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

Optimal. Leaf size=127 \[ -\frac{3 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^5}{5 d}+\frac{\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))^3}{d}-\frac{\left (a^2+b^2\right )^3 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac{(b \cos (c+d x)-a \sin (c+d x))^7}{7 d} \]

[Out]

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

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Rubi [A]  time = 0.0778204, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3072, 194} \[ -\frac{3 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^5}{5 d}+\frac{\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))^3}{d}-\frac{\left (a^2+b^2\right )^3 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac{(b \cos (c+d x)-a \sin (c+d x))^7}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3072

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int
[(a^2 + b^2 - x^2)^((n - 1)/2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 + b^2, 0] && IGtQ[(n - 1)/2, 0]

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 1.0378, size = 246, normalized size = 1.94 \[ \frac{1225 a \left (a^2+b^2\right )^3 \sin (c+d x)+245 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right )^2 \sin (3 (c+d x))+49 a \left (-9 a^4 b^2-5 a^2 b^4+a^6+5 b^6\right ) \sin (5 (c+d x))+5 a \left (-21 a^4 b^2+35 a^2 b^4+a^6-7 b^6\right ) \sin (7 (c+d x))-1225 b \left (a^2+b^2\right )^3 \cos (c+d x)+245 b \left (b^2-3 a^2\right ) \left (a^2+b^2\right )^2 \cos (3 (c+d x))-49 b \left (-5 a^4 b^2-9 a^2 b^4+5 a^6+b^6\right ) \cos (5 (c+d x))+5 b \left (35 a^4 b^2-21 a^2 b^4-7 a^6+b^6\right ) \cos (7 (c+d x))}{2240 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1225*b*(a^2 + b^2)^3*Cos[c + d*x] + 245*b*(-3*a^2 + b^2)*(a^2 + b^2)^2*Cos[3*(c + d*x)] - 49*b*(5*a^6 - 5*a^
4*b^2 - 9*a^2*b^4 + b^6)*Cos[5*(c + d*x)] + 5*b*(-7*a^6 + 35*a^4*b^2 - 21*a^2*b^4 + b^6)*Cos[7*(c + d*x)] + 12
25*a*(a^2 + b^2)^3*Sin[c + d*x] + 245*a*(a^2 - 3*b^2)*(a^2 + b^2)^2*Sin[3*(c + d*x)] + 49*a*(a^6 - 9*a^4*b^2 -
5*a^2*b^4 + 5*b^6)*Sin[5*(c + d*x)] + 5*a*(a^6 - 21*a^4*b^2 + 35*a^2*b^4 - 7*b^6)*Sin[7*(c + d*x)])/(2240*d)

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Maple [B]  time = 0.169, size = 321, normalized size = 2.5 \begin{align*}{\frac{1}{d} \left ( -{\frac{{b}^{7}\cos \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \sin \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+a{b}^{6} \left ( \sin \left ( dx+c \right ) \right ) ^{7}+21\,{a}^{2}{b}^{5} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{35}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{105}} \right ) +35\,{a}^{3}{b}^{4} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+1/35\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) +35\,{a}^{4}{b}^{3} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +21\,{a}^{5}{b}^{2} \left ( -1/7\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}+1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{a}^{6}b \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{{a}^{7}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-1/7*b^7*(16/5+sin(d*x+c)^6+6/5*sin(d*x+c)^4+8/5*sin(d*x+c)^2)*cos(d*x+c)+a*b^6*sin(d*x+c)^7+21*a^2*b^5*(
-1/7*sin(d*x+c)^4*cos(d*x+c)^3-4/35*sin(d*x+c)^2*cos(d*x+c)^3-8/105*cos(d*x+c)^3)+35*a^3*b^4*(-1/7*sin(d*x+c)^
3*cos(d*x+c)^4-3/35*sin(d*x+c)*cos(d*x+c)^4+1/35*(2+cos(d*x+c)^2)*sin(d*x+c))+35*a^4*b^3*(-1/7*sin(d*x+c)^2*co
s(d*x+c)^5-2/35*cos(d*x+c)^5)+21*a^5*b^2*(-1/7*sin(d*x+c)*cos(d*x+c)^6+1/35*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2
)*sin(d*x+c))-a^6*b*cos(d*x+c)^7+1/7*a^7*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [B]  time = 1.00866, size = 347, normalized size = 2.73 \begin{align*} -\frac{35 \, a^{6} b \cos \left (d x + c\right )^{7} - 35 \, a b^{6} \sin \left (d x + c\right )^{7} +{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{7} - 7 \,{\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{5} b^{2} - 35 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{4} b^{3} + 35 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{3} b^{4} + 7 \,{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b^{5} -{\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} b^{7}}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(35*a^6*b*cos(d*x + c)^7 - 35*a*b^6*sin(d*x + c)^7 + (5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x
+ c)^3 - 35*sin(d*x + c))*a^7 - 7*(15*sin(d*x + c)^7 - 42*sin(d*x + c)^5 + 35*sin(d*x + c)^3)*a^5*b^2 - 35*(5*
cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^4*b^3 + 35*(5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*a^3*b^4 + 7*(15*cos(d*x
+ c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^2*b^5 - (5*cos(d*x + c)^7 - 21*cos(d*x + c)^5 + 35*cos(d*x +
c)^3 - 35*cos(d*x + c))*b^7)/d

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Fricas [B]  time = 3.08474, size = 583, normalized size = 4.59 \begin{align*} -\frac{35 \, b^{7} \cos \left (d x + c\right ) + 5 \,{\left (7 \, a^{6} b - 35 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{7} + 7 \,{\left (35 \, a^{4} b^{3} - 42 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{5} + 35 \,{\left (7 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{3} -{\left (16 \, a^{7} + 56 \, a^{5} b^{2} + 70 \, a^{3} b^{4} + 35 \, a b^{6} + 5 \,{\left (a^{7} - 21 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \cos \left (d x + c\right )^{6} +{\left (6 \, a^{7} + 21 \, a^{5} b^{2} - 280 \, a^{3} b^{4} + 105 \, a b^{6}\right )} \cos \left (d x + c\right )^{4} +{\left (8 \, a^{7} + 28 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 105 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/35*(35*b^7*cos(d*x + c) + 5*(7*a^6*b - 35*a^4*b^3 + 21*a^2*b^5 - b^7)*cos(d*x + c)^7 + 7*(35*a^4*b^3 - 42*a
^2*b^5 + 3*b^7)*cos(d*x + c)^5 + 35*(7*a^2*b^5 - b^7)*cos(d*x + c)^3 - (16*a^7 + 56*a^5*b^2 + 70*a^3*b^4 + 35*
a*b^6 + 5*(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*cos(d*x + c)^6 + (6*a^7 + 21*a^5*b^2 - 280*a^3*b^4 + 105*a
*b^6)*cos(d*x + c)^4 + (8*a^7 + 28*a^5*b^2 + 35*a^3*b^4 - 105*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/d

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Sympy [A]  time = 10.1715, size = 461, normalized size = 3.63 \begin{align*} \begin{cases} \frac{16 a^{7} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 a^{7} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 a^{7} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{a^{7} \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{a^{6} b \cos ^{7}{\left (c + d x \right )}}{d} + \frac{8 a^{5} b^{2} \sin ^{7}{\left (c + d x \right )}}{5 d} + \frac{28 a^{5} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{7 a^{5} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{7 a^{4} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{d} - \frac{2 a^{4} b^{3} \cos ^{7}{\left (c + d x \right )}}{d} + \frac{2 a^{3} b^{4} \sin ^{7}{\left (c + d x \right )}}{d} + \frac{7 a^{3} b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - \frac{7 a^{2} b^{5} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac{28 a^{2} b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{8 a^{2} b^{5} \cos ^{7}{\left (c + d x \right )}}{5 d} + \frac{a b^{6} \sin ^{7}{\left (c + d x \right )}}{d} - \frac{b^{7} \sin ^{6}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} - \frac{2 b^{7} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac{8 b^{7} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac{16 b^{7} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + b \sin{\left (c \right )}\right )^{7} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((16*a**7*sin(c + d*x)**7/(35*d) + 8*a**7*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*a**7*sin(c + d*x)
**3*cos(c + d*x)**4/d + a**7*sin(c + d*x)*cos(c + d*x)**6/d - a**6*b*cos(c + d*x)**7/d + 8*a**5*b**2*sin(c + d
*x)**7/(5*d) + 28*a**5*b**2*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 7*a**5*b**2*sin(c + d*x)**3*cos(c + d*x)**
4/d - 7*a**4*b**3*sin(c + d*x)**2*cos(c + d*x)**5/d - 2*a**4*b**3*cos(c + d*x)**7/d + 2*a**3*b**4*sin(c + d*x)
**7/d + 7*a**3*b**4*sin(c + d*x)**5*cos(c + d*x)**2/d - 7*a**2*b**5*sin(c + d*x)**4*cos(c + d*x)**3/d - 28*a**
2*b**5*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 8*a**2*b**5*cos(c + d*x)**7/(5*d) + a*b**6*sin(c + d*x)**7/d -
b**7*sin(c + d*x)**6*cos(c + d*x)/d - 2*b**7*sin(c + d*x)**4*cos(c + d*x)**3/d - 8*b**7*sin(c + d*x)**2*cos(c
+ d*x)**5/(5*d) - 16*b**7*cos(c + d*x)**7/(35*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**7, True))

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Giac [B]  time = 1.13206, size = 427, normalized size = 3.36 \begin{align*} -\frac{{\left (7 \, a^{6} b - 35 \, a^{4} b^{3} + 21 \, a^{2} b^{5} - b^{7}\right )} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{7 \,{\left (5 \, a^{6} b - 5 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + b^{7}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac{7 \,{\left (3 \, a^{6} b + 5 \, a^{4} b^{3} + a^{2} b^{5} - b^{7}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac{35 \,{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )}{64 \, d} + \frac{{\left (a^{7} - 21 \, a^{5} b^{2} + 35 \, a^{3} b^{4} - 7 \, a b^{6}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{7 \,{\left (a^{7} - 9 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 5 \, a b^{6}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{7 \,{\left (a^{7} - a^{5} b^{2} - 5 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac{35 \,{\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/448*(7*a^6*b - 35*a^4*b^3 + 21*a^2*b^5 - b^7)*cos(7*d*x + 7*c)/d - 7/320*(5*a^6*b - 5*a^4*b^3 - 9*a^2*b^5 +
b^7)*cos(5*d*x + 5*c)/d - 7/64*(3*a^6*b + 5*a^4*b^3 + a^2*b^5 - b^7)*cos(3*d*x + 3*c)/d - 35/64*(a^6*b + 3*a^
4*b^3 + 3*a^2*b^5 + b^7)*cos(d*x + c)/d + 1/448*(a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*sin(7*d*x + 7*c)/d +
7/320*(a^7 - 9*a^5*b^2 - 5*a^3*b^4 + 5*a*b^6)*sin(5*d*x + 5*c)/d + 7/64*(a^7 - a^5*b^2 - 5*a^3*b^4 - 3*a*b^6)
*sin(3*d*x + 3*c)/d + 35/64*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*sin(d*x + c)/d