3.59 \(\int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx\)

Optimal. Leaf size=61 \[ \frac{1024 \cos ^{17}(a+b x)}{17 b}-\frac{1024 \cos ^{15}(a+b x)}{5 b}+\frac{3072 \cos ^{13}(a+b x)}{13 b}-\frac{1024 \cos ^{11}(a+b x)}{11 b} \]

[Out]

(-1024*Cos[a + b*x]^11)/(11*b) + (3072*Cos[a + b*x]^13)/(13*b) - (1024*Cos[a + b*x]^15)/(5*b) + (1024*Cos[a +
b*x]^17)/(17*b)

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Rubi [A]  time = 0.0722782, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2565, 270} \[ \frac{1024 \cos ^{17}(a+b x)}{17 b}-\frac{1024 \cos ^{15}(a+b x)}{5 b}+\frac{3072 \cos ^{13}(a+b x)}{13 b}-\frac{1024 \cos ^{11}(a+b x)}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^10,x]

[Out]

(-1024*Cos[a + b*x]^11)/(11*b) + (3072*Cos[a + b*x]^13)/(13*b) - (1024*Cos[a + b*x]^15)/(5*b) + (1024*Cos[a +
b*x]^17)/(17*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 ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx &=1024 \int \cos ^{10}(a+b x) \sin ^7(a+b x) \, dx\\ &=-\frac{1024 \operatorname{Subst}\left (\int x^{10} \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{1024 \operatorname{Subst}\left (\int \left (x^{10}-3 x^{12}+3 x^{14}-x^{16}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{1024 \cos ^{11}(a+b x)}{11 b}+\frac{3072 \cos ^{13}(a+b x)}{13 b}-\frac{1024 \cos ^{15}(a+b x)}{5 b}+\frac{1024 \cos ^{17}(a+b x)}{17 b}\\ \end{align*}

Mathematica [A]  time = 0.145892, size = 119, normalized size = 1.95 \[ -\frac{35 \cos (a+b x)}{32 b}-\frac{7 \cos (3 (a+b x))}{16 b}+\frac{7 \cos (5 (a+b x))}{80 b}+\frac{\cos (7 (a+b x))}{8 b}-\frac{5 \cos (11 (a+b x))}{176 b}-\frac{\cos (13 (a+b x))}{208 b}+\frac{\cos (15 (a+b x))}{320 b}+\frac{\cos (17 (a+b x))}{1088 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^10,x]

[Out]

(-35*Cos[a + b*x])/(32*b) - (7*Cos[3*(a + b*x)])/(16*b) + (7*Cos[5*(a + b*x)])/(80*b) + Cos[7*(a + b*x)]/(8*b)
- (5*Cos[11*(a + b*x)])/(176*b) - Cos[13*(a + b*x)]/(208*b) + Cos[15*(a + b*x)]/(320*b) + Cos[17*(a + b*x)]/(
1088*b)

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Maple [A]  time = 0.04, size = 71, normalized size = 1.2 \begin{align*} 1024\,{\frac{1}{b} \left ( -1/17\, \left ( \sin \left ( bx+a \right ) \right ) ^{6} \left ( \cos \left ( bx+a \right ) \right ) ^{11}-{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{11}}{85}}-{\frac{8\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{11}}{1105}}-{\frac{16\, \left ( \cos \left ( bx+a \right ) \right ) ^{11}}{12155}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x)

[Out]

1024/b*(-1/17*sin(b*x+a)^6*cos(b*x+a)^11-2/85*sin(b*x+a)^4*cos(b*x+a)^11-8/1105*sin(b*x+a)^2*cos(b*x+a)^11-16/
12155*cos(b*x+a)^11)

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Maxima [A]  time = 1.06209, size = 123, normalized size = 2.02 \begin{align*} \frac{715 \, \cos \left (17 \, b x + 17 \, a\right ) + 2431 \, \cos \left (15 \, b x + 15 \, a\right ) - 3740 \, \cos \left (13 \, b x + 13 \, a\right ) - 22100 \, \cos \left (11 \, b x + 11 \, a\right ) + 97240 \, \cos \left (7 \, b x + 7 \, a\right ) + 68068 \, \cos \left (5 \, b x + 5 \, a\right ) - 340340 \, \cos \left (3 \, b x + 3 \, a\right ) - 850850 \, \cos \left (b x + a\right )}{777920 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x, algorithm="maxima")

[Out]

1/777920*(715*cos(17*b*x + 17*a) + 2431*cos(15*b*x + 15*a) - 3740*cos(13*b*x + 13*a) - 22100*cos(11*b*x + 11*a
) + 97240*cos(7*b*x + 7*a) + 68068*cos(5*b*x + 5*a) - 340340*cos(3*b*x + 3*a) - 850850*cos(b*x + a))/b

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Fricas [A]  time = 0.569647, size = 142, normalized size = 2.33 \begin{align*} \frac{1024 \,{\left (715 \, \cos \left (b x + a\right )^{17} - 2431 \, \cos \left (b x + a\right )^{15} + 2805 \, \cos \left (b x + a\right )^{13} - 1105 \, \cos \left (b x + a\right )^{11}\right )}}{12155 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x, algorithm="fricas")

[Out]

1024/12155*(715*cos(b*x + a)^17 - 2431*cos(b*x + a)^15 + 2805*cos(b*x + a)^13 - 1105*cos(b*x + a)^11)/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)**3*sin(2*b*x+2*a)**10,x)

[Out]

Timed out

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Giac [B]  time = 2.30865, size = 424, normalized size = 6.95 \begin{align*} -\frac{32768 \,{\left (\frac{17 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{136 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{680 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{9775 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac{71825 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac{221000 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac{486200 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac{668525 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac{692835 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} + \frac{466752 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} + \frac{233376 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{11}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{11}} + \frac{65637 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{12}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{12}} + \frac{12155 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{13}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{13}} - 1\right )}}{12155 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x, algorithm="giac")

[Out]

-32768/12155*(17*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 136*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 680*(
cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 + 9775*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + 71825*(cos(b*x + a
) - 1)^5/(cos(b*x + a) + 1)^5 + 221000*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 + 486200*(cos(b*x + a) - 1)^7
/(cos(b*x + a) + 1)^7 + 668525*(cos(b*x + a) - 1)^8/(cos(b*x + a) + 1)^8 + 692835*(cos(b*x + a) - 1)^9/(cos(b*
x + a) + 1)^9 + 466752*(cos(b*x + a) - 1)^10/(cos(b*x + a) + 1)^10 + 233376*(cos(b*x + a) - 1)^11/(cos(b*x + a
) + 1)^11 + 65637*(cos(b*x + a) - 1)^12/(cos(b*x + a) + 1)^12 + 12155*(cos(b*x + a) - 1)^13/(cos(b*x + a) + 1)
^13 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^17)