3.1.38 \(\int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx\) [38]

Optimal. Leaf size=203 \[ \frac {11 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}-\frac {a^3 \cos ^9(c+d x)}{9 d}+\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]

[Out]

11*a^3*cos(d*x+c)/d+3*a^3*cos(d*x+c)^2/d-14/3*a^3*cos(d*x+c)^3/d-7/2*a^3*cos(d*x+c)^4/d+6/5*a^3*cos(d*x+c)^5/d
+11/6*a^3*cos(d*x+c)^6/d+1/7*a^3*cos(d*x+c)^7/d-3/8*a^3*cos(d*x+c)^8/d-1/9*a^3*cos(d*x+c)^9/d+a^3*ln(cos(d*x+c
))/d+3*a^3*sec(d*x+c)/d+1/2*a^3*sec(d*x+c)^2/d

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Rubi [A]
time = 0.14, antiderivative size = 203, 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^3 \cos ^9(c+d x)}{9 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^2(c+d x)}{d}+\frac {11 a^3 \cos (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \log (\cos (c+d x))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*a^3*Cos[c + d*x])/d + (3*a^3*Cos[c + d*x]^2)/d - (14*a^3*Cos[c + d*x]^3)/(3*d) - (7*a^3*Cos[c + d*x]^4)/(2
*d) + (6*a^3*Cos[c + d*x]^5)/(5*d) + (11*a^3*Cos[c + d*x]^6)/(6*d) + (a^3*Cos[c + d*x]^7)/(7*d) - (3*a^3*Cos[c
+ d*x]^8)/(8*d) - (a^3*Cos[c + d*x]^9)/(9*d) + (a^3*Log[Cos[c + d*x]])/d + (3*a^3*Sec[c + d*x])/d + (a^3*Sec[
c + d*x]^2)/(2*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 (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx &=-\int (-a-a \cos (c+d x))^3 \sin ^6(c+d x) \tan ^3(c+d x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {a^3 (-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^9 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-a-x)^4 (-a+x)^7}{x^3} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {\text {Subst}\left (\int \left (-11 a^8-\frac {a^{11}}{x^3}+\frac {3 a^{10}}{x^2}+\frac {a^9}{x}+6 a^7 x+14 a^6 x^2-14 a^5 x^3-6 a^4 x^4+11 a^3 x^5-a^2 x^6-3 a x^7+x^8\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {11 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {14 a^3 \cos ^3(c+d x)}{3 d}-\frac {7 a^3 \cos ^4(c+d x)}{2 d}+\frac {6 a^3 \cos ^5(c+d x)}{5 d}+\frac {11 a^3 \cos ^6(c+d x)}{6 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \cos ^8(c+d x)}{8 d}-\frac {a^3 \cos ^9(c+d x)}{9 d}+\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}\\ \end {align*}

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Mathematica [A]
time = 1.16, size = 148, normalized size = 0.73 \begin {gather*} \frac {a^3 (471450+11624760 \cos (c+d x)+2188872 \cos (3 (c+d x))+41160 \cos (4 (c+d x))-204156 \cos (5 (c+d x))-35805 \cos (6 (c+d x))+22972 \cos (7 (c+d x))+9030 \cos (8 (c+d x))-820 \cos (9 (c+d x))-945 \cos (10 (c+d x))-140 \cos (11 (c+d x))+645120 \log (\cos (c+d x))+210 \cos (2 (c+d x)) (-413+3072 \log (\cos (c+d x)))) \sec ^2(c+d x)}{1290240 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(471450 + 11624760*Cos[c + d*x] + 2188872*Cos[3*(c + d*x)] + 41160*Cos[4*(c + d*x)] - 204156*Cos[5*(c + d
*x)] - 35805*Cos[6*(c + d*x)] + 22972*Cos[7*(c + d*x)] + 9030*Cos[8*(c + d*x)] - 820*Cos[9*(c + d*x)] - 945*Co
s[10*(c + d*x)] - 140*Cos[11*(c + d*x)] + 645120*Log[Cos[c + d*x]] + 210*Cos[2*(c + d*x)]*(-413 + 3072*Log[Cos
[c + d*x]]))*Sec[c + d*x]^2)/(1290240*d)

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Maple [A]
time = 0.10, size = 252, normalized size = 1.24

method	result	size
derivativedivides	\(\frac {a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{2}+\frac {2 \left (\sin ^{6}\left (d x +c \right )\right )}{3}+\sin ^{4}\left (d x +c \right )+2 \left (\sin ^{2}\left (d x +c \right )\right )+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\)	\(252\)
default	\(\frac {a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{2 \cos \left (d x +c \right )^{2}}+\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{2}+\frac {2 \left (\sin ^{6}\left (d x +c \right )\right )}{3}+\sin ^{4}\left (d x +c \right )+2 \left (\sin ^{2}\left (d x +c \right )\right )+4 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin ^{10}\left (d x +c \right )}{\cos \left (d x +c \right )}+\left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\left (\sin ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\)	\(252\)
risch	\(-\frac {25 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{64 d}+\frac {1059 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{256 d}+\frac {57 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{256 d}+\frac {57 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{256 d}+\frac {1059 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{256 d}-\frac {25 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{64 d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {2 i a^{3} c}{d}-i a^{3} x -\frac {a^{3} \cos \left (9 d x +9 c \right )}{2304 d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {3 a^{3} \cos \left (8 d x +8 c \right )}{1024 d}-\frac {3 a^{3} \cos \left (7 d x +7 c \right )}{1792 d}+\frac {13 a^{3} \cos \left (6 d x +6 c \right )}{384 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{40 d}-\frac {45 a^{3} \cos \left (4 d x +4 c \right )}{256 d}\)	\(295\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(1/2*sin(d*x+c)^10/cos(d*x+c)^2+1/2*sin(d*x+c)^8+2/3*sin(d*x+c)^6+sin(d*x+c)^4+2*sin(d*x+c)^2+4*ln(co
s(d*x+c)))+3*a^3*(sin(d*x+c)^10/cos(d*x+c)+(128/35+sin(d*x+c)^8+8/7*sin(d*x+c)^6+48/35*sin(d*x+c)^4+64/35*sin(
d*x+c)^2)*cos(d*x+c))+3*a^3*(-1/8*sin(d*x+c)^8-1/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c)^2-ln(cos(d*x+c
)))-1/9*a^3*(128/35+sin(d*x+c)^8+8/7*sin(d*x+c)^6+48/35*sin(d*x+c)^4+64/35*sin(d*x+c)^2)*cos(d*x+c))

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Maxima [A]
time = 0.27, size = 158, normalized size = 0.78 \begin {gather*} -\frac {280 \, a^{3} \cos \left (d x + c\right )^{9} + 945 \, a^{3} \cos \left (d x + c\right )^{8} - 360 \, a^{3} \cos \left (d x + c\right )^{7} - 4620 \, a^{3} \cos \left (d x + c\right )^{6} - 3024 \, a^{3} \cos \left (d x + c\right )^{5} + 8820 \, a^{3} \cos \left (d x + c\right )^{4} + 11760 \, a^{3} \cos \left (d x + c\right )^{3} - 7560 \, a^{3} \cos \left (d x + c\right )^{2} - 27720 \, a^{3} \cos \left (d x + c\right ) - 2520 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{2520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(280*a^3*cos(d*x + c)^9 + 945*a^3*cos(d*x + c)^8 - 360*a^3*cos(d*x + c)^7 - 4620*a^3*cos(d*x + c)^6 -
3024*a^3*cos(d*x + c)^5 + 8820*a^3*cos(d*x + c)^4 + 11760*a^3*cos(d*x + c)^3 - 7560*a^3*cos(d*x + c)^2 - 27720
*a^3*cos(d*x + c) - 2520*a^3*log(cos(d*x + c)) - 1260*(6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d

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Fricas [A]
time = 2.92, size = 182, normalized size = 0.90 \begin {gather*} -\frac {35840 \, a^{3} \cos \left (d x + c\right )^{11} + 120960 \, a^{3} \cos \left (d x + c\right )^{10} - 46080 \, a^{3} \cos \left (d x + c\right )^{9} - 591360 \, a^{3} \cos \left (d x + c\right )^{8} - 387072 \, a^{3} \cos \left (d x + c\right )^{7} + 1128960 \, a^{3} \cos \left (d x + c\right )^{6} + 1505280 \, a^{3} \cos \left (d x + c\right )^{5} - 967680 \, a^{3} \cos \left (d x + c\right )^{4} - 3548160 \, a^{3} \cos \left (d x + c\right )^{3} - 322560 \, a^{3} \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 212205 \, a^{3} \cos \left (d x + c\right )^{2} - 967680 \, a^{3} \cos \left (d x + c\right ) - 161280 \, a^{3}}{322560 \, d \cos \left (d x + c\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/322560*(35840*a^3*cos(d*x + c)^11 + 120960*a^3*cos(d*x + c)^10 - 46080*a^3*cos(d*x + c)^9 - 591360*a^3*cos(
d*x + c)^8 - 387072*a^3*cos(d*x + c)^7 + 1128960*a^3*cos(d*x + c)^6 + 1505280*a^3*cos(d*x + c)^5 - 967680*a^3*
cos(d*x + c)^4 - 3548160*a^3*cos(d*x + c)^3 - 322560*a^3*cos(d*x + c)^2*log(-cos(d*x + c)) + 212205*a^3*cos(d*
x + c)^2 - 967680*a^3*cos(d*x + c) - 161280*a^3)/(d*cos(d*x + c)^2)

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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((a+a*sec(d*x+c))**3*sin(d*x+c)**9,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 8570 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (187) = 374\).
time = 0.65, size = 396, normalized size = 1.95 \begin {gather*} -\frac {2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {1260 \, {\left (9 \, a^{3} + \frac {2 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {3 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} + \frac {45257 \, a^{3} - \frac {392193 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {1467972 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {3001908 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3232782 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {2359854 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {1190196 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {397764 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {79281 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2520*(2520*a^3*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 2520*a^3*log(abs(-(cos(d*x + c) - 1)/
(cos(d*x + c) + 1) - 1)) - 1260*(9*a^3 + 2*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 3*a^3*(cos(d*x + c) - 1
)^2/(cos(d*x + c) + 1)^2)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)^2 + (45257*a^3 - 392193*a^3*(cos(d*x + c
) - 1)/(cos(d*x + c) + 1) + 1467972*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 3001908*a^3*(cos(d*x + c)
- 1)^3/(cos(d*x + c) + 1)^3 + 3232782*a^3*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 2359854*a^3*(cos(d*x + c
) - 1)^5/(cos(d*x + c) + 1)^5 + 1190196*a^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 397764*a^3*(cos(d*x +
c) - 1)^7/(cos(d*x + c) + 1)^7 + 79281*a^3*(cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 - 7129*a^3*(cos(d*x + c)
- 1)^9/(cos(d*x + c) + 1)^9)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)/d

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Mupad [B]
time = 0.96, size = 157, normalized size = 0.77 \begin {gather*} \frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+11\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-\frac {14\,a^3\,{\cos \left (c+d\,x\right )}^3}{3}-\frac {7\,a^3\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a^3\,{\cos \left (c+d\,x\right )}^5}{5}+\frac {11\,a^3\,{\cos \left (c+d\,x\right )}^6}{6}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}-\frac {3\,a^3\,{\cos \left (c+d\,x\right )}^8}{8}-\frac {a^3\,{\cos \left (c+d\,x\right )}^9}{9}+a^3\,\ln \left (\cos \left (c+d\,x\right )\right )}{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]

((3*a^3*cos(c + d*x) + a^3/2)/cos(c + d*x)^2 + 11*a^3*cos(c + d*x) + 3*a^3*cos(c + d*x)^2 - (14*a^3*cos(c + d*
x)^3)/3 - (7*a^3*cos(c + d*x)^4)/2 + (6*a^3*cos(c + d*x)^5)/5 + (11*a^3*cos(c + d*x)^6)/6 + (a^3*cos(c + d*x)^
7)/7 - (3*a^3*cos(c + d*x)^8)/8 - (a^3*cos(c + d*x)^9)/9 + a^3*log(cos(c + d*x)))/d

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