3.8 \(\int \sin ^8(a+b x) \, dx\)

Optimal. Leaf size=88 \[ -\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}-\frac {7 \sin ^5(a+b x) \cos (a+b x)}{48 b}-\frac {35 \sin ^3(a+b x) \cos (a+b x)}{192 b}-\frac {35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac {35 x}{128} \]

[Out]

35/128*x-35/128*cos(b*x+a)*sin(b*x+a)/b-35/192*cos(b*x+a)*sin(b*x+a)^3/b-7/48*cos(b*x+a)*sin(b*x+a)^5/b-1/8*co
s(b*x+a)*sin(b*x+a)^7/b

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Rubi [A]  time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2635, 8} \[ -\frac {\sin ^7(a+b x) \cos (a+b x)}{8 b}-\frac {7 \sin ^5(a+b x) \cos (a+b x)}{48 b}-\frac {35 \sin ^3(a+b x) \cos (a+b x)}{192 b}-\frac {35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac {35 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^8,x]

[Out]

(35*x)/128 - (35*Cos[a + b*x]*Sin[a + b*x])/(128*b) - (35*Cos[a + b*x]*Sin[a + b*x]^3)/(192*b) - (7*Cos[a + b*
x]*Sin[a + b*x]^5)/(48*b) - (Cos[a + b*x]*Sin[a + b*x]^7)/(8*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rubi steps

\begin {align*} \int \sin ^8(a+b x) \, dx &=-\frac {\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac {7}{8} \int \sin ^6(a+b x) \, dx\\ &=-\frac {7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac {\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac {35}{48} \int \sin ^4(a+b x) \, dx\\ &=-\frac {35 \cos (a+b x) \sin ^3(a+b x)}{192 b}-\frac {7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac {\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac {35}{64} \int \sin ^2(a+b x) \, dx\\ &=-\frac {35 \cos (a+b x) \sin (a+b x)}{128 b}-\frac {35 \cos (a+b x) \sin ^3(a+b x)}{192 b}-\frac {7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac {\cos (a+b x) \sin ^7(a+b x)}{8 b}+\frac {35 \int 1 \, dx}{128}\\ &=\frac {35 x}{128}-\frac {35 \cos (a+b x) \sin (a+b x)}{128 b}-\frac {35 \cos (a+b x) \sin ^3(a+b x)}{192 b}-\frac {7 \cos (a+b x) \sin ^5(a+b x)}{48 b}-\frac {\cos (a+b x) \sin ^7(a+b x)}{8 b}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 55, normalized size = 0.62 \[ \frac {-672 \sin (2 (a+b x))+168 \sin (4 (a+b x))-32 \sin (6 (a+b x))+3 \sin (8 (a+b x))+840 a+840 b x}{3072 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^8,x]

[Out]

(840*a + 840*b*x - 672*Sin[2*(a + b*x)] + 168*Sin[4*(a + b*x)] - 32*Sin[6*(a + b*x)] + 3*Sin[8*(a + b*x)])/(30
72*b)

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fricas [A]  time = 0.45, size = 56, normalized size = 0.64 \[ \frac {105 \, b x + {\left (48 \, \cos \left (b x + a\right )^{7} - 200 \, \cos \left (b x + a\right )^{5} + 326 \, \cos \left (b x + a\right )^{3} - 279 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^8,x, algorithm="fricas")

[Out]

1/384*(105*b*x + (48*cos(b*x + a)^7 - 200*cos(b*x + a)^5 + 326*cos(b*x + a)^3 - 279*cos(b*x + a))*sin(b*x + a)
)/b

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giac [A]  time = 0.13, size = 60, normalized size = 0.68 \[ \frac {35}{128} \, x + \frac {\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac {\sin \left (6 \, b x + 6 \, a\right )}{96 \, b} + \frac {7 \, \sin \left (4 \, b x + 4 \, a\right )}{128 \, b} - \frac {7 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^8,x, algorithm="giac")

[Out]

35/128*x + 1/1024*sin(8*b*x + 8*a)/b - 1/96*sin(6*b*x + 6*a)/b + 7/128*sin(4*b*x + 4*a)/b - 7/32*sin(2*b*x + 2
*a)/b

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maple [A]  time = 0.10, size = 58, normalized size = 0.66 \[ \frac {-\frac {\left (\sin ^{7}\left (b x +a \right )+\frac {7 \left (\sin ^{5}\left (b x +a \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (b x +a \right )\right )}{24}+\frac {35 \sin \left (b x +a \right )}{16}\right ) \cos \left (b x +a \right )}{8}+\frac {35 b x}{128}+\frac {35 a}{128}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)^8,x)

[Out]

1/b*(-1/8*(sin(b*x+a)^7+7/6*sin(b*x+a)^5+35/24*sin(b*x+a)^3+35/16*sin(b*x+a))*cos(b*x+a)+35/128*b*x+35/128*a)

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maxima [A]  time = 0.34, size = 59, normalized size = 0.67 \[ \frac {128 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 840 \, b x + 840 \, a + 3 \, \sin \left (8 \, b x + 8 \, a\right ) + 168 \, \sin \left (4 \, b x + 4 \, a\right ) - 768 \, \sin \left (2 \, b x + 2 \, a\right )}{3072 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)^8,x, algorithm="maxima")

[Out]

1/3072*(128*sin(2*b*x + 2*a)^3 + 840*b*x + 840*a + 3*sin(8*b*x + 8*a) + 168*sin(4*b*x + 4*a) - 768*sin(2*b*x +
2*a))/b

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mupad [B]  time = 1.50, size = 90, normalized size = 1.02 \[ \frac {35\,x}{128}-\frac {\frac {93\,{\mathrm {tan}\left (a+b\,x\right )}^7}{128}+\frac {511\,{\mathrm {tan}\left (a+b\,x\right )}^5}{384}+\frac {385\,{\mathrm {tan}\left (a+b\,x\right )}^3}{384}+\frac {35\,\mathrm {tan}\left (a+b\,x\right )}{128}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^8+4\,{\mathrm {tan}\left (a+b\,x\right )}^6+6\,{\mathrm {tan}\left (a+b\,x\right )}^4+4\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)^8,x)

[Out]

(35*x)/128 - ((35*tan(a + b*x))/128 + (385*tan(a + b*x)^3)/384 + (511*tan(a + b*x)^5)/384 + (93*tan(a + b*x)^7
)/128)/(b*(4*tan(a + b*x)^2 + 6*tan(a + b*x)^4 + 4*tan(a + b*x)^6 + tan(a + b*x)^8 + 1))

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sympy [A]  time = 10.28, size = 184, normalized size = 2.09 \[ \begin {cases} \frac {35 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac {35 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac {105 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac {35 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac {35 x \cos ^{8}{\left (a + b x \right )}}{128} - \frac {93 \sin ^{7}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{128 b} - \frac {511 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} - \frac {385 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac {35 \sin {\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \sin ^{8}{\relax (a )} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)**8,x)

[Out]

Piecewise((35*x*sin(a + b*x)**8/128 + 35*x*sin(a + b*x)**6*cos(a + b*x)**2/32 + 105*x*sin(a + b*x)**4*cos(a +
b*x)**4/64 + 35*x*sin(a + b*x)**2*cos(a + b*x)**6/32 + 35*x*cos(a + b*x)**8/128 - 93*sin(a + b*x)**7*cos(a + b
*x)/(128*b) - 511*sin(a + b*x)**5*cos(a + b*x)**3/(384*b) - 385*sin(a + b*x)**3*cos(a + b*x)**5/(384*b) - 35*s
in(a + b*x)*cos(a + b*x)**7/(128*b), Ne(b, 0)), (x*sin(a)**8, True))

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