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

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

[Out]

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

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Rubi [A]  time = 0.0163204, 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} \[ -\frac{\sin ^7(a+b x)}{7 b}+\frac{3 \sin ^5(a+b x)}{5 b}-\frac{\sin ^3(a+b x)}{b}+\frac{\sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^7,x]

[Out]

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

Mathematica [A]  time = 0.0132348, size = 54, normalized size = 1. \[ -\frac{\sin ^7(a+b x)}{7 b}+\frac{3 \sin ^5(a+b x)}{5 b}-\frac{\sin ^3(a+b x)}{b}+\frac{\sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^7,x]

[Out]

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

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Maple [A]  time = 0.026, size = 42, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( bx+a \right ) }{7\,b} \left ({\frac{16}{5}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.63903, size = 59, normalized size = 1.09 \begin{align*} -\frac{5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.68887, size = 112, normalized size = 2.07 \begin{align*} \frac{{\left (5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/35*(5*cos(b*x + a)^6 + 6*cos(b*x + a)^4 + 8*cos(b*x + a)^2 + 16)*sin(b*x + a)/b

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Sympy [A]  time = 6.8842, size = 78, normalized size = 1.44 \begin{align*} \begin{cases} \frac{16 \sin ^{7}{\left (a + b x \right )}}{35 b} + \frac{8 \sin ^{5}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{5 b} + \frac{2 \sin ^{3}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{b} + \frac{\sin{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cos ^{7}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.51886, size = 59, normalized size = 1.09 \begin{align*} -\frac{5 \, \sin \left (b x + a\right )^{7} - 21 \, \sin \left (b x + a\right )^{5} + 35 \, \sin \left (b x + a\right )^{3} - 35 \, \sin \left (b x + a\right )}{35 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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