3.91 \(\int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=139 \[ -\frac{(a-a \cos (c+d x))^{11}}{11 a^{14} d}+\frac{7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac{19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac{25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac{16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac{2 (a-a \cos (c+d x))^6}{3 a^9 d} \]

[Out]

(2*(a - a*Cos[c + d*x])^6)/(3*a^9*d) - (16*(a - a*Cos[c + d*x])^7)/(7*a^10*d) + (25*(a - a*Cos[c + d*x])^8)/(8
*a^11*d) - (19*(a - a*Cos[c + d*x])^9)/(9*a^12*d) + (7*(a - a*Cos[c + d*x])^10)/(10*a^13*d) - (a - a*Cos[c + d
*x])^11/(11*a^14*d)

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Rubi [A]  time = 0.194717, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3872, 2836, 12, 88} \[ -\frac{(a-a \cos (c+d x))^{11}}{11 a^{14} d}+\frac{7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac{19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac{25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac{16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac{2 (a-a \cos (c+d x))^6}{3 a^9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]

[Out]

(2*(a - a*Cos[c + d*x])^6)/(3*a^9*d) - (16*(a - a*Cos[c + d*x])^7)/(7*a^10*d) + (25*(a - a*Cos[c + d*x])^8)/(8
*a^11*d) - (19*(a - a*Cos[c + d*x])^9)/(9*a^12*d) + (7*(a - a*Cos[c + d*x])^10)/(10*a^13*d) - (a - a*Cos[c + d
*x])^11/(11*a^14*d)

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2836

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d*x)/b
)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac{\cos ^3(c+d x) \sin ^{11}(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-a-x)^5 x^3 (-a+x)^2}{a^3} \, dx,x,-a \cos (c+d x)\right )}{a^{11} d}\\ &=\frac{\operatorname{Subst}\left (\int (-a-x)^5 x^3 (-a+x)^2 \, dx,x,-a \cos (c+d x)\right )}{a^{14} d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-4 a^5 (-a-x)^5-16 a^4 (-a-x)^6-25 a^3 (-a-x)^7-19 a^2 (-a-x)^8-7 a (-a-x)^9-(-a-x)^{10}\right ) \, dx,x,-a \cos (c+d x)\right )}{a^{14} d}\\ &=\frac{2 (a-a \cos (c+d x))^6}{3 a^9 d}-\frac{16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac{25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac{19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac{7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac{(a-a \cos (c+d x))^{11}}{11 a^{14} d}\\ \end{align*}

Mathematica [A]  time = 4.27029, size = 120, normalized size = 0.86 \[ \frac{2273040 \cos (c+d x)-1496880 \cos (2 (c+d x))+535920 \cos (3 (c+d x))+110880 \cos (4 (c+d x))-293832 \cos (5 (c+d x))+212520 \cos (6 (c+d x))-67320 \cos (7 (c+d x))-27720 \cos (8 (c+d x))+40040 \cos (9 (c+d x))-16632 \cos (10 (c+d x))+2520 \cos (11 (c+d x))-1615571}{28385280 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]

[Out]

(-1615571 + 2273040*Cos[c + d*x] - 1496880*Cos[2*(c + d*x)] + 535920*Cos[3*(c + d*x)] + 110880*Cos[4*(c + d*x)
] - 293832*Cos[5*(c + d*x)] + 212520*Cos[6*(c + d*x)] - 67320*Cos[7*(c + d*x)] - 27720*Cos[8*(c + d*x)] + 4004
0*Cos[9*(c + d*x)] - 16632*Cos[10*(c + d*x)] + 2520*Cos[11*(c + d*x)])/(28385280*a^3*d)

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Maple [A]  time = 0.12, size = 90, normalized size = 0.7 \begin{align*} -{\frac{1}{d{a}^{3}} \left ( -{\frac{5}{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{8}}}+{\frac{1}{6\, \left ( \sec \left ( dx+c \right ) \right ) ^{6}}}-{\frac{1}{11\, \left ( \sec \left ( dx+c \right ) \right ) ^{11}}}+{\frac{1}{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}}+{\frac{3}{10\, \left ( \sec \left ( dx+c \right ) \right ) ^{10}}}-{\frac{3}{5\, \left ( \sec \left ( dx+c \right ) \right ) ^{5}}}+{\frac{5}{7\, \left ( \sec \left ( dx+c \right ) \right ) ^{7}}}-{\frac{1}{9\, \left ( \sec \left ( dx+c \right ) \right ) ^{9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^11/(a+a*sec(d*x+c))^3,x)

[Out]

-1/d/a^3*(-5/8/sec(d*x+c)^8+1/6/sec(d*x+c)^6-1/11/sec(d*x+c)^11+1/4/sec(d*x+c)^4+3/10/sec(d*x+c)^10-3/5/sec(d*
x+c)^5+5/7/sec(d*x+c)^7-1/9/sec(d*x+c)^9)

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Maxima [A]  time = 0.998433, size = 120, normalized size = 0.86 \begin{align*} \frac{2520 \, \cos \left (d x + c\right )^{11} - 8316 \, \cos \left (d x + c\right )^{10} + 3080 \, \cos \left (d x + c\right )^{9} + 17325 \, \cos \left (d x + c\right )^{8} - 19800 \, \cos \left (d x + c\right )^{7} - 4620 \, \cos \left (d x + c\right )^{6} + 16632 \, \cos \left (d x + c\right )^{5} - 6930 \, \cos \left (d x + c\right )^{4}}{27720 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

1/27720*(2520*cos(d*x + c)^11 - 8316*cos(d*x + c)^10 + 3080*cos(d*x + c)^9 + 17325*cos(d*x + c)^8 - 19800*cos(
d*x + c)^7 - 4620*cos(d*x + c)^6 + 16632*cos(d*x + c)^5 - 6930*cos(d*x + c)^4)/(a^3*d)

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Fricas [A]  time = 1.80858, size = 267, normalized size = 1.92 \begin{align*} \frac{2520 \, \cos \left (d x + c\right )^{11} - 8316 \, \cos \left (d x + c\right )^{10} + 3080 \, \cos \left (d x + c\right )^{9} + 17325 \, \cos \left (d x + c\right )^{8} - 19800 \, \cos \left (d x + c\right )^{7} - 4620 \, \cos \left (d x + c\right )^{6} + 16632 \, \cos \left (d x + c\right )^{5} - 6930 \, \cos \left (d x + c\right )^{4}}{27720 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/27720*(2520*cos(d*x + c)^11 - 8316*cos(d*x + c)^10 + 3080*cos(d*x + c)^9 + 17325*cos(d*x + c)^8 - 19800*cos(
d*x + c)^7 - 4620*cos(d*x + c)^6 + 16632*cos(d*x + c)^5 - 6930*cos(d*x + c)^4)/(a^3*d)

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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(sin(d*x+c)**11/(a+a*sec(d*x+c))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.3818, size = 279, normalized size = 2.01 \begin{align*} \frac{32 \,{\left (\frac{209 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{1045 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3135 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{6270 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{8778 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac{13398 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{2310 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{9240 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 19\right )}}{3465 \, a^{3} d{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^11/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

32/3465*(209*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1045*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 3135*(co
s(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 6270*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 8778*(cos(d*x + c) -
1)^5/(cos(d*x + c) + 1)^5 - 13398*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 2310*(cos(d*x + c) - 1)^7/(cos(
d*x + c) + 1)^7 - 9240*(cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 - 19)/(a^3*d*((cos(d*x + c) - 1)/(cos(d*x + c
) + 1) - 1)^11)