3.34 \(\int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx\)

Optimal. Leaf size=102 \[ \frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^4 x}{2} \]

[Out]

(7*a^4*x)/2 + (8*a^4*Sin[c + d*x])/d + (7*a^4*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^4*Cos[c + d*x]^3*Sin[c + d
*x])/d - (8*a^4*Sin[c + d*x]^3)/(3*d) + (a^4*Sin[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.106915, antiderivative size = 114, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2751, 2645, 2637, 2635, 8, 2633} \[ -\frac{16 a^4 \sin ^3(c+d x)}{15 d}+\frac{32 a^4 \sin (c+d x)}{5 d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{5 d}+\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{10 d}+\frac{7 a^4 x}{2}+\frac{\sin (c+d x) (a \cos (c+d x)+a)^4}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]*(a + a*Cos[c + d*x])^4,x]

[Out]

(7*a^4*x)/2 + (32*a^4*Sin[c + d*x])/(5*d) + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(10*d) + (a^4*Cos[c + d*x]^3*Si
n[c + d*x])/(5*d) + ((a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d) - (16*a^4*Sin[c + d*x]^3)/(15*d)

Rule 2751

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(d
*Cos[e + f*x]*(a + b*Sin[e + f*x])^m)/(f*(m + 1)), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*Sin
[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m,
-2^(-1)]

Rule 2645

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

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

Rubi steps

\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x))^4 \, dx &=\frac{(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{4}{5} \int (a+a \cos (c+d x))^4 \, dx\\ &=\frac{(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{4}{5} \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx\\ &=\frac{4 a^4 x}{5}+\frac{(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} \left (4 a^4\right ) \int \cos ^4(c+d x) \, dx+\frac{1}{5} \left (16 a^4\right ) \int \cos (c+d x) \, dx+\frac{1}{5} \left (16 a^4\right ) \int \cos ^3(c+d x) \, dx+\frac{1}{5} \left (24 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{4 a^4 x}{5}+\frac{16 a^4 \sin (c+d x)}{5 d}+\frac{12 a^4 \cos (c+d x) \sin (c+d x)}{5 d}+\frac{a^4 \cos ^3(c+d x) \sin (c+d x)}{5 d}+\frac{(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}+\frac{1}{5} \left (3 a^4\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{5} \left (12 a^4\right ) \int 1 \, dx-\frac{\left (16 a^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=\frac{16 a^4 x}{5}+\frac{32 a^4 \sin (c+d x)}{5 d}+\frac{27 a^4 \cos (c+d x) \sin (c+d x)}{10 d}+\frac{a^4 \cos ^3(c+d x) \sin (c+d x)}{5 d}+\frac{(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac{16 a^4 \sin ^3(c+d x)}{15 d}+\frac{1}{10} \left (3 a^4\right ) \int 1 \, dx\\ &=\frac{7 a^4 x}{2}+\frac{32 a^4 \sin (c+d x)}{5 d}+\frac{27 a^4 \cos (c+d x) \sin (c+d x)}{10 d}+\frac{a^4 \cos ^3(c+d x) \sin (c+d x)}{5 d}+\frac{(a+a \cos (c+d x))^4 \sin (c+d x)}{5 d}-\frac{16 a^4 \sin ^3(c+d x)}{15 d}\\ \end{align*}

Mathematica [A]  time = 0.141688, size = 63, normalized size = 0.62 \[ \frac{a^4 (1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x))+840 d x)}{240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]*(a + a*Cos[c + d*x])^4,x]

[Out]

(a^4*(840*d*x + 1470*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 145*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] + 3*Sin[
5*(c + d*x)]))/(240*d)

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Maple [A]  time = 0.042, size = 133, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+4\,{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +2\,{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) +4\,{a}^{4} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{4}\sin \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)*(a+cos(d*x+c)*a)^4,x)

[Out]

1/d*(1/5*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c
)+3/8*d*x+3/8*c)+2*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+4*a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+a^4*sin(d*x
+c))

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Maxima [A]  time = 1.14421, size = 173, normalized size = 1.7 \begin{align*} \frac{8 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{4} - 240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{4} + 15 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} + 120 \, a^{4} \sin \left (d x + c\right )}{120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/120*(8*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^4 - 240*(sin(d*x + c)^3 - 3*sin(d*x + c))*
a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^4 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*a^
4 + 120*a^4*sin(d*x + c))/d

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Fricas [A]  time = 1.6471, size = 190, normalized size = 1.86 \begin{align*} \frac{105 \, a^{4} d x +{\left (6 \, a^{4} \cos \left (d x + c\right )^{4} + 30 \, a^{4} \cos \left (d x + c\right )^{3} + 68 \, a^{4} \cos \left (d x + c\right )^{2} + 105 \, a^{4} \cos \left (d x + c\right ) + 166 \, a^{4}\right )} \sin \left (d x + c\right )}{30 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/30*(105*a^4*d*x + (6*a^4*cos(d*x + c)^4 + 30*a^4*cos(d*x + c)^3 + 68*a^4*cos(d*x + c)^2 + 105*a^4*cos(d*x +
c) + 166*a^4)*sin(d*x + c))/d

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Sympy [A]  time = 2.50715, size = 280, normalized size = 2.75 \begin{align*} \begin{cases} \frac{3 a^{4} x \sin ^{4}{\left (c + d x \right )}}{2} + 3 a^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} + 2 a^{4} x \sin ^{2}{\left (c + d x \right )} + \frac{3 a^{4} x \cos ^{4}{\left (c + d x \right )}}{2} + 2 a^{4} x \cos ^{2}{\left (c + d x \right )} + \frac{8 a^{4} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{3 a^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{4 a^{4} \sin ^{3}{\left (c + d x \right )}}{d} + \frac{a^{4} \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{5 a^{4} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac{6 a^{4} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{2 a^{4} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{a^{4} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right )^{4} \cos{\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)*(a+a*cos(d*x+c))**4,x)

[Out]

Piecewise((3*a**4*x*sin(c + d*x)**4/2 + 3*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 2*a**4*x*sin(c + d*x)**2 +
3*a**4*x*cos(c + d*x)**4/2 + 2*a**4*x*cos(c + d*x)**2 + 8*a**4*sin(c + d*x)**5/(15*d) + 4*a**4*sin(c + d*x)**3
*cos(c + d*x)**2/(3*d) + 3*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 4*a**4*sin(c + d*x)**3/d + a**4*sin(c + d
*x)*cos(c + d*x)**4/d + 5*a**4*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 6*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 2*
a**4*sin(c + d*x)*cos(c + d*x)/d + a**4*sin(c + d*x)/d, Ne(d, 0)), (x*(a*cos(c) + a)**4*cos(c), True))

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Giac [A]  time = 1.2418, size = 120, normalized size = 1.18 \begin{align*} \frac{7}{2} \, a^{4} x + \frac{a^{4} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{a^{4} \sin \left (4 \, d x + 4 \, c\right )}{8 \, d} + \frac{29 \, a^{4} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{2 \, a^{4} \sin \left (2 \, d x + 2 \, c\right )}{d} + \frac{49 \, a^{4} \sin \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)*(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

7/2*a^4*x + 1/80*a^4*sin(5*d*x + 5*c)/d + 1/8*a^4*sin(4*d*x + 4*c)/d + 29/48*a^4*sin(3*d*x + 3*c)/d + 2*a^4*si
n(2*d*x + 2*c)/d + 49/8*a^4*sin(d*x + c)/d