3.7 \(\int \sin ^7(a+b x) \, dx\)

Optimal. Leaf size=54 \[ \frac{\cos ^7(a+b x)}{7 b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^3(a+b x)}{b}-\frac{\cos (a+b x)}{b} \]

[Out]

-(Cos[a + b*x]/b) + Cos[a + b*x]^3/b - (3*Cos[a + b*x]^5)/(5*b) + Cos[a + b*x]^7/(7*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0157233, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ \frac{\cos ^7(a+b x)}{7 b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^3(a+b x)}{b}-\frac{\cos (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^7,x]

[Out]

-(Cos[a + b*x]/b) + Cos[a + b*x]^3/b - (3*Cos[a + b*x]^5)/(5*b) + Cos[a + b*x]^7/(7*b)

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 \sin ^7(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\cos (a+b x)}{b}+\frac{\cos ^3(a+b x)}{b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^7(a+b x)}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0092732, size = 59, normalized size = 1.09 \[ -\frac{35 \cos (a+b x)}{64 b}+\frac{7 \cos (3 (a+b x))}{64 b}-\frac{7 \cos (5 (a+b x))}{320 b}+\frac{\cos (7 (a+b x))}{448 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^7,x]

[Out]

(-35*Cos[a + b*x])/(64*b) + (7*Cos[3*(a + b*x)])/(64*b) - (7*Cos[5*(a + b*x)])/(320*b) + Cos[7*(a + b*x)]/(448
*b)

________________________________________________________________________________________

Maple [A]  time = 0.038, size = 42, normalized size = 0.8 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{7\,b} \left ({\frac{16}{5}}+ \left ( \sin \left ( bx+a \right ) \right ) ^{6}+{\frac{6\, \left ( \sin \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \sin \left ( bx+a \right ) \right ) ^{2}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^7,x)

[Out]

-1/7/b*(16/5+sin(b*x+a)^6+6/5*sin(b*x+a)^4+8/5*sin(b*x+a)^2)*cos(b*x+a)

________________________________________________________________________________________

Maxima [A]  time = 0.979021, size = 59, normalized size = 1.09 \begin{align*} \frac{5 \, \cos \left (b x + a\right )^{7} - 21 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^7,x, algorithm="maxima")

[Out]

1/35*(5*cos(b*x + a)^7 - 21*cos(b*x + a)^5 + 35*cos(b*x + a)^3 - 35*cos(b*x + a))/b

________________________________________________________________________________________

Fricas [A]  time = 2.27506, size = 115, normalized size = 2.13 \begin{align*} \frac{5 \, \cos \left (b x + a\right )^{7} - 21 \, \cos \left (b x + a\right )^{5} + 35 \, \cos \left (b x + a\right )^{3} - 35 \, \cos \left (b x + a\right )}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^7,x, algorithm="fricas")

[Out]

1/35*(5*cos(b*x + a)^7 - 21*cos(b*x + a)^5 + 35*cos(b*x + a)^3 - 35*cos(b*x + a))/b

________________________________________________________________________________________

Sympy [A]  time = 6.82139, size = 80, normalized size = 1.48 \begin{align*} \begin{cases} - \frac{\sin ^{6}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{b} - \frac{2 \sin ^{4}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac{8 \sin ^{2}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac{16 \cos ^{7}{\left (a + b x \right )}}{35 b} & \text{for}\: b \neq 0 \\x \sin ^{7}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)**7,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.13322, size = 68, normalized size = 1.26 \begin{align*} \frac{\cos \left (b x + a\right )^{7}}{7 \, b} - \frac{3 \, \cos \left (b x + a\right )^{5}}{5 \, b} + \frac{\cos \left (b x + a\right )^{3}}{b} - \frac{\cos \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^7,x, algorithm="giac")

[Out]

1/7*cos(b*x + a)^7/b - 3/5*cos(b*x + a)^5/b + cos(b*x + a)^3/b - cos(b*x + a)/b