3.1.1 \(\int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx\) [1]

Optimal. Leaf size=61 \[ \frac {128 \sin ^9(a+b x)}{9 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {128 \sin ^{15}(a+b x)}{15 b} \]

[Out]

128/9*sin(b*x+a)^9/b-384/11*sin(b*x+a)^11/b+384/13*sin(b*x+a)^13/b-128/15*sin(b*x+a)^15/b

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Rubi [A]
time = 0.04, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4373, 2644, 276} \begin {gather*} -\frac {128 \sin ^{15}(a+b x)}{15 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {128 \sin ^9(a+b x)}{9 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]*Sin[2*a + 2*b*x]^7,x]

[Out]

(128*Sin[a + b*x]^9)/(9*b) - (384*Sin[a + b*x]^11)/(11*b) + (384*Sin[a + b*x]^13)/(13*b) - (128*Sin[a + b*x]^1
5)/(15*b)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 4373

Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rubi steps

\begin {align*} \int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^7(a+b x) \sin ^8(a+b x) \, dx\\ &=\frac {128 \text {Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {128 \text {Subst}\left (\int \left (x^8-3 x^{10}+3 x^{12}-x^{14}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac {128 \sin ^9(a+b x)}{9 b}-\frac {384 \sin ^{11}(a+b x)}{11 b}+\frac {384 \sin ^{13}(a+b x)}{13 b}-\frac {128 \sin ^{15}(a+b x)}{15 b}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 47, normalized size = 0.77 \begin {gather*} \frac {4 (8330+10755 \cos (2 (a+b x))+3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))) \sin ^9(a+b x)}{6435 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^7,x]

[Out]

(4*(8330 + 10755*Cos[2*(a + b*x)] + 3366*Cos[4*(a + b*x)] + 429*Cos[6*(a + b*x)])*Sin[a + b*x]^9)/(6435*b)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs. \(2(53)=106\).
time = 0.18, size = 111, normalized size = 1.82

method	result	size
default	\(\frac {35 \sin \left (x b +a \right )}{128 b}-\frac {35 \sin \left (3 x b +3 a \right )}{384 b}-\frac {21 \sin \left (5 x b +5 a \right )}{640 b}+\frac {3 \sin \left (7 x b +7 a \right )}{128 b}+\frac {7 \sin \left (9 x b +9 a \right )}{1152 b}-\frac {7 \sin \left (11 x b +11 a \right )}{1408 b}-\frac {\sin \left (13 x b +13 a \right )}{1664 b}+\frac {\sin \left (15 x b +15 a \right )}{1920 b}\)	\(111\)
risch	\(\frac {35 \sin \left (x b +a \right )}{128 b}-\frac {35 \sin \left (3 x b +3 a \right )}{384 b}-\frac {21 \sin \left (5 x b +5 a \right )}{640 b}+\frac {3 \sin \left (7 x b +7 a \right )}{128 b}+\frac {7 \sin \left (9 x b +9 a \right )}{1152 b}-\frac {7 \sin \left (11 x b +11 a \right )}{1408 b}-\frac {\sin \left (13 x b +13 a \right )}{1664 b}+\frac {\sin \left (15 x b +15 a \right )}{1920 b}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x+a)*sin(2*b*x+2*a)^7,x,method=_RETURNVERBOSE)

[Out]

35/128*sin(b*x+a)/b-35/384*sin(3*b*x+3*a)/b-21/640/b*sin(5*b*x+5*a)+3/128/b*sin(7*b*x+7*a)+7/1152/b*sin(9*b*x+
9*a)-7/1408/b*sin(11*b*x+11*a)-1/1664/b*sin(13*b*x+13*a)+1/1920/b*sin(15*b*x+15*a)

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Maxima [A]
time = 0.30, size = 91, normalized size = 1.49 \begin {gather*} \frac {429 \, \sin \left (15 \, b x + 15 \, a\right ) - 495 \, \sin \left (13 \, b x + 13 \, a\right ) - 4095 \, \sin \left (11 \, b x + 11 \, a\right ) + 5005 \, \sin \left (9 \, b x + 9 \, a\right ) + 19305 \, \sin \left (7 \, b x + 7 \, a\right ) - 27027 \, \sin \left (5 \, b x + 5 \, a\right ) - 75075 \, \sin \left (3 \, b x + 3 \, a\right ) + 225225 \, \sin \left (b x + a\right )}{823680 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="maxima")

[Out]

1/823680*(429*sin(15*b*x + 15*a) - 495*sin(13*b*x + 13*a) - 4095*sin(11*b*x + 11*a) + 5005*sin(9*b*x + 9*a) +
19305*sin(7*b*x + 7*a) - 27027*sin(5*b*x + 5*a) - 75075*sin(3*b*x + 3*a) + 225225*sin(b*x + a))/b

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Fricas [A]
time = 3.91, size = 83, normalized size = 1.36 \begin {gather*} \frac {128 \, {\left (429 \, \cos \left (b x + a\right )^{14} - 1518 \, \cos \left (b x + a\right )^{12} + 1854 \, \cos \left (b x + a\right )^{10} - 800 \, \cos \left (b x + a\right )^{8} + 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 )}{6435 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="fricas")

[Out]

128/6435*(429*cos(b*x + a)^14 - 1518*cos(b*x + a)^12 + 1854*cos(b*x + a)^10 - 800*cos(b*x + a)^8 + 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (53) = 106\).
time = 34.30, size = 269, normalized size = 4.41 \begin {gather*} \begin {cases} - \frac {3838 \sin {\left (a + b x \right )} \sin ^{6}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{6435 b} - \frac {1648 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {768 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{715 b} - \frac {2048 \sin {\left (a + b x \right )} \cos ^{7}{\left (2 a + 2 b x \right )}}{6435 b} + \frac {1241 \sin ^{7}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{6435 b} + \frac {376 \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{715 b} + \frac {640 \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1287 b} + \frac {1024 \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{6}{\left (2 a + 2 b x \right )}}{6435 b} & \text {for}\: b \neq 0 \\x \sin {\left (a \right )} \sin ^{7}{\left (2 a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)*sin(2*b*x+2*a)**7,x)

[Out]

Piecewise((-3838*sin(a + b*x)*sin(2*a + 2*b*x)**6*cos(2*a + 2*b*x)/(6435*b) - 1648*sin(a + b*x)*sin(2*a + 2*b*
x)**4*cos(2*a + 2*b*x)**3/(1287*b) - 768*sin(a + b*x)*sin(2*a + 2*b*x)**2*cos(2*a + 2*b*x)**5/(715*b) - 2048*s
in(a + b*x)*cos(2*a + 2*b*x)**7/(6435*b) + 1241*sin(2*a + 2*b*x)**7*cos(a + b*x)/(6435*b) + 376*sin(2*a + 2*b*
x)**5*cos(a + b*x)*cos(2*a + 2*b*x)**2/(715*b) + 640*sin(2*a + 2*b*x)**3*cos(a + b*x)*cos(2*a + 2*b*x)**4/(128
7*b) + 1024*sin(2*a + 2*b*x)*cos(a + b*x)*cos(2*a + 2*b*x)**6/(6435*b), Ne(b, 0)), (x*sin(a)*sin(2*a)**7, True
))

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Giac [A]
time = 0.43, size = 46, normalized size = 0.75 \begin {gather*} -\frac {128 \, {\left (429 \, \sin \left (b x + a\right )^{15} - 1485 \, \sin \left (b x + a\right )^{13} + 1755 \, \sin \left (b x + a\right )^{11} - 715 \, \sin \left (b x + a\right )^{9}\right )}}{6435 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="giac")

[Out]

-128/6435*(429*sin(b*x + a)^15 - 1485*sin(b*x + a)^13 + 1755*sin(b*x + a)^11 - 715*sin(b*x + a)^9)/b

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Mupad [B]
time = 0.08, size = 45, normalized size = 0.74 \begin {gather*} \frac {-\frac {128\,{\sin \left (a+b\,x\right )}^{15}}{15}+\frac {384\,{\sin \left (a+b\,x\right )}^{13}}{13}-\frac {384\,{\sin \left (a+b\,x\right )}^{11}}{11}+\frac {128\,{\sin \left (a+b\,x\right )}^9}{9}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x)*sin(2*a + 2*b*x)^7,x)

[Out]

((128*sin(a + b*x)^9)/9 - (384*sin(a + b*x)^11)/11 + (384*sin(a + b*x)^13)/13 - (128*sin(a + b*x)^15)/15)/b

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