3.1.91 \(\int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [91]

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

[Out]

2/3*(a-a*cos(d*x+c))^6/a^9/d-16/7*(a-a*cos(d*x+c))^7/a^10/d+25/8*(a-a*cos(d*x+c))^8/a^11/d-19/9*(a-a*cos(d*x+c
))^9/a^12/d+7/10*(a-a*cos(d*x+c))^10/a^13/d-1/11*(a-a*cos(d*x+c))^11/a^14/d

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Rubi [A]
time = 0.14, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12, 90} \begin {gather*} -\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} \end {gather*}

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 12

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

Rule 90

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]))

Rule 2915

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/b)*x
)^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 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[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]

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 {\text {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 {\text {Subst}\left (\int (-a-x)^5 x^3 (-a+x)^2 \, dx,x,-a \cos (c+d x)\right )}{a^{14} d}\\ &=\frac {\text {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*}

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Mathematica [A]
time = 2.69, size = 120, normalized size = 0.86 \begin {gather*} \frac {-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))+40040 \cos (9 (c+d x))-16632 \cos (10 (c+d x))+2520 \cos (11 (c+d x))}{28385280 a^3 d} \end {gather*}

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 = 0.15, size = 90, normalized size = 0.65

method	result	size
derivativedivides	\(-\frac {-\frac {5}{8 \sec \left (d x +c \right )^{8}}+\frac {3}{10 \sec \left (d x +c \right )^{10}}+\frac {5}{7 \sec \left (d x +c \right )^{7}}-\frac {1}{11 \sec \left (d x +c \right )^{11}}+\frac {1}{4 \sec \left (d x +c \right )^{4}}-\frac {3}{5 \sec \left (d x +c \right )^{5}}+\frac {1}{6 \sec \left (d x +c \right )^{6}}-\frac {1}{9 \sec \left (d x +c \right )^{9}}}{d \,a^{3}}\)	\(90\)
default	\(-\frac {-\frac {5}{8 \sec \left (d x +c \right )^{8}}+\frac {3}{10 \sec \left (d x +c \right )^{10}}+\frac {5}{7 \sec \left (d x +c \right )^{7}}-\frac {1}{11 \sec \left (d x +c \right )^{11}}+\frac {1}{4 \sec \left (d x +c \right )^{4}}-\frac {3}{5 \sec \left (d x +c \right )^{5}}+\frac {1}{6 \sec \left (d x +c \right )^{6}}-\frac {1}{9 \sec \left (d x +c \right )^{9}}}{d \,a^{3}}\)	\(90\)
risch	\(\frac {41 \cos \left (d x +c \right )}{512 a^{3} d}+\frac {\cos \left (11 d x +11 c \right )}{11264 d \,a^{3}}-\frac {3 \cos \left (10 d x +10 c \right )}{5120 d \,a^{3}}+\frac {13 \cos \left (9 d x +9 c \right )}{9216 d \,a^{3}}-\frac {\cos \left (8 d x +8 c \right )}{1024 d \,a^{3}}-\frac {17 \cos \left (7 d x +7 c \right )}{7168 d \,a^{3}}+\frac {23 \cos \left (6 d x +6 c \right )}{3072 d \,a^{3}}-\frac {53 \cos \left (5 d x +5 c \right )}{5120 d \,a^{3}}+\frac {\cos \left (4 d x +4 c \right )}{256 d \,a^{3}}+\frac {29 \cos \left (3 d x +3 c \right )}{1536 d \,a^{3}}-\frac {27 \cos \left (2 d x +2 c \right )}{512 d \,a^{3}}\)	\(186\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 89, normalized size = 0.64 \begin {gather*} \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 {gather*}

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 = 4.21, size = 89, normalized size = 0.64 \begin {gather*} \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 {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 6191 deep

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Giac [A]
time = 0.55, size = 207, normalized size = 1.49 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.09, size = 110, normalized size = 0.79 \begin {gather*} -\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{6\,a^3}+\frac {5\,{\cos \left (c+d\,x\right )}^7}{7\,a^3}-\frac {5\,{\cos \left (c+d\,x\right )}^8}{8\,a^3}-\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^{10}}{10\,a^3}-\frac {{\cos \left (c+d\,x\right )}^{11}}{11\,a^3}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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