3.355 \(\int (d \cos (a+b x))^n \sin ^5(a+b x) \, dx\)

Optimal. Leaf size=76 \[ \frac{2 (d \cos (a+b x))^{n+3}}{b d^3 (n+3)}-\frac{(d \cos (a+b x))^{n+5}}{b d^5 (n+5)}-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)} \]

[Out]

-((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n))) + (2*(d*Cos[a + b*x])^(3 + n))/(b*d^3*(3 + n)) - (d*Cos[a + b*x])^(5
+ n)/(b*d^5*(5 + n))

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Rubi [A]  time = 0.0655888, antiderivative size = 76, 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 = {2565, 270} \[ \frac{2 (d \cos (a+b x))^{n+3}}{b d^3 (n+3)}-\frac{(d \cos (a+b x))^{n+5}}{b d^5 (n+5)}-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^n*Sin[a + b*x]^5,x]

[Out]

-((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n))) + (2*(d*Cos[a + b*x])^(3 + n))/(b*d^3*(3 + n)) - (d*Cos[a + b*x])^(5
+ n)/(b*d^5*(5 + n))

Rule 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])

Rule 270

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

Rubi steps

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

Mathematica [A]  time = 0.299578, size = 83, normalized size = 1.09 \[ -\frac{\cos (a+b x) \left (-4 \left (n^2+8 n+7\right ) \cos (2 (a+b x))+\left (n^2+4 n+3\right ) \cos (4 (a+b x))+3 n^2+28 n+89\right ) (d \cos (a+b x))^n}{8 b (n+1) (n+3) (n+5)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^5,x]

[Out]

-(Cos[a + b*x]*(d*Cos[a + b*x])^n*(89 + 28*n + 3*n^2 - 4*(7 + 8*n + n^2)*Cos[2*(a + b*x)] + (3 + 4*n + n^2)*Co
s[4*(a + b*x)]))/(8*b*(1 + n)*(3 + n)*(5 + n))

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Maple [F]  time = 0.893, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( \sin \left ( bx+a \right ) \right ) ^{5}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^n*sin(b*x+a)^5,x)

[Out]

int((d*cos(b*x+a))^n*sin(b*x+a)^5,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.7898, size = 209, normalized size = 2.75 \begin{align*} -\frac{{\left ({\left (n^{2} + 4 \, n + 3\right )} \cos \left (b x + a\right )^{5} - 2 \,{\left (n^{2} + 6 \, n + 5\right )} \cos \left (b x + a\right )^{3} +{\left (n^{2} + 8 \, n + 15\right )} \cos \left (b x + a\right )\right )} \left (d \cos \left (b x + a\right )\right )^{n}}{b n^{3} + 9 \, b n^{2} + 23 \, b n + 15 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

-((n^2 + 4*n + 3)*cos(b*x + a)^5 - 2*(n^2 + 6*n + 5)*cos(b*x + a)^3 + (n^2 + 8*n + 15)*cos(b*x + a))*(d*cos(b*
x + a))^n/(b*n^3 + 9*b*n^2 + 23*b*n + 15*b)

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Sympy [A]  time = 86.7078, size = 2462, normalized size = 32.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))**n*sin(b*x+a)**5,x)

[Out]

Piecewise((x*(d*cos(a))**n*sin(a)**5, Eq(b, 0)), ((-log(cos(a + b*x))/b + sin(a + b*x)**4/(4*b*cos(a + b*x)**4
) - sin(a + b*x)**2/(2*b*cos(a + b*x)**2))/d**5, Eq(n, -5)), ((2*log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**8
/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b) - 4*log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**4/(b*ta
n(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b) + 2*log(tan(a/2 + b*x/2) - 1)/(b*tan(a/2 + b*x/2)**8 - 2*b*ta
n(a/2 + b*x/2)**4 + b) + 2*log(tan(a/2 + b*x/2) + 1)*tan(a/2 + b*x/2)**8/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2
+ b*x/2)**4 + b) - 4*log(tan(a/2 + b*x/2) + 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/
2)**4 + b) + 2*log(tan(a/2 + b*x/2) + 1)/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b) - 2*log(tan(a/2
+ b*x/2)**2 + 1)*tan(a/2 + b*x/2)**8/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b) + 4*log(tan(a/2 +
b*x/2)**2 + 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b) - 2*log(tan(a/2 + b*x
/2)**2 + 1)/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b) + 4*tan(a/2 + b*x/2)**6/(b*tan(a/2 + b*x/2)*
*8 - 2*b*tan(a/2 + b*x/2)**4 + b) + 4*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**8 - 2*b*tan(a/2 + b*x/2)**4 + b
))/d**3, Eq(n, -3)), ((-log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**8/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b
*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 4*log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/
2)**6/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b
) - 6*log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan
(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 4*log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**2/(b*tan(a/2 +
b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - log(tan(a/2 +
b*x/2) - 1)/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)*
*2 + b) - log(tan(a/2 + b*x/2) + 1)*tan(a/2 + b*x/2)**8/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b
*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 4*log(tan(a/2 + b*x/2) + 1)*tan(a/2 + b*x/2)**6/(b*tan(a
/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 6*log(tan(
a/2 + b*x/2) + 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)*
*4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 4*log(tan(a/2 + b*x/2) + 1)*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**8 + 4
*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - log(tan(a/2 + b*x/2) + 1)/(b
*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) + log(
tan(a/2 + b*x/2)**2 + 1)*tan(a/2 + b*x/2)**8/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 +
b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) + 4*log(tan(a/2 + b*x/2)**2 + 1)*tan(a/2 + b*x/2)**6/(b*tan(a/2 + b*x
/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) + 6*log(tan(a/2 + b*
x/2)**2 + 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 +
4*b*tan(a/2 + b*x/2)**2 + b) + 4*log(tan(a/2 + b*x/2)**2 + 1)*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**8 + 4*b
*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) + log(tan(a/2 + b*x/2)**2 + 1)/(
b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 2*t
an(a/2 + b*x/2)**6/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 +
b*x/2)**2 + b) - 8*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 + 6*b*tan(a/2 + b*x/2)
**4 + 4*b*tan(a/2 + b*x/2)**2 + b) - 2*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**8 + 4*b*tan(a/2 + b*x/2)**6 +
6*b*tan(a/2 + b*x/2)**4 + 4*b*tan(a/2 + b*x/2)**2 + b))/d, Eq(n, -1)), (-d**n*n**2*sin(a + b*x)**4*cos(a + b*x
)*cos(a + b*x)**n/(b*n**3 + 9*b*n**2 + 23*b*n + 15*b) - 8*d**n*n*sin(a + b*x)**4*cos(a + b*x)*cos(a + b*x)**n/
(b*n**3 + 9*b*n**2 + 23*b*n + 15*b) - 4*d**n*n*sin(a + b*x)**2*cos(a + b*x)**3*cos(a + b*x)**n/(b*n**3 + 9*b*n
**2 + 23*b*n + 15*b) - 15*d**n*sin(a + b*x)**4*cos(a + b*x)*cos(a + b*x)**n/(b*n**3 + 9*b*n**2 + 23*b*n + 15*b
) - 20*d**n*sin(a + b*x)**2*cos(a + b*x)**3*cos(a + b*x)**n/(b*n**3 + 9*b*n**2 + 23*b*n + 15*b) - 8*d**n*cos(a
+ b*x)**5*cos(a + b*x)**n/(b*n**3 + 9*b*n**2 + 23*b*n + 15*b), True))

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Giac [B]  time = 1.1605, size = 336, normalized size = 4.42 \begin{align*} -\frac{\left (d \cos \left (b x + a\right )\right )^{n} d^{5} n^{2} \cos \left (b x + a\right )^{5} + 4 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} n \cos \left (b x + a\right )^{5} - 2 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} n^{2} \cos \left (b x + a\right )^{3} + 3 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} \cos \left (b x + a\right )^{5} - 12 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} n \cos \left (b x + a\right )^{3} + \left (d \cos \left (b x + a\right )\right )^{n} d^{5} n^{2} \cos \left (b x + a\right ) - 10 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} \cos \left (b x + a\right )^{3} + 8 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} n \cos \left (b x + a\right ) + 15 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{5} \cos \left (b x + a\right )}{{\left (d^{4} n^{3} + 9 \, d^{4} n^{2} + 23 \, d^{4} n + 15 \, d^{4}\right )} b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*cos(b*x+a))^n*sin(b*x+a)^5,x, algorithm="giac")

[Out]

-((d*cos(b*x + a))^n*d^5*n^2*cos(b*x + a)^5 + 4*(d*cos(b*x + a))^n*d^5*n*cos(b*x + a)^5 - 2*(d*cos(b*x + a))^n
*d^5*n^2*cos(b*x + a)^3 + 3*(d*cos(b*x + a))^n*d^5*cos(b*x + a)^5 - 12*(d*cos(b*x + a))^n*d^5*n*cos(b*x + a)^3
+ (d*cos(b*x + a))^n*d^5*n^2*cos(b*x + a) - 10*(d*cos(b*x + a))^n*d^5*cos(b*x + a)^3 + 8*(d*cos(b*x + a))^n*d
^5*n*cos(b*x + a) + 15*(d*cos(b*x + a))^n*d^5*cos(b*x + a))/((d^4*n^3 + 9*d^4*n^2 + 23*d^4*n + 15*d^4)*b*d)