3.75 \(\int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=114 \[ \frac {(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {4 (a-a \cos (c+d x))^5}{5 a^7 d} \]

[Out]

4/5*(a-a*cos(d*x+c))^5/a^7/d-2*(a-a*cos(d*x+c))^6/a^8/d+13/7*(a-a*cos(d*x+c))^7/a^9/d-3/4*(a-a*cos(d*x+c))^8/a
^10/d+1/9*(a-a*cos(d*x+c))^9/a^11/d

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Rubi [A]  time = 0.18, antiderivative size = 114, 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 = {3872, 2836, 12, 88} \[ \frac {(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac {3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac {13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac {2 (a-a \cos (c+d x))^6}{a^8 d}+\frac {4 (a-a \cos (c+d x))^5}{5 a^7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^2,x]

[Out]

(4*(a - a*Cos[c + d*x])^5)/(5*a^7*d) - (2*(a - a*Cos[c + d*x])^6)/(a^8*d) + (13*(a - a*Cos[c + d*x])^7)/(7*a^9
*d) - (3*(a - a*Cos[c + d*x])^8)/(4*a^10*d) + (a - a*Cos[c + d*x])^9/(9*a^11*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 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]))

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 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]

Rubi steps

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

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Mathematica [A]  time = 3.95, size = 62, normalized size = 0.54 \[ \frac {2 \sin ^{10}\left (\frac {1}{2} (c+d x)\right ) (1615 \cos (c+d x)+970 \cos (2 (c+d x))+385 \cos (3 (c+d x))+70 \cos (4 (c+d x))+992)}{315 a^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^2,x]

[Out]

(2*(992 + 1615*Cos[c + d*x] + 970*Cos[2*(c + d*x)] + 385*Cos[3*(c + d*x)] + 70*Cos[4*(c + d*x)])*Sin[(c + d*x)
/2]^10)/(315*a^2*d)

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fricas [A]  time = 0.60, size = 79, normalized size = 0.69 \[ -\frac {140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/1260*(140*cos(d*x + c)^9 - 315*cos(d*x + c)^8 - 180*cos(d*x + c)^7 + 840*cos(d*x + c)^6 - 252*cos(d*x + c)^
5 - 630*cos(d*x + c)^4 + 420*cos(d*x + c)^3)/(a^2*d)

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giac [A]  time = 0.29, size = 141, normalized size = 1.24 \[ -\frac {64 \, {\left (\frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {126 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {210 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 1\right )}}{315 \, a^{2} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

-64/315*(9*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 36*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 84*(cos(d*x
+ c) - 1)^3/(cos(d*x + c) + 1)^3 - 126*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 210*(cos(d*x + c) - 1)^6/(c
os(d*x + c) + 1)^6 - 1)/(a^2*d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)

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maple [A]  time = 0.63, size = 79, normalized size = 0.69 \[ \frac {-\frac {2}{3 \sec \left (d x +c \right )^{6}}+\frac {1}{7 \sec \left (d x +c \right )^{7}}-\frac {1}{9 \sec \left (d x +c \right )^{9}}-\frac {1}{3 \sec \left (d x +c \right )^{3}}+\frac {1}{2 \sec \left (d x +c \right )^{4}}+\frac {1}{4 \sec \left (d x +c \right )^{8}}+\frac {1}{5 \sec \left (d x +c \right )^{5}}}{d \,a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^9/(a+a*sec(d*x+c))^2,x)

[Out]

1/d/a^2*(-2/3/sec(d*x+c)^6+1/7/sec(d*x+c)^7-1/9/sec(d*x+c)^9-1/3/sec(d*x+c)^3+1/2/sec(d*x+c)^4+1/4/sec(d*x+c)^
8+1/5/sec(d*x+c)^5)

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maxima [A]  time = 0.33, size = 79, normalized size = 0.69 \[ -\frac {140 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 180 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 252 \, \cos \left (d x + c\right )^{5} - 630 \, \cos \left (d x + c\right )^{4} + 420 \, \cos \left (d x + c\right )^{3}}{1260 \, a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/1260*(140*cos(d*x + c)^9 - 315*cos(d*x + c)^8 - 180*cos(d*x + c)^7 + 840*cos(d*x + c)^6 - 252*cos(d*x + c)^
5 - 630*cos(d*x + c)^4 + 420*cos(d*x + c)^3)/(a^2*d)

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mupad [B]  time = 0.91, size = 96, normalized size = 0.84 \[ \frac {\frac {{\cos \left (c+d\,x\right )}^4}{2\,a^2}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a^2}+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a^2}-\frac {2\,{\cos \left (c+d\,x\right )}^6}{3\,a^2}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2}+\frac {{\cos \left (c+d\,x\right )}^8}{4\,a^2}-\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^2}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^9/(a + a/cos(c + d*x))^2,x)

[Out]

(cos(c + d*x)^4/(2*a^2) - cos(c + d*x)^3/(3*a^2) + cos(c + d*x)^5/(5*a^2) - (2*cos(c + d*x)^6)/(3*a^2) + cos(c
+ d*x)^7/(7*a^2) + cos(c + d*x)^8/(4*a^2) - cos(c + d*x)^9/(9*a^2))/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**9/(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

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