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

Optimal. Leaf size=65 \[ -\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2669, 2635, 8} \[ -\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rubi steps

\begin {align*} \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx &=-\frac {a \cos ^5(c+d x)}{5 d}+a \int \cos ^4(c+d x) \, dx\\ &=-\frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (3 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac {a \cos ^5(c+d x)}{5 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} (3 a) \int 1 \, dx\\ &=\frac {3 a x}{8}-\frac {a \cos ^5(c+d x)}{5 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 62, normalized size = 0.95 \[ \frac {3 a (c+d x)}{8 d}+\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \sin (4 (c+d x))}{32 d}-\frac {a \cos ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*(c + d*x))/(8*d) - (a*Cos[c + d*x]^5)/(5*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)

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fricas [A]  time = 0.61, size = 51, normalized size = 0.78 \[ -\frac {8 \, a \cos \left (d x + c\right )^{5} - 15 \, a d x - 5 \, {\left (2 \, a \cos \left (d x + c\right )^{3} + 3 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{40 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40*(8*a*cos(d*x + c)^5 - 15*a*d*x - 5*(2*a*cos(d*x + c)^3 + 3*a*cos(d*x + c))*sin(d*x + c))/d

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giac [A]  time = 0.69, size = 77, normalized size = 1.18 \[ \frac {3}{8} \, a x - \frac {a \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac {a \cos \left (d x + c\right )}{8 \, d} + \frac {a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*a*x - 1/80*a*cos(5*d*x + 5*c)/d - 1/16*a*cos(3*d*x + 3*c)/d - 1/8*a*cos(d*x + c)/d + 1/32*a*sin(4*d*x + 4*
c)/d + 1/4*a*sin(2*d*x + 2*c)/d

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maple [A]  time = 0.14, size = 52, normalized size = 0.80 \[ \frac {-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) a}{5}+a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.33, size = 48, normalized size = 0.74 \[ -\frac {32 \, a \cos \left (d x + c\right )^{5} - 5 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{160 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/160*(32*a*cos(d*x + c)^5 - 5*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a)/d

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mupad [B]  time = 7.99, size = 165, normalized size = 2.54 \[ \frac {3\,a\,x}{8}+\frac {-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\left (\frac {a\,\left (75\,c+75\,d\,x-80\right )}{40}-\frac {15\,a\,\left (c+d\,x\right )}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\left (\frac {a\,\left (150\,c+150\,d\,x-160\right )}{40}-\frac {15\,a\,\left (c+d\,x\right )}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {a\,\left (15\,c+15\,d\,x-16\right )}{40}-\frac {3\,a\,\left (c+d\,x\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a*x)/8 + ((a*(15*c + 15*d*x - 16))/40 + (5*a*tan(c/2 + (d*x)/2))/4 - (3*a*(c + d*x))/8 + tan(c/2 + (d*x)/2)
^8*((a*(75*c + 75*d*x - 80))/40 - (15*a*(c + d*x))/8) + tan(c/2 + (d*x)/2)^4*((a*(150*c + 150*d*x - 160))/40 -
(15*a*(c + d*x))/4) + (a*tan(c/2 + (d*x)/2)^3)/2 - (a*tan(c/2 + (d*x)/2)^7)/2 - (5*a*tan(c/2 + (d*x)/2)^9)/4)
/(d*(tan(c/2 + (d*x)/2)^2 + 1)^5)

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sympy [A]  time = 2.18, size = 124, normalized size = 1.91 \[ \begin {cases} \frac {3 a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {a \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \cos ^{4}{\relax (c )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*(a+a*sin(d*x+c)),x)

[Out]

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

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