3.34 \(\int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx\)

Optimal. Leaf size=61 \[ -\frac{128 \sin ^{13}(a+b x)}{13 b}+\frac{384 \sin ^{11}(a+b x)}{11 b}-\frac{128 \sin ^9(a+b x)}{3 b}+\frac{128 \sin ^7(a+b x)}{7 b} \]

[Out]

(128*Sin[a + b*x]^7)/(7*b) - (128*Sin[a + b*x]^9)/(3*b) + (384*Sin[a + b*x]^11)/(11*b) - (128*Sin[a + b*x]^13)
/(13*b)

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Rubi [A]  time = 0.0606673, antiderivative size = 61, normalized size of antiderivative = 1., 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 = {4288, 2564, 270} \[ -\frac{128 \sin ^{13}(a+b x)}{13 b}+\frac{384 \sin ^{11}(a+b x)}{11 b}-\frac{128 \sin ^9(a+b x)}{3 b}+\frac{128 \sin ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(128*Sin[a + b*x]^7)/(7*b) - (128*Sin[a + b*x]^9)/(3*b) + (384*Sin[a + b*x]^11)/(11*b) - (128*Sin[a + b*x]^13)
/(13*b)

Rule 4288

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]

Rule 2564

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 270

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]

Rubi steps

\begin{align*} \int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^7(a+b x) \sin ^6(a+b x) \, dx\\ &=\frac{128 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{128 \operatorname{Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{128 \sin ^7(a+b x)}{7 b}-\frac{128 \sin ^9(a+b x)}{3 b}+\frac{384 \sin ^{11}(a+b x)}{11 b}-\frac{128 \sin ^{13}(a+b x)}{13 b}\\ \end{align*}

Mathematica [A]  time = 0.213718, size = 48, normalized size = 0.79 \[ \frac{128 \left (-231 \sin ^{13}(a+b x)+819 \sin ^{11}(a+b x)-1001 \sin ^9(a+b x)+429 \sin ^7(a+b x)\right )}{3003 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(128*(429*Sin[a + b*x]^7 - 1001*Sin[a + b*x]^9 + 819*Sin[a + b*x]^11 - 231*Sin[a + b*x]^13))/(3003*b)

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Maple [A]  time = 0.06, size = 97, normalized size = 1.6 \begin{align*} 128\,{\frac{1}{b} \left ( -1/13\, \left ( \sin \left ( bx+a \right ) \right ) ^{5} \left ( \cos \left ( bx+a \right ) \right ) ^{8}-{\frac{5\, \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{8}}{143}}-{\frac{5\,\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{8}}{429}}+{\frac{5\,\sin \left ( bx+a \right ) }{3003} \left ({\frac{16}{5}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

128/b*(-1/13*sin(b*x+a)^5*cos(b*x+a)^8-5/143*sin(b*x+a)^3*cos(b*x+a)^8-5/429*sin(b*x+a)*cos(b*x+a)^8+5/3003*(1
6/5+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 = 1.2241, size = 108, normalized size = 1.77 \begin{align*} -\frac{231 \, \sin \left (13 \, b x + 13 \, a\right ) + 273 \, \sin \left (11 \, b x + 11 \, a\right ) - 2002 \, \sin \left (9 \, b x + 9 \, a\right ) - 2574 \, \sin \left (7 \, b x + 7 \, a\right ) + 9009 \, \sin \left (5 \, b x + 5 \, a\right ) + 15015 \, \sin \left (3 \, b x + 3 \, a\right ) - 60060 \, \sin \left (b x + a\right )}{96096 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/96096*(231*sin(13*b*x + 13*a) + 273*sin(11*b*x + 11*a) - 2002*sin(9*b*x + 9*a) - 2574*sin(7*b*x + 7*a) + 90
09*sin(5*b*x + 5*a) + 15015*sin(3*b*x + 3*a) - 60060*sin(b*x + a))/b

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Fricas [A]  time = 0.520104, size = 207, normalized size = 3.39 \begin{align*} -\frac{128 \,{\left (231 \, \cos \left (b x + a\right )^{12} - 567 \, \cos \left (b x + a\right )^{10} + 371 \, \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 )}{3003 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128/3003*(231*cos(b*x + a)^12 - 567*cos(b*x + a)^10 + 371*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 [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.82603, size = 62, normalized size = 1.02 \begin{align*} -\frac{128 \,{\left (231 \, \sin \left (b x + a\right )^{13} - 819 \, \sin \left (b x + a\right )^{11} + 1001 \, \sin \left (b x + a\right )^{9} - 429 \, \sin \left (b x + a\right )^{7}\right )}}{3003 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128/3003*(231*sin(b*x + a)^13 - 819*sin(b*x + a)^11 + 1001*sin(b*x + a)^9 - 429*sin(b*x + a)^7)/b