3.1.7 \(\int \sin ^7(a+b x) \, dx\) [7]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, 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 = {2713} \begin {gather*} \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} \end {gather*}

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 2713

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 {\text {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*}

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Mathematica [A]
time = 0.01, size = 59, normalized size = 1.09 \begin {gather*} -\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} \end {gather*}

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)

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Maple [A]
time = 0.11, size = 42, normalized size = 0.78

method	result	size
derivativedivides	\(-\frac {\left (\frac {16}{5}+\sin ^{6}\left (b x +a \right )+\frac {6 \left (\sin ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (b x +a \right )\right )}{5}\right ) \cos \left (b x +a \right )}{7 b}\)	\(42\)
default	\(-\frac {\left (\frac {16}{5}+\sin ^{6}\left (b x +a \right )+\frac {6 \left (\sin ^{4}\left (b x +a \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (b x +a \right )\right )}{5}\right ) \cos \left (b x +a \right )}{7 b}\)	\(42\)
risch	\(-\frac {35 \cos \left (b x +a \right )}{64 b}+\frac {\cos \left (7 b x +7 a \right )}{448 b}-\frac {7 \cos \left (5 b x +5 a \right )}{320 b}+\frac {7 \cos \left (3 b x +3 a \right )}{64 b}\)	\(55\)
norman	\(\frac {-\frac {32 \left (\tan ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b}-\frac {32}{35 b}-\frac {32 \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}-\frac {96 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 b}}{\left (1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )^{7}}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]
time = 0.33, size = 44, normalized size = 0.81 \begin {gather*} \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 {gather*}

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

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Fricas [A]
time = 0.37, size = 44, normalized size = 0.81 \begin {gather*} \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 {gather*}

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

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Sympy [A]
time = 0.61, size = 80, normalized size = 1.48 \begin {gather*} \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 {gather*}

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))

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Giac [A]
time = 4.34, size = 50, normalized size = 0.93 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.38, size = 43, normalized size = 0.80 \begin {gather*} \frac {\cos \left (a+b\,x\right )\,\left (5\,{\cos \left (a+b\,x\right )}^6-21\,{\cos \left (a+b\,x\right )}^4+35\,{\cos \left (a+b\,x\right )}^2-35\right )}{35\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^7,x)

[Out]

(cos(a + b*x)*(35*cos(a + b*x)^2 - 21*cos(a + b*x)^4 + 5*cos(a + b*x)^6 - 35))/(35*b)

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