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

Optimal. Leaf size=144 \[ \frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac {2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac {(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac {a (a+b \sin (c+d x))^{12}}{3 b^5 d} \]

[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

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Rubi [A]  time = 0.22, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2668, 697} \[ \frac {2 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac {2 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac {\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac {(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac {a (a+b \sin (c+d x))^{12}}{3 b^5 d} \]

Antiderivative was successfully verified.

[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 697

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 2668

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*} \int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx &=\frac {\operatorname {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 {\operatorname {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*}

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Mathematica [A]  time = 1.99, size = 120, normalized size = 0.83 \[ \frac {\frac {2}{11} \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{11}+\frac {1}{9} \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^9+\frac {1}{13} (a+b \sin (c+d x))^{13}-\frac {1}{3} a (a+b \sin (c+d x))^{12}-\frac {2}{5} a (a-b) (a+b) (a+b \sin (c+d x))^{10}}{b^5 d} \]

Antiderivative was successfully verified.

[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)

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fricas [B]  time = 0.56, size = 356, normalized size = 2.47 \[ \frac {4290 \, a b^{7} \cos \left (d x + c\right )^{12} - 5148 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{10} + 6435 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{8} - 8580 \, {\left (a^{7} b + 7 \, a^{5} b^{3} + 7 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{6} + {\left (495 \, b^{8} \cos \left (d x + c\right )^{12} - 180 \, {\left (91 \, a^{2} b^{6} + 10 \, b^{8}\right )} \cos \left (d x + c\right )^{10} + 10 \, {\left (5005 \, a^{4} b^{4} + 4186 \, a^{2} b^{6} + 229 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 3432 \, a^{8} + 13728 \, a^{6} b^{2} + 11440 \, a^{4} b^{4} + 2080 \, a^{2} b^{6} + 40 \, b^{8} - 20 \, {\left (1287 \, a^{6} b^{2} + 3575 \, a^{4} b^{4} + 1469 \, a^{2} b^{6} + 53 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 3 \, {\left (429 \, a^{8} + 1716 \, a^{6} b^{2} + 1430 \, a^{4} b^{4} + 260 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (429 \, a^{8} + 1716 \, a^{6} b^{2} + 1430 \, a^{4} b^{4} + 260 \, a^{2} b^{6} + 5 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6435 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [B]  time = 3.94, size = 464, normalized size = 3.22 \[ \frac {a b^{7} \cos \left (12 \, d x + 12 \, c\right )}{3072 \, d} + \frac {b^{8} \sin \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {{\left (14 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (10 \, d x + 10 \, c\right )}{1280 \, d} + \frac {{\left (28 \, a^{5} b^{3} - a b^{7}\right )} \cos \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {{\left (32 \, a^{7} b - 112 \, a^{5} b^{3} - 70 \, a^{3} b^{5} - 5 \, a b^{7}\right )} \cos \left (6 \, d x + 6 \, c\right )}{768 \, d} - \frac {{\left (256 \, a^{7} b + 224 \, a^{5} b^{3} - 5 \, a b^{7}\right )} \cos \left (4 \, d x + 4 \, c\right )}{1024 \, d} - \frac {{\left (80 \, a^{7} b + 168 \, a^{5} b^{3} + 70 \, a^{3} b^{5} + 5 \, a b^{7}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {{\left (112 \, a^{2} b^{6} + 3 \, b^{8}\right )} \sin \left (11 \, d x + 11 \, c\right )}{45056 \, d} + \frac {{\left (560 \, a^{4} b^{4} + 56 \, a^{2} b^{6} - b^{8}\right )} \sin \left (9 \, d x + 9 \, c\right )}{18432 \, d} - \frac {{\left (128 \, a^{6} b^{2} - 80 \, a^{4} b^{4} - 40 \, a^{2} b^{6} - b^{8}\right )} \sin \left (7 \, d x + 7 \, c\right )}{2048 \, d} + \frac {{\left (256 \, a^{8} - 5376 \, a^{6} b^{2} - 4480 \, a^{4} b^{4} - 560 \, a^{2} b^{6} - 5 \, b^{8}\right )} \sin \left (5 \, d x + 5 \, c\right )}{20480 \, d} + \frac {{\left (1280 \, a^{8} - 1792 \, a^{6} b^{2} - 4480 \, a^{4} b^{4} - 1120 \, a^{2} b^{6} - 25 \, b^{8}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12288 \, d} + \frac {5 \, {\left (128 \, a^{8} + 448 \, a^{6} b^{2} + 336 \, a^{4} b^{4} + 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )}{1024 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [B]  time = 0.27, size = 530, normalized size = 3.68 \[ \frac {b^{8} \left (-\frac {\left (\sin ^{7}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{13}-\frac {7 \left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{143}-\frac {35 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{1287}-\frac {5 \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{429}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{429}\right )+8 a \,b^{7} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{20}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{40}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{120}\right )+28 a^{2} b^{6} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{11}-\frac {5 \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{99}-\frac {5 \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{231}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{231}\right )+56 a^{3} b^{5} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{20}-\frac {\left (\cos ^{6}\left (d x +c \right )\right )}{60}\right )+70 a^{4} b^{4} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{6}\left (d x +c \right )\right )}{9}-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{21}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )+56 a^{5} 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 )+28 a^{6} b^{2} \left (-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {4 a^{7} b \left (\cos ^{6}\left (d x +c \right )\right )}{3}+\frac {a^{8} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^8*(-1/13*sin(d*x+c)^7*cos(d*x+c)^6-7/143*sin(d*x+c)^5*cos(d*x+c)^6-35/1287*sin(d*x+c)^3*cos(d*x+c)^6-5/
429*cos(d*x+c)^6*sin(d*x+c)+1/429*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+8*a*b^7*(-1/12*sin(d*x+c)^6*
cos(d*x+c)^6-1/20*sin(d*x+c)^4*cos(d*x+c)^6-1/40*sin(d*x+c)^2*cos(d*x+c)^6-1/120*cos(d*x+c)^6)+28*a^2*b^6*(-1/
11*sin(d*x+c)^5*cos(d*x+c)^6-5/99*sin(d*x+c)^3*cos(d*x+c)^6-5/231*cos(d*x+c)^6*sin(d*x+c)+1/231*(8/3+cos(d*x+c
)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+56*a^3*b^5*(-1/10*sin(d*x+c)^4*cos(d*x+c)^6-1/20*sin(d*x+c)^2*cos(d*x+c)^6-1
/60*cos(d*x+c)^6)+70*a^4*b^4*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*cos(d*x+c)^6*sin(d*x+c)+1/105*(8/3+cos(d*x+c
)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))+56*a^5*b^3*(-1/8*sin(d*x+c)^2*cos(d*x+c)^6-1/24*cos(d*x+c)^6)+28*a^6*b^2*(-1
/7*cos(d*x+c)^6*sin(d*x+c)+1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-4/3*a^7*b*cos(d*x+c)^6+1/5*a^8
*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))

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maxima [B]  time = 0.33, size = 311, normalized size = 2.16 \[ \frac {495 \, b^{8} \sin \left (d x + c\right )^{13} + 4290 \, a b^{7} \sin \left (d x + c\right )^{12} + 1170 \, {\left (14 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{11} + 5148 \, {\left (7 \, a^{3} b^{5} - 2 \, a b^{7}\right )} \sin \left (d x + c\right )^{10} + 25740 \, a^{7} b \sin \left (d x + c\right )^{2} + 715 \, {\left (70 \, a^{4} b^{4} - 56 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{9} + 6435 \, a^{8} \sin \left (d x + c\right ) + 6435 \, {\left (7 \, a^{5} b^{3} - 14 \, a^{3} b^{5} + a b^{7}\right )} \sin \left (d x + c\right )^{8} + 25740 \, {\left (a^{6} b^{2} - 5 \, a^{4} b^{4} + a^{2} b^{6}\right )} \sin \left (d x + c\right )^{7} + 8580 \, {\left (a^{7} b - 14 \, a^{5} b^{3} + 7 \, a^{3} b^{5}\right )} \sin \left (d x + c\right )^{6} + 1287 \, {\left (a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4}\right )} \sin \left (d x + c\right )^{5} - 12870 \, {\left (2 \, a^{7} b - 7 \, a^{5} b^{3}\right )} \sin \left (d x + c\right )^{4} - 4290 \, {\left (a^{8} - 14 \, a^{6} b^{2}\right )} \sin \left (d x + c\right )^{3}}{6435 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 5.45, size = 306, normalized size = 2.12 \[ \frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {a^8}{5}-\frac {56\,a^6\,b^2}{5}+14\,a^4\,b^4\right )+{\sin \left (c+d\,x\right )}^9\,\left (\frac {70\,a^4\,b^4}{9}-\frac {56\,a^2\,b^6}{9}+\frac {b^8}{9}\right )+a^8\,\sin \left (c+d\,x\right )+\frac {b^8\,{\sin \left (c+d\,x\right )}^{13}}{13}-{\sin \left (c+d\,x\right )}^4\,\left (4\,a^7\,b-14\,a^5\,b^3\right )-{\sin \left (c+d\,x\right )}^{10}\,\left (\frac {8\,a\,b^7}{5}-\frac {28\,a^3\,b^5}{5}\right )-\frac {2\,a^6\,{\sin \left (c+d\,x\right )}^3\,\left (a^2-14\,b^2\right )}{3}+4\,a^7\,b\,{\sin \left (c+d\,x\right )}^2+\frac {2\,a\,b^7\,{\sin \left (c+d\,x\right )}^{12}}{3}+\frac {2\,b^6\,{\sin \left (c+d\,x\right )}^{11}\,\left (14\,a^2-b^2\right )}{11}+\frac {4\,a^3\,b\,{\sin \left (c+d\,x\right )}^6\,\left (a^4-14\,a^2\,b^2+7\,b^4\right )}{3}+a\,b^3\,{\sin \left (c+d\,x\right )}^8\,\left (7\,a^4-14\,a^2\,b^2+b^4\right )+4\,a^2\,b^2\,{\sin \left (c+d\,x\right )}^7\,\left (a^4-5\,a^2\,b^2+b^4\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 119.11, size = 614, normalized size = 4.26 \[ \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 {\relax (c )}\right )^{8} \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+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))

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