3.1.92 \(\int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx\) [92]

Optimal. Leaf size=109 \[ -\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{3 a^3 d}-\frac {2 \cos ^7(c+d x)}{7 a^3 d}+\frac {3 \cos ^8(c+d x)}{8 a^3 d}-\frac {\cos ^9(c+d x)}{9 a^3 d} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 109, 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, 76} \begin {gather*} -\frac {\cos ^9(c+d x)}{9 a^3 d}+\frac {3 \cos ^8(c+d x)}{8 a^3 d}-\frac {2 \cos ^7(c+d x)}{7 a^3 d}-\frac {\cos ^6(c+d x)}{3 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^4(c+d x)}{4 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/4*Cos[c + d*x]^4/(a^3*d) + (3*Cos[c + d*x]^5)/(5*a^3*d) - Cos[c + d*x]^6/(3*a^3*d) - (2*Cos[c + d*x]^7)/(7*
a^3*d) + (3*Cos[c + d*x]^8)/(8*a^3*d) - Cos[c + d*x]^9/(9*a^3*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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

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

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Mathematica [A]
time = 1.85, size = 100, normalized size = 0.92 \begin {gather*} -\frac {34771-52920 \cos (c+d x)+37800 \cos (2 (c+d x))-18480 \cos (3 (c+d x))+3780 \cos (4 (c+d x))+3024 \cos (5 (c+d x))-4200 \cos (6 (c+d x))+2700 \cos (7 (c+d x))-945 \cos (8 (c+d x))+140 \cos (9 (c+d x))}{322560 a^3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^3,x]

[Out]

-1/322560*(34771 - 52920*Cos[c + d*x] + 37800*Cos[2*(c + d*x)] - 18480*Cos[3*(c + d*x)] + 3780*Cos[4*(c + d*x)
] + 3024*Cos[5*(c + d*x)] - 4200*Cos[6*(c + d*x)] + 2700*Cos[7*(c + d*x)] - 945*Cos[8*(c + d*x)] + 140*Cos[9*(
c + d*x)])/(a^3*d)

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Maple [A]
time = 0.19, size = 69, normalized size = 0.63

method	result	size
derivativedivides	\(\frac {-\frac {1}{9 \sec \left (d x +c \right )^{9}}-\frac {1}{3 \sec \left (d x +c \right )^{6}}+\frac {3}{8 \sec \left (d x +c \right )^{8}}-\frac {1}{4 \sec \left (d x +c \right )^{4}}+\frac {3}{5 \sec \left (d x +c \right )^{5}}-\frac {2}{7 \sec \left (d x +c \right )^{7}}}{d \,a^{3}}\)	\(69\)
default	\(\frac {-\frac {1}{9 \sec \left (d x +c \right )^{9}}-\frac {1}{3 \sec \left (d x +c \right )^{6}}+\frac {3}{8 \sec \left (d x +c \right )^{8}}-\frac {1}{4 \sec \left (d x +c \right )^{4}}+\frac {3}{5 \sec \left (d x +c \right )^{5}}-\frac {2}{7 \sec \left (d x +c \right )^{7}}}{d \,a^{3}}\)	\(69\)
risch	\(\frac {21 \cos \left (d x +c \right )}{128 a^{3} d}-\frac {\cos \left (9 d x +9 c \right )}{2304 d \,a^{3}}+\frac {3 \cos \left (8 d x +8 c \right )}{1024 d \,a^{3}}-\frac {15 \cos \left (7 d x +7 c \right )}{1792 d \,a^{3}}+\frac {5 \cos \left (6 d x +6 c \right )}{384 d \,a^{3}}-\frac {3 \cos \left (5 d x +5 c \right )}{320 d \,a^{3}}-\frac {3 \cos \left (4 d x +4 c \right )}{256 d \,a^{3}}+\frac {11 \cos \left (3 d x +3 c \right )}{192 d \,a^{3}}-\frac {15 \cos \left (2 d x +2 c \right )}{128 d \,a^{3}}\)	\(152\)
norman	\(\frac {\frac {128}{315 a d}+\frac {32 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {32 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {64 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {128 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a d}+\frac {512 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 a d}+\frac {512 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 a d}+\frac {256 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9} a^{2}}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^9/(a+a*sec(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 69, normalized size = 0.63 \begin {gather*} -\frac {280 \, \cos \left (d x + c\right )^{9} - 945 \, \cos \left (d x + c\right )^{8} + 720 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 1512 \, \cos \left (d x + c\right )^{5} + 630 \, \cos \left (d x + c\right )^{4}}{2520 \, a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/2520*(280*cos(d*x + c)^9 - 945*cos(d*x + c)^8 + 720*cos(d*x + c)^7 + 840*cos(d*x + c)^6 - 1512*cos(d*x + c)
^5 + 630*cos(d*x + c)^4)/(a^3*d)

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Fricas [A]
time = 2.75, size = 69, normalized size = 0.63 \begin {gather*} -\frac {280 \, \cos \left (d x + c\right )^{9} - 945 \, \cos \left (d x + c\right )^{8} + 720 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 1512 \, \cos \left (d x + c\right )^{5} + 630 \, \cos \left (d x + c\right )^{4}}{2520 \, a^{3} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/2520*(280*cos(d*x + c)^9 - 945*cos(d*x + c)^8 + 720*cos(d*x + c)^7 + 840*cos(d*x + c)^6 - 1512*cos(d*x + c)
^5 + 630*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)**9/(a+a*sec(d*x+c))**3,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3006 deep

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Giac [A]
time = 0.58, size = 185, normalized size = 1.70 \begin {gather*} \frac {32 \, {\left (\frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {144 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {336 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {504 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {630 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {105 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {315 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 4\right )}}{315 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^9/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

32/315*(36*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 144*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 336*(cos(d*
x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 504*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 630*(cos(d*x + c) - 1)^5/
(cos(d*x + c) + 1)^5 - 105*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 315*(cos(d*x + c) - 1)^7/(cos(d*x + c)
+ 1)^7 - 4)/(a^3*d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)

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Mupad [B]
time = 0.92, size = 84, normalized size = 0.77 \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}{3\,a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^7}{7\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^8}{8\,a^3}+\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^3}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^9/(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/(3*a^3) + (2*cos(c + d*x)^7)/(7*a^3) -
(3*cos(c + d*x)^8)/(8*a^3) + cos(c + d*x)^9/(9*a^3))/d

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