3.35 \(\int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx\)

Optimal. Leaf size=46 \[ -\frac{64 \cos ^{11}(a+b x)}{11 b}+\frac{128 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^7(a+b x)}{7 b} \]

[Out]

(-64*Cos[a + b*x]^7)/(7*b) + (128*Cos[a + b*x]^9)/(9*b) - (64*Cos[a + b*x]^11)/(11*b)

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Rubi [A]  time = 0.0559452, antiderivative size = 46, 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, 2565, 270} \[ -\frac{64 \cos ^{11}(a+b x)}{11 b}+\frac{128 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Sin[2*a + 2*b*x]^6,x]

[Out]

(-64*Cos[a + b*x]^7)/(7*b) + (128*Cos[a + b*x]^9)/(9*b) - (64*Cos[a + b*x]^11)/(11*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 2565

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(a*f)^(-1), Subst[
Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2]
&&  !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[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 ^6(2 a+2 b x) \, dx &=64 \int \cos ^6(a+b x) \sin ^5(a+b x) \, dx\\ &=-\frac{64 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{64 \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{64 \cos ^7(a+b x)}{7 b}+\frac{128 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^{11}(a+b x)}{11 b}\\ \end{align*}

Mathematica [A]  time = 0.0579333, size = 89, normalized size = 1.93 \[ -\frac{5 \cos (a+b x)}{8 b}-\frac{5 \cos (3 (a+b x))}{24 b}+\frac{\cos (5 (a+b x))}{16 b}+\frac{5 \cos (7 (a+b x))}{112 b}-\frac{\cos (9 (a+b x))}{144 b}-\frac{\cos (11 (a+b x))}{176 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^6,x]

[Out]

(-5*Cos[a + b*x])/(8*b) - (5*Cos[3*(a + b*x)])/(24*b) + Cos[5*(a + b*x)]/(16*b) + (5*Cos[7*(a + b*x)])/(112*b)
- Cos[9*(a + b*x)]/(144*b) - Cos[11*(a + b*x)]/(176*b)

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Maple [A]  time = 0.029, size = 53, normalized size = 1.2 \begin{align*} 64\,{\frac{1}{b} \left ( -1/11\, \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{7}-{\frac{4\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{99}}-{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{693}} \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)^6,x)

[Out]

64/b*(-1/11*sin(b*x+a)^4*cos(b*x+a)^7-4/99*sin(b*x+a)^2*cos(b*x+a)^7-8/693*cos(b*x+a)^7)

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Maxima [A]  time = 1.1505, size = 93, normalized size = 2.02 \begin{align*} -\frac{63 \, \cos \left (11 \, b x + 11 \, a\right ) + 77 \, \cos \left (9 \, b x + 9 \, a\right ) - 495 \, \cos \left (7 \, b x + 7 \, a\right ) - 693 \, \cos \left (5 \, b x + 5 \, a\right ) + 2310 \, \cos \left (3 \, b x + 3 \, a\right ) + 6930 \, \cos \left (b x + a\right )}{11088 \, 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)^6,x, algorithm="maxima")

[Out]

-1/11088*(63*cos(11*b*x + 11*a) + 77*cos(9*b*x + 9*a) - 495*cos(7*b*x + 7*a) - 693*cos(5*b*x + 5*a) + 2310*cos
(3*b*x + 3*a) + 6930*cos(b*x + a))/b

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Fricas [A]  time = 0.503029, size = 99, normalized size = 2.15 \begin{align*} -\frac{64 \,{\left (63 \, \cos \left (b x + a\right )^{11} - 154 \, \cos \left (b x + a\right )^{9} + 99 \, \cos \left (b x + a\right )^{7}\right )}}{693 \, 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)^6,x, algorithm="fricas")

[Out]

-64/693*(63*cos(b*x + a)^11 - 154*cos(b*x + a)^9 + 99*cos(b*x + a)^7)/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)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.59592, size = 275, normalized size = 5.98 \begin{align*} -\frac{1024 \,{\left (\frac{11 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{55 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} - \frac{297 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} - \frac{1485 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} - \frac{2079 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} - \frac{2541 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} - \frac{1155 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - \frac{462 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} - 1\right )}}{693 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(2*b*x+2*a)^6,x, algorithm="giac")

[Out]

-1024/693*(11*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 55*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 - 297*(cos(
b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 - 1485*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 - 2079*(cos(b*x + a) - 1
)^5/(cos(b*x + a) + 1)^5 - 2541*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 - 1155*(cos(b*x + a) - 1)^7/(cos(b*x
+ a) + 1)^7 - 462*(cos(b*x + a) - 1)^8/(cos(b*x + a) + 1)^8 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) -
1)^11)