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

Optimal. Leaf size=87 \[ -\frac{(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac{8 (a \sin (c+d x)+a)^5}{5 a^4 d} \]

[Out]

(8*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(a + a*Sin[c + d*x])^6)/(a^5*d) + (6*(a + a*Sin[c + d*x])^7)/(7*a^6*
d) - (a + a*Sin[c + d*x])^8/(8*a^7*d)

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Rubi [A]  time = 0.0555053, antiderivative size = 87, 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 = {2667, 43} \[ -\frac{(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac{8 (a \sin (c+d x)+a)^5}{5 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(a + a*Sin[c + d*x])^6)/(a^5*d) + (6*(a + a*Sin[c + d*x])^7)/(7*a^6*
d) - (a + a*Sin[c + d*x])^8/(8*a^7*d)

Rule 2667

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 + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (8 a^3 (a+x)^4-12 a^2 (a+x)^5+6 a (a+x)^6-(a+x)^7\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{8 (a+a \sin (c+d x))^5}{5 a^4 d}-\frac{2 (a+a \sin (c+d x))^6}{a^5 d}+\frac{6 (a+a \sin (c+d x))^7}{7 a^6 d}-\frac{(a+a \sin (c+d x))^8}{8 a^7 d}\\ \end{align*}

Mathematica [A]  time = 0.0464023, size = 74, normalized size = 0.85 \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}-\frac{a \cos ^8(c+d x)}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Cos[c + d*x]^8)/(8*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c
+ d*x]^7)/(7*d)

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Maple [A]  time = 0.029, size = 56, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{8}}+{\frac{a\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(cos(d*x+c)^7*(a+a*sin(d*x+c)),x)

[Out]

1/d*(-1/8*a*cos(d*x+c)^8+1/7*a*(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 [A]  time = 0.94879, size = 124, normalized size = 1.43 \begin{align*} -\frac{35 \, a \sin \left (d x + c\right )^{8} + 40 \, a \sin \left (d x + c\right )^{7} - 140 \, a \sin \left (d x + c\right )^{6} - 168 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} + 280 \, a \sin \left (d x + c\right )^{3} - 140 \, a \sin \left (d x + c\right )^{2} - 280 \, a \sin \left (d x + c\right )}{280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/280*(35*a*sin(d*x + c)^8 + 40*a*sin(d*x + c)^7 - 140*a*sin(d*x + c)^6 - 168*a*sin(d*x + c)^5 + 210*a*sin(d*
x + c)^4 + 280*a*sin(d*x + c)^3 - 140*a*sin(d*x + c)^2 - 280*a*sin(d*x + c))/d

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Fricas [A]  time = 1.71391, size = 161, normalized size = 1.85 \begin{align*} -\frac{35 \, a \cos \left (d x + c\right )^{8} - 8 \,{\left (5 \, a \cos \left (d x + c\right )^{6} + 6 \, a \cos \left (d x + c\right )^{4} + 8 \, a \cos \left (d x + c\right )^{2} + 16 \, a\right )} \sin \left (d x + c\right )}{280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/280*(35*a*cos(d*x + c)^8 - 8*(5*a*cos(d*x + c)^6 + 6*a*cos(d*x + c)^4 + 8*a*cos(d*x + c)^2 + 16*a)*sin(d*x
+ c))/d

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Sympy [A]  time = 12.6511, size = 105, normalized size = 1.21 \begin{align*} \begin{cases} \frac{16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{a \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac{a \cos ^{8}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \cos ^{7}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.18415, size = 159, normalized size = 1.83 \begin{align*} -\frac{a \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac{a \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac{7 \, a \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{7 \, a \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac{a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac{35 \, a \sin \left (d x + c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024*a*cos(8*d*x + 8*c)/d - 1/128*a*cos(6*d*x + 6*c)/d - 7/256*a*cos(4*d*x + 4*c)/d - 7/128*a*cos(2*d*x + 2
*c)/d + 1/448*a*sin(7*d*x + 7*c)/d + 7/320*a*sin(5*d*x + 5*c)/d + 7/64*a*sin(3*d*x + 3*c)/d + 35/64*a*sin(d*x
+ c)/d