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3.356 \(\int (d \cos (a+b x))^n \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=50 \[ \frac{(d \cos (a+b x))^{n+3}}{b d^3 (n+3)}-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0500521, antiderivative size = 50, 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, 14} \[ \frac{(d \cos (a+b x))^{n+3}}{b d^3 (n+3)}-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)} \]

Antiderivative was successfully verified.

[In]

Int[(d*Cos[a + b*x])^n*Sin[a + b*x]^3,x]

[Out]

-((d*Cos[a + b*x])^(1 + n)/(b*d*(1 + n))) + (d*Cos[a + b*x])^(3 + n)/(b*d^3*(3 + 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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
 &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.117085, size = 50, normalized size = 1. \[ \frac{\cos (a+b x) ((n+1) \cos (2 (a+b x))-n-5) (d \cos (a+b x))^n}{2 b (n+1) (n+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^3,x]

[Out]

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

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Maple [C]  time = 1.467, size = 1076, normalized size = 21.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*cos(b*x+a))^n*sin(b*x+a)^3,x)

[Out]

-1/8*(n+9)/b/(1+n)/(3+n)*(exp(2*I*(b*x+a))+1)^n*(1/2)^n*d^n*exp(I*(b*x+a))^(-n)*exp(-1/2*I*(Pi*n*csgn(I*cos(b*
x+a))^3-Pi*n*csgn(I*exp(-I*(b*x+a)))*csgn(I*cos(b*x+a))^2-Pi*n*csgn(I*(exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))
^2-Pi*n*csgn(I*cos(b*x+a))*csgn(I*d*cos(b*x+a))^2+Pi*n*csgn(I*d)*csgn(I*cos(b*x+a))*csgn(I*d*cos(b*x+a))+Pi*n*
csgn(I*exp(-I*(b*x+a)))*csgn(I*(exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))+Pi*n*csgn(I*d*cos(b*x+a))^3-Pi*n*csgn(
I*d)*csgn(I*d*cos(b*x+a))^2+2*b*x+2*a))-1/8*(n+9)/b/(1+n)/(3+n)*(exp(2*I*(b*x+a))+1)^n*(1/2)^n*d^n*exp(I*(b*x+
a))^(-n)*exp(-1/2*I*(Pi*n*csgn(I*cos(b*x+a))^3-Pi*n*csgn(I*exp(-I*(b*x+a)))*csgn(I*cos(b*x+a))^2-Pi*n*csgn(I*(
exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))^2-Pi*n*csgn(I*cos(b*x+a))*csgn(I*d*cos(b*x+a))^2+Pi*n*csgn(I*d)*csgn(I
*cos(b*x+a))*csgn(I*d*cos(b*x+a))+Pi*n*csgn(I*exp(-I*(b*x+a)))*csgn(I*(exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))
+Pi*n*csgn(I*d*cos(b*x+a))^3-Pi*n*csgn(I*d)*csgn(I*d*cos(b*x+a))^2-2*b*x-2*a))+1/8/(b*n+3*b)*(exp(2*I*(b*x+a))
+1)^n*(1/2)^n*d^n*exp(I*(b*x+a))^(-n)*exp(-1/2*I*(Pi*n*csgn(I*cos(b*x+a))^3-Pi*n*csgn(I*exp(-I*(b*x+a)))*csgn(
I*cos(b*x+a))^2-Pi*n*csgn(I*(exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))^2-Pi*n*csgn(I*cos(b*x+a))*csgn(I*d*cos(b*
x+a))^2+Pi*n*csgn(I*d)*csgn(I*cos(b*x+a))*csgn(I*d*cos(b*x+a))+Pi*n*csgn(I*exp(-I*(b*x+a)))*csgn(I*(exp(2*I*(b
*x+a))+1))*csgn(I*cos(b*x+a))+Pi*n*csgn(I*d*cos(b*x+a))^3-Pi*n*csgn(I*d)*csgn(I*d*cos(b*x+a))^2+6*b*x+6*a))+1/
8/(b*n+3*b)*(exp(2*I*(b*x+a))+1)^n*(1/2)^n*d^n*exp(I*(b*x+a))^(-n)*exp(-1/2*I*(Pi*n*csgn(I*cos(b*x+a))^3-Pi*n*
csgn(I*exp(-I*(b*x+a)))*csgn(I*cos(b*x+a))^2-Pi*n*csgn(I*(exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))^2-Pi*n*csgn(
I*cos(b*x+a))*csgn(I*d*cos(b*x+a))^2+Pi*n*csgn(I*d)*csgn(I*cos(b*x+a))*csgn(I*d*cos(b*x+a))+Pi*n*csgn(I*exp(-I
*(b*x+a)))*csgn(I*(exp(2*I*(b*x+a))+1))*csgn(I*cos(b*x+a))+Pi*n*csgn(I*d*cos(b*x+a))^3-Pi*n*csgn(I*d)*csgn(I*d
*cos(b*x+a))^2-6*b*x-6*a))

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.26167, size = 122, normalized size = 2.44 \begin{align*} \frac{{\left ({\left (n + 1\right )} \cos \left (b x + a\right )^{3} -{\left (n + 3\right )} \cos \left (b x + a\right )\right )} \left (d \cos \left (b x + a\right )\right )^{n}}{b n^{2} + 4 \, b n + 3 \, 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)^3,x, algorithm="fricas")

[Out]

((n + 1)*cos(b*x + a)^3 - (n + 3)*cos(b*x + a))*(d*cos(b*x + a))^n/(b*n^2 + 4*b*n + 3*b)

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Sympy [A]  time = 14.6605, size = 694, normalized size = 13.88 \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)**3,x)

[Out]

Piecewise((x*(d*cos(a))**n*sin(a)**3, Eq(b, 0)), ((log(cos(a + b*x))/b + sin(a + b*x)**2/(2*b*cos(a + b*x)**2)
)/d**3, Eq(n, -3)), ((-log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*
x/2)**2 + b) - 2*log(tan(a/2 + b*x/2) - 1)*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**
2 + b) - log(tan(a/2 + b*x/2) - 1)/(b*tan(a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**2 + b) - log(tan(a/2 + b*x/2
) + 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**2 + b) - 2*log(tan(a/2 + b*x/2) + 1)
*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**2 + b) - log(tan(a/2 + b*x/2) + 1)/(b*tan(
a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**2 + b) + log(tan(a/2 + b*x/2)**2 + 1)*tan(a/2 + b*x/2)**4/(b*tan(a/2 +
 b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**2 + b) + 2*log(tan(a/2 + b*x/2)**2 + 1)*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*
x/2)**4 + 2*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)**4 + 2*b*tan(a/2 + b
*x/2)**2 + b) - 2*tan(a/2 + b*x/2)**2/(b*tan(a/2 + b*x/2)**4 + 2*b*tan(a/2 + b*x/2)**2 + b))/d, Eq(n, -1)), (-
d**n*n*sin(a + b*x)**2*cos(a + b*x)*cos(a + b*x)**n/(b*n**2 + 4*b*n + 3*b) - 3*d**n*sin(a + b*x)**2*cos(a + b*
x)*cos(a + b*x)**n/(b*n**2 + 4*b*n + 3*b) - 2*d**n*cos(a + b*x)**3*cos(a + b*x)**n/(b*n**2 + 4*b*n + 3*b), Tru
e))

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Giac [B]  time = 1.16013, size = 158, normalized size = 3.16 \begin{align*} \frac{\left (d \cos \left (b x + a\right )\right )^{n} d^{3} n \cos \left (b x + a\right )^{3} + \left (d \cos \left (b x + a\right )\right )^{n} d^{3} \cos \left (b x + a\right )^{3} - \left (d \cos \left (b x + a\right )\right )^{n} d^{3} n \cos \left (b x + a\right ) - 3 \, \left (d \cos \left (b x + a\right )\right )^{n} d^{3} \cos \left (b x + a\right )}{{\left (d^{2} n^{2} + 4 \, d^{2} n + 3 \, d^{2}\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)^3,x, algorithm="giac")

[Out]

((d*cos(b*x + a))^n*d^3*n*cos(b*x + a)^3 + (d*cos(b*x + a))^n*d^3*cos(b*x + a)^3 - (d*cos(b*x + a))^n*d^3*n*co
s(b*x + a) - 3*(d*cos(b*x + a))^n*d^3*cos(b*x + a))/((d^2*n^2 + 4*d^2*n + 3*d^2)*b*d)

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