3.630 \(\int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx\)

Optimal. Leaf size=254 \[ -\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac {6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac {(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)} \]

[Out]

-(a^2-b^2)^3*(a+b*sin(d*x+c))^(1+m)/b^7/d/(1+m)+6*a*(a^2-b^2)^2*(a+b*sin(d*x+c))^(2+m)/b^7/d/(2+m)-3*(5*a^4-6*
a^2*b^2+b^4)*(a+b*sin(d*x+c))^(3+m)/b^7/d/(3+m)+4*a*(5*a^2-3*b^2)*(a+b*sin(d*x+c))^(4+m)/b^7/d/(4+m)-3*(5*a^2-
b^2)*(a+b*sin(d*x+c))^(5+m)/b^7/d/(5+m)+6*a*(a+b*sin(d*x+c))^(6+m)/b^7/d/(6+m)-(a+b*sin(d*x+c))^(7+m)/b^7/d/(7
+m)

________________________________________________________________________________________

Rubi [A]  time = 0.16, antiderivative size = 254, 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} \[ -\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}-\frac {3 \left (-6 a^2 b^2+5 a^4+b^4\right ) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}+\frac {6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac {(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]

[Out]

-(((a^2 - b^2)^3*(a + b*Sin[c + d*x])^(1 + m))/(b^7*d*(1 + m))) + (6*a*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(2 +
m))/(b^7*d*(2 + m)) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*(a + b*Sin[c + d*x])^(3 + m))/(b^7*d*(3 + m)) + (4*a*(5*a^
2 - 3*b^2)*(a + b*Sin[c + d*x])^(4 + m))/(b^7*d*(4 + m)) - (3*(5*a^2 - b^2)*(a + b*Sin[c + d*x])^(5 + m))/(b^7
*d*(5 + m)) + (6*a*(a + b*Sin[c + d*x])^(6 + m))/(b^7*d*(6 + m)) - (a + b*Sin[c + d*x])^(7 + m)/(b^7*d*(7 + m)
)

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 ^7(c+d x) (a+b \sin (c+d x))^m \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^m \left (b^2-x^2\right )^3 \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\left (a^2-b^2\right )^3 (a+x)^m+6 a \left (a^2-b^2\right )^2 (a+x)^{1+m}-3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+x)^{2+m}+4 a \left (5 a^2-3 b^2\right ) (a+x)^{3+m}-3 \left (5 a^2-b^2\right ) (a+x)^{4+m}+6 a (a+x)^{5+m}-(a+x)^{6+m}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=-\frac {\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{1+m}}{b^7 d (1+m)}+\frac {6 a \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{2+m}}{b^7 d (2+m)}-\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right ) (a+b \sin (c+d x))^{3+m}}{b^7 d (3+m)}+\frac {4 a \left (5 a^2-3 b^2\right ) (a+b \sin (c+d x))^{4+m}}{b^7 d (4+m)}-\frac {3 \left (5 a^2-b^2\right ) (a+b \sin (c+d x))^{5+m}}{b^7 d (5+m)}+\frac {6 a (a+b \sin (c+d x))^{6+m}}{b^7 d (6+m)}-\frac {(a+b \sin (c+d x))^{7+m}}{b^7 d (7+m)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 6.11, size = 459, normalized size = 1.81 \[ \frac {\frac {6 \left (\left (b^2-a^2\right ) \left (\frac {4 \left (\left (b^2-a^2\right ) \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+1}}{m+1}+\frac {2 a (a+b \sin (c+d x))^{m+2}}{m+2}-\frac {(a+b \sin (c+d x))^{m+3}}{m+3}\right )+a \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+2}}{m+2}+\frac {2 a (a+b \sin (c+d x))^{m+3}}{m+3}-\frac {(a+b \sin (c+d x))^{m+4}}{m+4}\right )\right )}{m+5}+\frac {b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{m+1}}{m+5}\right )+a \left (\frac {4 \left (\left (b^2-a^2\right ) \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+2}}{m+2}+\frac {2 a (a+b \sin (c+d x))^{m+3}}{m+3}-\frac {(a+b \sin (c+d x))^{m+4}}{m+4}\right )+a \left (-\frac {\left (a^2-b^2\right ) (a+b \sin (c+d x))^{m+3}}{m+3}+\frac {2 a (a+b \sin (c+d x))^{m+4}}{m+4}-\frac {(a+b \sin (c+d x))^{m+5}}{m+5}\right )\right )}{m+6}+\frac {b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{m+2}}{m+6}\right )\right )}{m+7}+\frac {b^6 \cos ^6(c+d x) (a+b \sin (c+d x))^{m+1}}{m+7}}{b^7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]

[Out]

((b^6*Cos[c + d*x]^6*(a + b*Sin[c + d*x])^(1 + m))/(7 + m) + (6*((-a^2 + b^2)*((b^4*Cos[c + d*x]^4*(a + b*Sin[
c + d*x])^(1 + m))/(5 + m) + (4*((-a^2 + b^2)*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(1 + m))/(1 + m)) + (2*a*(a
+ b*Sin[c + d*x])^(2 + m))/(2 + m) - (a + b*Sin[c + d*x])^(3 + m)/(3 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c +
d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 + m)/(4 + m)))
)/(5 + m)) + a*((b^4*Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(2 + m))/(6 + m) + (4*((-a^2 + b^2)*(-(((a^2 - b^2)*(
a + b*Sin[c + d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 +
m)/(4 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(3 + m))/(3 + m)) + (2*a*(a + b*Sin[c + d*x])^(4 + m))/(
4 + m) - (a + b*Sin[c + d*x])^(5 + m)/(5 + m))))/(6 + m))))/(7 + m))/(b^7*d)

________________________________________________________________________________________

fricas [B]  time = 0.92, size = 814, normalized size = 3.20 \[ -\frac {{\left (720 \, a^{7} - 3024 \, a^{5} b^{2} + 5040 \, a^{3} b^{4} - 5040 \, a b^{6} - {\left (a b^{6} m^{6} + 15 \, a b^{6} m^{5} + 85 \, a b^{6} m^{4} + 225 \, a b^{6} m^{3} + 274 \, a b^{6} m^{2} + 120 \, a b^{6} m\right )} \cos \left (d x + c\right )^{6} - 6 \, {\left (2 \, a b^{6} m^{5} - {\left (5 \, a^{3} b^{4} - 23 \, a b^{6}\right )} m^{4} - 2 \, {\left (15 \, a^{3} b^{4} - 44 \, a b^{6}\right )} m^{3} - {\left (55 \, a^{3} b^{4} - 133 \, a b^{6}\right )} m^{2} - 6 \, {\left (5 \, a^{3} b^{4} - 11 \, a b^{6}\right )} m\right )} \cos \left (d x + c\right )^{4} - 192 \, {\left (a^{3} b^{4} + a b^{6}\right )} m^{3} + 288 \, {\left (a^{5} b^{2} - 2 \, a^{3} b^{4} - 7 \, a b^{6}\right )} m^{2} - 24 \, {\left ({\left (a^{3} b^{4} + 3 \, a b^{6}\right )} m^{4} - 6 \, {\left (a^{3} b^{4} - 5 \, a b^{6}\right )} m^{3} + {\left (15 \, a^{5} b^{2} - 55 \, a^{3} b^{4} + 84 \, a b^{6}\right )} m^{2} + 3 \, {\left (5 \, a^{5} b^{2} - 16 \, a^{3} b^{4} + 19 \, a b^{6}\right )} m\right )} \cos \left (d x + c\right )^{2} - 192 \, {\left (3 \, a^{5} b^{2} - 13 \, a^{3} b^{4} + 32 \, a b^{6}\right )} m - {\left (2304 \, b^{7} + {\left (b^{7} m^{6} + 21 \, b^{7} m^{5} + 175 \, b^{7} m^{4} + 735 \, b^{7} m^{3} + 1624 \, b^{7} m^{2} + 1764 \, b^{7} m + 720 \, b^{7}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (144 \, b^{7} + {\left (a^{2} b^{5} + b^{7}\right )} m^{5} + 2 \, {\left (5 \, a^{2} b^{5} + 8 \, b^{7}\right )} m^{4} + 5 \, {\left (7 \, a^{2} b^{5} + 19 \, b^{7}\right )} m^{3} + 10 \, {\left (5 \, a^{2} b^{5} + 26 \, b^{7}\right )} m^{2} + 12 \, {\left (2 \, a^{2} b^{5} + 27 \, b^{7}\right )} m\right )} \cos \left (d x + c\right )^{4} + 48 \, {\left (a^{4} b^{3} + 6 \, a^{2} b^{5} + b^{7}\right )} m^{3} - 576 \, {\left (a^{4} b^{3} - 4 \, a^{2} b^{5} - b^{7}\right )} m^{2} + 24 \, {\left (48 \, b^{7} + {\left (3 \, a^{2} b^{5} + b^{7}\right )} m^{4} - {\left (5 \, a^{4} b^{3} - 24 \, a^{2} b^{5} - 13 \, b^{7}\right )} m^{3} - {\left (15 \, a^{4} b^{3} - 51 \, a^{2} b^{5} - 56 \, b^{7}\right )} m^{2} - 2 \, {\left (5 \, a^{4} b^{3} - 15 \, a^{2} b^{5} - 46 \, b^{7}\right )} m\right )} \cos \left (d x + c\right )^{2} + 48 \, {\left (15 \, a^{6} b - 58 \, a^{4} b^{3} + 87 \, a^{2} b^{5} + 44 \, b^{7}\right )} m\right )} \sin \left (d x + c\right )\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{b^{7} d m^{7} + 28 \, b^{7} d m^{6} + 322 \, b^{7} d m^{5} + 1960 \, b^{7} d m^{4} + 6769 \, b^{7} d m^{3} + 13132 \, b^{7} d m^{2} + 13068 \, b^{7} d m + 5040 \, b^{7} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="fricas")

[Out]

-(720*a^7 - 3024*a^5*b^2 + 5040*a^3*b^4 - 5040*a*b^6 - (a*b^6*m^6 + 15*a*b^6*m^5 + 85*a*b^6*m^4 + 225*a*b^6*m^
3 + 274*a*b^6*m^2 + 120*a*b^6*m)*cos(d*x + c)^6 - 6*(2*a*b^6*m^5 - (5*a^3*b^4 - 23*a*b^6)*m^4 - 2*(15*a^3*b^4
- 44*a*b^6)*m^3 - (55*a^3*b^4 - 133*a*b^6)*m^2 - 6*(5*a^3*b^4 - 11*a*b^6)*m)*cos(d*x + c)^4 - 192*(a^3*b^4 + a
*b^6)*m^3 + 288*(a^5*b^2 - 2*a^3*b^4 - 7*a*b^6)*m^2 - 24*((a^3*b^4 + 3*a*b^6)*m^4 - 6*(a^3*b^4 - 5*a*b^6)*m^3
+ (15*a^5*b^2 - 55*a^3*b^4 + 84*a*b^6)*m^2 + 3*(5*a^5*b^2 - 16*a^3*b^4 + 19*a*b^6)*m)*cos(d*x + c)^2 - 192*(3*
a^5*b^2 - 13*a^3*b^4 + 32*a*b^6)*m - (2304*b^7 + (b^7*m^6 + 21*b^7*m^5 + 175*b^7*m^4 + 735*b^7*m^3 + 1624*b^7*
m^2 + 1764*b^7*m + 720*b^7)*cos(d*x + c)^6 + 6*(144*b^7 + (a^2*b^5 + b^7)*m^5 + 2*(5*a^2*b^5 + 8*b^7)*m^4 + 5*
(7*a^2*b^5 + 19*b^7)*m^3 + 10*(5*a^2*b^5 + 26*b^7)*m^2 + 12*(2*a^2*b^5 + 27*b^7)*m)*cos(d*x + c)^4 + 48*(a^4*b
^3 + 6*a^2*b^5 + b^7)*m^3 - 576*(a^4*b^3 - 4*a^2*b^5 - b^7)*m^2 + 24*(48*b^7 + (3*a^2*b^5 + b^7)*m^4 - (5*a^4*
b^3 - 24*a^2*b^5 - 13*b^7)*m^3 - (15*a^4*b^3 - 51*a^2*b^5 - 56*b^7)*m^2 - 2*(5*a^4*b^3 - 15*a^2*b^5 - 46*b^7)*
m)*cos(d*x + c)^2 + 48*(15*a^6*b - 58*a^4*b^3 + 87*a^2*b^5 + 44*b^7)*m)*sin(d*x + c))*(b*sin(d*x + c) + a)^m/(
b^7*d*m^7 + 28*b^7*d*m^6 + 322*b^7*d*m^5 + 1960*b^7*d*m^4 + 6769*b^7*d*m^3 + 13132*b^7*d*m^2 + 13068*b^7*d*m +
5040*b^7*d)

________________________________________________________________________________________

giac [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)^7*(a+b*sin(d*x+c))^m,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{7}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x)

[Out]

int(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x)

________________________________________________________________________________________

maxima [B]  time = 1.00, size = 558, normalized size = 2.20 \[ \frac {\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{m + 1}}{b {\left (m + 1\right )}} - \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} b^{3} \sin \left (d x + c\right )^{3} + {\left (m^{2} + m\right )} a b^{2} \sin \left (d x + c\right )^{2} - 2 \, a^{2} b m \sin \left (d x + c\right ) + 2 \, a^{3}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} b^{3}} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} b^{5} \sin \left (d x + c\right )^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a b^{4} \sin \left (d x + c\right )^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{2} b^{3} \sin \left (d x + c\right )^{3} + 12 \, {\left (m^{2} + m\right )} a^{3} b^{2} \sin \left (d x + c\right )^{2} - 24 \, a^{4} b m \sin \left (d x + c\right ) + 24 \, a^{5}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} b^{5}} - \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} b^{7} \sin \left (d x + c\right )^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} a b^{6} \sin \left (d x + c\right )^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} a^{2} b^{5} \sin \left (d x + c\right )^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} a^{3} b^{4} \sin \left (d x + c\right )^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} a^{4} b^{3} \sin \left (d x + c\right )^{3} + 360 \, {\left (m^{2} + m\right )} a^{5} b^{2} \sin \left (d x + c\right )^{2} - 720 \, a^{6} b m \sin \left (d x + c\right ) + 720 \, a^{7}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{m}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} b^{7}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*sin(d*x+c))^m,x, algorithm="maxima")

[Out]

((b*sin(d*x + c) + a)^(m + 1)/(b*(m + 1)) - 3*((m^2 + 3*m + 2)*b^3*sin(d*x + c)^3 + (m^2 + m)*a*b^2*sin(d*x +
c)^2 - 2*a^2*b*m*sin(d*x + c) + 2*a^3)*(b*sin(d*x + c) + a)^m/((m^3 + 6*m^2 + 11*m + 6)*b^3) + 3*((m^4 + 10*m^
3 + 35*m^2 + 50*m + 24)*b^5*sin(d*x + c)^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*a*b^4*sin(d*x + c)^4 - 4*(m^3 + 3*m^
2 + 2*m)*a^2*b^3*sin(d*x + c)^3 + 12*(m^2 + m)*a^3*b^2*sin(d*x + c)^2 - 24*a^4*b*m*sin(d*x + c) + 24*a^5)*(b*s
in(d*x + c) + a)^m/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*b^5) - ((m^6 + 21*m^5 + 175*m^4 + 735*m^3
+ 1624*m^2 + 1764*m + 720)*b^7*sin(d*x + c)^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*a*b^6*sin(
d*x + c)^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*a^2*b^5*sin(d*x + c)^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m
)*a^3*b^4*sin(d*x + c)^4 - 120*(m^3 + 3*m^2 + 2*m)*a^4*b^3*sin(d*x + c)^3 + 360*(m^2 + m)*a^5*b^2*sin(d*x + c)
^2 - 720*a^6*b*m*sin(d*x + c) + 720*a^7)*(b*sin(d*x + c) + a)^m/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3
+ 13132*m^2 + 13068*m + 5040)*b^7))/d

________________________________________________________________________________________

mupad [B]  time = 19.09, size = 1196, normalized size = 4.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7*(a + b*sin(c + d*x))^m,x)

[Out]

((a + b*sin(c + d*x))^m*(a*b^6*645120i - a^7*92160i - a^3*b^4*645120i + a^5*b^2*387072i - a^3*b^4*m*401856i +
a^5*b^2*m*96768i + a*b^6*m^2*436336i + a*b^6*m^3*105000i + a*b^6*m^4*14632i + a*b^6*m^5*1176i + a*b^6*m^6*40i
- a^3*b^4*m^2*26592i - a^5*b^2*m^2*13824i + a^3*b^4*m^3*6720i + a^3*b^4*m^4*96i + a*b^6*m*897792i))/(128*b^7*d
*(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (sin(7*c + 7*d*x)*(a
+ b*sin(c + d*x))^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720)*1i)/(64*d*(m*13068i + m^2*13
132i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (sin(c + d*x)*(a + b*sin(c + d*x))^m*(1
94868*b^7*m + 176400*b^7 + 78968*b^7*m^2 + 16299*b^7*m^3 + 2027*b^7*m^4 + 153*b^7*m^5 + 5*b^7*m^6 + 279936*a^2
*b^5*m - 182016*a^4*b^3*m + 169440*a^2*b^5*m^2 - 42624*a^4*b^3*m^2 + 29328*a^2*b^5*m^3 + 1152*a^4*b^3*m^3 + 16
32*a^2*b^5*m^4 + 48*a^2*b^5*m^5 + 46080*a^6*b*m)*1i)/(64*b^7*d*(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i
+ m^5*322i + m^6*28i + m^7*1i + 5040i)) + (sin(3*c + 3*d*x)*(a + b*sin(c + d*x))^m*(3*m + m^2 + 2)*(3602*b^4*m
- 640*a^4*m + 5880*b^4 + 797*b^4*m^2 + 78*b^4*m^3 + 3*b^4*m^4 + 2208*a^2*b^2*m + 552*a^2*b^2*m^2 + 24*a^2*b^2
*m^3)*3i)/(64*b^4*d*(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (
sin(5*c + 5*d*x)*(a + b*sin(c + d*x))^m*(24*a^2*m + 79*b^2*m + 294*b^2 + 5*b^2*m^2)*(50*m + 35*m^2 + 10*m^3 +
m^4 + 24)*1i)/(64*b^2*d*(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i))
+ (a*m*cos(6*c + 6*d*x)*(a + b*sin(c + d*x))^m*(m*274i + m^2*225i + m^3*85i + m^4*15i + m^5*1i + 120i))/(32*b
*d*(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (3*a*m*cos(4*c + 4
*d*x)*(a + b*sin(c + d*x))^m*(b^2*m*17i - a^2*20i + b^2*64i + b^2*m^2*1i)*(11*m + 6*m^2 + m^3 + 6))/(16*b^3*d*
(m*13068i + m^2*13132i + m^3*6769i + m^4*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i)) + (3*a*m*cos(2*c + 2*d*
x)*(m + 1)*(a + b*sin(c + d*x))^m*(b^4*m*6370i + a^4*1920i + b^4*10008i - a^2*b^2*7104i + b^4*m^2*1411i + b^4*
m^3*134i + b^4*m^4*5i - a^2*b^2*m*1696i - a^2*b^2*m^2*32i))/(32*b^5*d*(m*13068i + m^2*13132i + m^3*6769i + m^4
*1960i + m^5*322i + m^6*28i + m^7*1i + 5040i))

________________________________________________________________________________________

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)**7*(a+b*sin(d*x+c))**m,x)

[Out]

Timed out

________________________________________________________________________________________