c)^3+4*sin(d*x+c)^2*a^7*b+a^8*sin(d*x+c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (134) = 268\).

Time = 0.34 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.47 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {4290 \, a b^{7} \cos \left (d x + c\right )^{12} - 5148 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{10} + 6435 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{8} - 8580 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{6} + {\left (495 \, b^{8} \cos \left (d x + c\right )^{12} - 180 \, {\left (91 \, a^{2} b^{6} + 10 \, b^{8}\right )} \cos \left (d x + c\right )^{10} + 10 \, {\left (5005 \, a^{4} b^{4} + 4186 \, a^{2} b^{6} + 229 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 3432 \, a^{8} + 13728 \, a^{6} b^{2} + 11440 \, a^{4} b^{4} + 2080 \, a^{2} b^{6} + 40 \, b^{8} - 20 \, {\left (1287 \, a^{6} b^{2} + 3575 \, a^{4} b^{4} + 1469 \, a^{2} b^{6} + 53 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (429 \, a^{8} + 1716 \, a^{6} b^{2} + 1430 \, a^{4} b^{4} + 260 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (429 \, a^{8} + 1716 \, a^{6} b^{2} + 1430 \, a^{4} b^{4} + 260 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6435 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c))^8,x, algorithm="fricas")

[Out]

1/6435*(4290*a*b^7*cos(d*x + c)^12 - 5148*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^10 + 6435*(7*a^5*b^3 + 14*a^3*b^5
+ 3*a*b^7)*cos(d*x + c)^8 - 8580*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*cos(d*x + c)^6 + (495*b^8*cos(d*x +
c)^12 - 180*(91*a^2*b^6 + 10*b^8)*cos(d*x + c)^10 + 10*(5005*a^4*b^4 + 4186*a^2*b^6 + 229*b^8)*cos(d*x + c)^8
+ 3432*a^8 + 13728*a^6*b^2 + 11440*a^4*b^4 + 2080*a^2*b^6 + 40*b^8 - 20*(1287*a^6*b^2 + 3575*a^4*b^4 + 1469*a^
2*b^6 + 53*b^8)*cos(d*x + c)^6 + 3*(429*a^8 + 1716*a^6*b^2 + 1430*a^4*b^4 + 260*a^2*b^6 + 5*b^8)*cos(d*x + c)^
4 + 4*(429*a^8 + 1716*a^6*b^2 + 1430*a^4*b^4 + 260*a^2*b^6 + 5*b^8)*cos(d*x + c)^2)*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (124) = 248\).

Time = 3.58 (sec) , antiderivative size = 614, normalized size of antiderivative = 4.26 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\begin {cases} \frac {8 a^{8} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{8} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a^{8} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {4 a^{7} b \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {32 a^{6} b^{2} \sin ^{7}{\left (c + d x \right )}}{15 d} + \frac {112 a^{6} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {28 a^{6} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {28 a^{5} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {7 a^{5} b^{3} \cos ^{8}{\left (c + d x \right )}}{3 d} + \frac {16 a^{4} b^{4} \sin ^{9}{\left (c + d x \right )}}{9 d} + \frac {8 a^{4} b^{4} \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {14 a^{4} b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {28 a^{3} b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {14 a^{3} b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{3 d} - \frac {14 a^{3} b^{5} \cos ^{10}{\left (c + d x \right )}}{15 d} + \frac {32 a^{2} b^{6} \sin ^{11}{\left (c + d x \right )}}{99 d} + \frac {16 a^{2} b^{6} \sin ^{9}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{9 d} + \frac {4 a^{2} b^{6} \sin ^{7}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {4 a b^{7} \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} - \frac {a b^{7} \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{d} - \frac {2 a b^{7} \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{5 d} - \frac {a b^{7} \cos ^{12}{\left (c + d x \right )}}{15 d} + \frac {8 b^{8} \sin ^{13}{\left (c + d x \right )}}{1287 d} + \frac {4 b^{8} \sin ^{11}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{99 d} + \frac {b^{8} \sin ^{9}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{8} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**5*(a+b*sin(d*x+c))**8,x)

[Out]

Piecewise((8*a**8*sin(c + d*x)**5/(15*d) + 4*a**8*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + a**8*sin(c + d*x)*co
s(c + d*x)**4/d - 4*a**7*b*cos(c + d*x)**6/(3*d) + 32*a**6*b**2*sin(c + d*x)**7/(15*d) + 112*a**6*b**2*sin(c +
d*x)**5*cos(c + d*x)**2/(15*d) + 28*a**6*b**2*sin(c + d*x)**3*cos(c + d*x)**4/(3*d) - 28*a**5*b**3*sin(c + d*
x)**2*cos(c + d*x)**6/(3*d) - 7*a**5*b**3*cos(c + d*x)**8/(3*d) + 16*a**4*b**4*sin(c + d*x)**9/(9*d) + 8*a**4*
b**4*sin(c + d*x)**7*cos(c + d*x)**2/d + 14*a**4*b**4*sin(c + d*x)**5*cos(c + d*x)**4/d - 28*a**3*b**5*sin(c +
d*x)**4*cos(c + d*x)**6/(3*d) - 14*a**3*b**5*sin(c + d*x)**2*cos(c + d*x)**8/(3*d) - 14*a**3*b**5*cos(c + d*x
)**10/(15*d) + 32*a**2*b**6*sin(c + d*x)**11/(99*d) + 16*a**2*b**6*sin(c + d*x)**9*cos(c + d*x)**2/(9*d) + 4*a
**2*b**6*sin(c + d*x)**7*cos(c + d*x)**4/d - 4*a*b**7*sin(c + d*x)**6*cos(c + d*x)**6/(3*d) - a*b**7*sin(c + d
*x)**4*cos(c + d*x)**8/d - 2*a*b**7*sin(c + d*x)**2*cos(c + d*x)**10/(5*d) - a*b**7*cos(c + d*x)**12/(15*d) +
8*b**8*sin(c + d*x)**13/(1287*d) + 4*b**8*sin(c + d*x)**11*cos(c + d*x)**2/(99*d) + b**8*sin(c + d*x)**9*cos(c
+ d*x)**4/(9*d), Ne(d, 0)), (x*(a + b*sin(c))**8*cos(c)**5, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (134) = 268\).

Time = 0.18 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.16 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {495 \, b^{8} \sin \left (d x + c\right )^{13} + 4290 \, a b^{7} \sin \left (d x + c\right )^{12} + 1170 \, {\left (14 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{11} + 5148 \, {\left (7 \, a^{3} b^{5} - 2 \, a b^{7}\right )} \sin \left (d x + c\right )^{10} + 25740 \, a^{7} b \sin \left (d x + c\right )^{2} + 715 \, {\left (70 \, a^{4} b^{4} - 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{9} + 6435 \, a^{8} \sin \left (d x + c\right ) + 6435 \, {\left (7 \, a^{5} b^{3} - 14 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{8} + 25740 \, {\left (a^{6} b^{2} - 5 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{7} + 8580 \, {\left (a^{7} b - 14 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \sin \left (d x + c\right )^{6} + 1287 \, {\left (a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4}\right )} \sin \left (d x + c\right )^{5} - 12870 \, {\left (2 \, a^{7} b - 7 \, a^{5} b^{3}\right )} \sin \left (d x + c\right )^{4} - 4290 \, {\left (a^{8} - 14 \, a^{6} b^{2}\right )} \sin \left (d x + c\right )^{3}}{6435 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c))^8,x, algorithm="maxima")

[Out]

1/6435*(495*b^8*sin(d*x + c)^13 + 4290*a*b^7*sin(d*x + c)^12 + 1170*(14*a^2*b^6 - b^8)*sin(d*x + c)^11 + 5148*
(7*a^3*b^5 - 2*a*b^7)*sin(d*x + c)^10 + 25740*a^7*b*sin(d*x + c)^2 + 715*(70*a^4*b^4 - 56*a^2*b^6 + b^8)*sin(d
*x + c)^9 + 6435*a^8*sin(d*x + c) + 6435*(7*a^5*b^3 - 14*a^3*b^5 + a*b^7)*sin(d*x + c)^8 + 25740*(a^6*b^2 - 5*
a^4*b^4 + a^2*b^6)*sin(d*x + c)^7 + 8580*(a^7*b - 14*a^5*b^3 + 7*a^3*b^5)*sin(d*x + c)^6 + 1287*(a^8 - 56*a^6*
b^2 + 70*a^4*b^4)*sin(d*x + c)^5 - 12870*(2*a^7*b - 7*a^5*b^3)*sin(d*x + c)^4 - 4290*(a^8 - 14*a^6*b^2)*sin(d*
x + c)^3)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (134) = 268\).

Time = 0.44 (sec) , antiderivative size = 464, normalized size of antiderivative = 3.22 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {a b^{7} \cos \left (12 \, d x + 12 \, c\right )}{3072 \, d} + \frac {b^{8} \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {{\left (14 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (10 \, d x + 10 \, c\right )}{1280 \, d} + \frac {{\left (28 \, a^{5} b^{3} - a b^{7}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {{\left (32 \, a^{7} b - 112 \, a^{5} b^{3} - 70 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (6 \, d x + 6 \, c\right )}{768 \, d} - \frac {{\left (256 \, a^{7} b + 224 \, a^{5} b^{3} - 5 \, a b^{7}\right )} \cos \left (4 \, d x + 4 \, c\right )}{1024 \, d} - \frac {{\left (80 \, a^{7} b + 168 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {{\left (112 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac {{\left (560 \, a^{4} b^{4} + 56 \, a^{2} b^{6} - b^{8}\right )} \sin \left (9 \, d x + 9 \, c\right )}{18432 \, d} - \frac {{\left (128 \, a^{6} b^{2} - 80 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - b^{8}\right )} \sin \left (7 \, d x + 7 \, c\right )}{2048 \, d} + \frac {{\left (256 \, a^{8} - 5376 \, a^{6} b^{2} - 4480 \, a^{4} b^{4} - 560 \, a^{2} b^{6} - 5 \, b^{8}\right )} \sin \left (5 \, d x + 5 \, c\right )}{20480 \, d} + \frac {{\left (1280 \, a^{8} - 1792 \, a^{6} b^{2} - 4480 \, a^{4} b^{4} - 1120 \, a^{2} b^{6} - 25 \, b^{8}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12288 \, d} + \frac {5 \, {\left (128 \, a^{8} + 448 \, a^{6} b^{2} + 336 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{1024 \, d} \]

[In]

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c))^8,x, algorithm="giac")

[Out]

1/3072*a*b^7*cos(12*d*x + 12*c)/d + 1/53248*b^8*sin(13*d*x + 13*c)/d - 1/1280*(14*a^3*b^5 + a*b^7)*cos(10*d*x
+ 10*c)/d + 1/512*(28*a^5*b^3 - a*b^7)*cos(8*d*x + 8*c)/d - 1/768*(32*a^7*b - 112*a^5*b^3 - 70*a^3*b^5 - 5*a*b
^7)*cos(6*d*x + 6*c)/d - 1/1024*(256*a^7*b + 224*a^5*b^3 - 5*a*b^7)*cos(4*d*x + 4*c)/d - 1/128*(80*a^7*b + 168
*a^5*b^3 + 70*a^3*b^5 + 5*a*b^7)*cos(2*d*x + 2*c)/d - 1/45056*(112*a^2*b^6 + 3*b^8)*sin(11*d*x + 11*c)/d + 1/1
8432*(560*a^4*b^4 + 56*a^2*b^6 - b^8)*sin(9*d*x + 9*c)/d - 1/2048*(128*a^6*b^2 - 80*a^4*b^4 - 40*a^2*b^6 - b^8
)*sin(7*d*x + 7*c)/d + 1/20480*(256*a^8 - 5376*a^6*b^2 - 4480*a^4*b^4 - 560*a^2*b^6 - 5*b^8)*sin(5*d*x + 5*c)/
d + 1/12288*(1280*a^8 - 1792*a^6*b^2 - 4480*a^4*b^4 - 1120*a^2*b^6 - 25*b^8)*sin(3*d*x + 3*c)/d + 5/1024*(128*
a^8 + 448*a^6*b^2 + 336*a^4*b^4 + 56*a^2*b^6 + b^8)*sin(d*x + c)/d

Mupad [B] (verification not implemented)

Time = 4.72 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.12 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {a^8}{5}-\frac {56\,a^6\,b^2}{5}+14\,a^4\,b^4\right )+{\sin \left (c+d\,x\right )}^9\,\left (\frac {70\,a^4\,b^4}{9}-\frac {56\,a^2\,b^6}{9}+\frac {b^8}{9}\right )+a^8\,\sin \left (c+d\,x\right )+\frac {b^8\,{\sin \left (c+d\,x\right )}^{13}}{13}-{\sin \left (c+d\,x\right )}^4\,\left (4\,a^7\,b-14\,a^5\,b^3\right )-{\sin \left (c+d\,x\right )}^{10}\,\left (\frac {8\,a\,b^7}{5}-\frac {28\,a^3\,b^5}{5}\right )-\frac {2\,a^6\,{\sin \left (c+d\,x\right )}^3\,\left (a^2-14\,b^2\right )}{3}+4\,a^7\,b\,{\sin \left (c+d\,x\right )}^2+\frac {2\,a\,b^7\,{\sin \left (c+d\,x\right )}^{12}}{3}+\frac {2\,b^6\,{\sin \left (c+d\,x\right )}^{11}\,\left (14\,a^2-b^2\right )}{11}+\frac {4\,a^3\,b\,{\sin \left (c+d\,x\right )}^6\,\left (a^4-14\,a^2\,b^2+7\,b^4\right )}{3}+a\,b^3\,{\sin \left (c+d\,x\right )}^8\,\left (7\,a^4-14\,a^2\,b^2+b^4\right )+4\,a^2\,b^2\,{\sin \left (c+d\,x\right )}^7\,\left (a^4-5\,a^2\,b^2+b^4\right )}{d} \]

[In]

int(cos(c + d*x)^5*(a + b*sin(c + d*x))^8,x)

[Out]

(sin(c + d*x)^5*(a^8/5 + 14*a^4*b^4 - (56*a^6*b^2)/5) + sin(c + d*x)^9*(b^8/9 - (56*a^2*b^6)/9 + (70*a^4*b^4)/
9) + a^8*sin(c + d*x) + (b^8*sin(c + d*x)^13)/13 - sin(c + d*x)^4*(4*a^7*b - 14*a^5*b^3) - sin(c + d*x)^10*((8
*a*b^7)/5 - (28*a^3*b^5)/5) - (2*a^6*sin(c + d*x)^3*(a^2 - 14*b^2))/3 + 4*a^7*b*sin(c + d*x)^2 + (2*a*b^7*sin(
c + d*x)^12)/3 + (2*b^6*sin(c + d*x)^11*(14*a^2 - b^2))/11 + (4*a^3*b*sin(c + d*x)^6*(a^4 + 7*b^4 - 14*a^2*b^2
))/3 + a*b^3*sin(c + d*x)^8*(7*a^4 + b^4 - 14*a^2*b^2) + 4*a^2*b^2*sin(c + d*x)^7*(a^4 + b^4 - 5*a^2*b^2))/d