\(\int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx\) [994]

Optimal result

Integrand size = 31, antiderivative size = 162 \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3 (5 A-3 B) \text {arctanh}(\sin (c+d x))}{32 d}+\frac {a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac {a^6 A}{12 d (a-a \sin (c+d x))^3}+\frac {a^5 (3 A-B)}{32 d (a-a \sin (c+d x))^2}+\frac {a^4 (2 A-B)}{16 d (a-a \sin (c+d x))}-\frac {a^4 (A-B)}{32 d (a+a \sin (c+d x))} \]

[Out]

1/32*a^3*(5*A-3*B)*arctanh(sin(d*x+c))/d+1/16*a^7*(A+B)/d/(a-a*sin(d*x+c))^4+1/12*a^6*A/d/(a-a*sin(d*x+c))^3+1
/32*a^5*(3*A-B)/d/(a-a*sin(d*x+c))^2+1/16*a^4*(2*A-B)/d/(a-a*sin(d*x+c))-1/32*a^4*(A-B)/d/(a+a*sin(d*x+c))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2915, 78, 212} \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac {a^6 A}{12 d (a-a \sin (c+d x))^3}+\frac {a^5 (3 A-B)}{32 d (a-a \sin (c+d x))^2}+\frac {a^4 (2 A-B)}{16 d (a-a \sin (c+d x))}-\frac {a^4 (A-B)}{32 d (a \sin (c+d x)+a)}+\frac {a^3 (5 A-3 B) \text {arctanh}(\sin (c+d x))}{32 d} \]

[In]

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

[Out]

(a^3*(5*A - 3*B)*ArcTanh[Sin[c + d*x]])/(32*d) + (a^7*(A + B))/(16*d*(a - a*Sin[c + d*x])^4) + (a^6*A)/(12*d*(
a - a*Sin[c + d*x])^3) + (a^5*(3*A - B))/(32*d*(a - a*Sin[c + d*x])^2) + (a^4*(2*A - B))/(16*d*(a - a*Sin[c +
d*x])) - (a^4*(A - B))/(32*d*(a + a*Sin[c + d*x]))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 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]

Rubi steps \begin{align*} \text {integral}& = \frac {a^9 \text {Subst}\left (\int \frac {A+\frac {B x}{a}}{(a-x)^5 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^9 \text {Subst}\left (\int \left (\frac {A+B}{4 a^2 (a-x)^5}+\frac {A}{4 a^3 (a-x)^4}+\frac {3 A-B}{16 a^4 (a-x)^3}+\frac {2 A-B}{16 a^5 (a-x)^2}+\frac {A-B}{32 a^5 (a+x)^2}+\frac {5 A-3 B}{32 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d} \\ & = \frac {a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac {a^6 A}{12 d (a-a \sin (c+d x))^3}+\frac {a^5 (3 A-B)}{32 d (a-a \sin (c+d x))^2}+\frac {a^4 (2 A-B)}{16 d (a-a \sin (c+d x))}-\frac {a^4 (A-B)}{32 d (a+a \sin (c+d x))}+\frac {\left (a^4 (5 A-3 B)\right ) \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{32 d} \\ & = \frac {a^3 (5 A-3 B) \text {arctanh}(\sin (c+d x))}{32 d}+\frac {a^7 (A+B)}{16 d (a-a \sin (c+d x))^4}+\frac {a^6 A}{12 d (a-a \sin (c+d x))^3}+\frac {a^5 (3 A-B)}{32 d (a-a \sin (c+d x))^2}+\frac {a^4 (2 A-B)}{16 d (a-a \sin (c+d x))}-\frac {a^4 (A-B)}{32 d (a+a \sin (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.93 \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^9 \left (\frac {(5 A-3 B) \text {arctanh}(\sin (c+d x))}{32 a^6}+\frac {A+B}{16 a^2 (a-a \sin (c+d x))^4}+\frac {A}{12 a^3 (a-a \sin (c+d x))^3}+\frac {3 A-B}{32 a^4 (a-a \sin (c+d x))^2}+\frac {2 A-B}{16 a^5 (a-a \sin (c+d x))}-\frac {A-B}{32 a^5 (a+a \sin (c+d x))}\right )}{d} \]

[In]

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

[Out]

(a^9*(((5*A - 3*B)*ArcTanh[Sin[c + d*x]])/(32*a^6) + (A + B)/(16*a^2*(a - a*Sin[c + d*x])^4) + A/(12*a^3*(a -
a*Sin[c + d*x])^3) + (3*A - B)/(32*a^4*(a - a*Sin[c + d*x])^2) + (2*A - B)/(16*a^5*(a - a*Sin[c + d*x])) - (A
- B)/(32*a^5*(a + a*Sin[c + d*x]))))/d

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.80

method	result	size
parallelrisch	\(-\frac {65 \left (\left (A -\frac {3 B}{5}\right ) \left (\frac {14 \sin \left (d x +c \right )}{13}+\sin \left (3 d x +3 c \right )-\frac {8 \cos \left (2 d x +2 c \right )}{13}-\frac {\sin \left (5 d x +5 c \right )}{13}+\frac {6 \cos \left (4 d x +4 c \right )}{13}-\frac {14}{13}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (A -\frac {3 B}{5}\right ) \left (\frac {14 \sin \left (d x +c \right )}{13}+\sin \left (3 d x +3 c \right )-\frac {8 \cos \left (2 d x +2 c \right )}{13}-\frac {\sin \left (5 d x +5 c \right )}{13}+\frac {6 \cos \left (4 d x +4 c \right )}{13}-\frac {14}{13}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {16 \left (\frac {3 A}{5}-B \right ) \cos \left (2 d x +2 c \right )}{13}+\frac {6 \left (9 A +B \right ) \cos \left (4 d x +4 c \right )}{65}+\frac {4 \left (\frac {59 A}{3}+9 B \right ) \sin \left (3 d x +3 c \right )}{65}-\frac {32 A \sin \left (5 d x +5 c \right )}{195}+\frac {4 \left (-3 B +\frac {187 A}{15}\right ) \sin \left (d x +c \right )}{13}-\frac {102 A}{65}+\frac {74 B}{65}\right ) a^{3}}{32 d \left (-\sin \left (5 d x +5 c \right )+14 \sin \left (d x +c \right )+6 \cos \left (4 d x +4 c \right )-14+13 \sin \left (3 d x +3 c \right )-8 \cos \left (2 d x +2 c \right )\right )}\)	\(292\)
risch	\(-\frac {i a^{3} {\mathrm e}^{i \left (d x +c \right )} \left (-54 i B \,{\mathrm e}^{i \left (d x +c \right )}+15 A \,{\mathrm e}^{8 i \left (d x +c \right )}+150 i A \,{\mathrm e}^{5 i \left (d x +c \right )}-9 B \,{\mathrm e}^{8 i \left (d x +c \right )}-90 i B \,{\mathrm e}^{5 i \left (d x +c \right )}-200 A \,{\mathrm e}^{6 i \left (d x +c \right )}-90 i A \,{\mathrm e}^{7 i \left (d x +c \right )}+120 B \,{\mathrm e}^{6 i \left (d x +c \right )}-150 i A \,{\mathrm e}^{3 i \left (d x +c \right )}-142 A \,{\mathrm e}^{4 i \left (d x +c \right )}+90 i B \,{\mathrm e}^{3 i \left (d x +c \right )}-222 B \,{\mathrm e}^{4 i \left (d x +c \right )}+54 i B \,{\mathrm e}^{7 i \left (d x +c \right )}-200 A \,{\mathrm e}^{2 i \left (d x +c \right )}+90 i A \,{\mathrm e}^{i \left (d x +c \right )}+120 B \,{\mathrm e}^{2 i \left (d x +c \right )}+15 A -9 B \right )}{48 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{8} d}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{32 d}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{32 d}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{32 d}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{32 d}\)	\(371\)
derivativedivides	\(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{5}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{5}\left (d x +c \right )}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{128 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+\frac {3 A \,a^{3}}{8 \cos \left (d x +c \right )^{8}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+A \,a^{3} \left (-\left (-\frac {\left (\sec ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sec ^{5}\left (d x +c \right )\right )}{48}-\frac {35 \left (\sec ^{3}\left (d x +c \right )\right )}{192}-\frac {35 \sec \left (d x +c \right )}{128}\right ) \tan \left (d x +c \right )+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+\frac {B \,a^{3}}{8 \cos \left (d x +c \right )^{8}}}{d}\)	\(542\)
default	\(\frac {A \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+B \,a^{3} \left (\frac {\sin ^{5}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{5}\left (d x +c \right )}{16 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{5}\left (d x +c \right )}{64 \cos \left (d x +c \right )^{4}}-\frac {\sin ^{5}\left (d x +c \right )}{128 \cos \left (d x +c \right )^{2}}-\frac {\left (\sin ^{3}\left (d x +c \right )\right )}{128}-\frac {3 \sin \left (d x +c \right )}{128}+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+3 A \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+3 B \,a^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )+\frac {3 A \,a^{3}}{8 \cos \left (d x +c \right )^{8}}+3 B \,a^{3} \left (\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{48 \cos \left (d x +c \right )^{6}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{64 \cos \left (d x +c \right )^{4}}+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{128 \cos \left (d x +c \right )^{2}}+\frac {5 \sin \left (d x +c \right )}{128}-\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+A \,a^{3} \left (-\left (-\frac {\left (\sec ^{7}\left (d x +c \right )\right )}{8}-\frac {7 \left (\sec ^{5}\left (d x +c \right )\right )}{48}-\frac {35 \left (\sec ^{3}\left (d x +c \right )\right )}{192}-\frac {35 \sec \left (d x +c \right )}{128}\right ) \tan \left (d x +c \right )+\frac {35 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{128}\right )+\frac {B \,a^{3}}{8 \cos \left (d x +c \right )^{8}}}{d}\)	\(542\)

[In]

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

[Out]

-65/32*((A-3/5*B)*(14/13*sin(d*x+c)+sin(3*d*x+3*c)-8/13*cos(2*d*x+2*c)-1/13*sin(5*d*x+5*c)+6/13*cos(4*d*x+4*c)
-14/13)*ln(tan(1/2*d*x+1/2*c)-1)-(A-3/5*B)*(14/13*sin(d*x+c)+sin(3*d*x+3*c)-8/13*cos(2*d*x+2*c)-1/13*sin(5*d*x
+5*c)+6/13*cos(4*d*x+4*c)-14/13)*ln(tan(1/2*d*x+1/2*c)+1)+16/13*(3/5*A-B)*cos(2*d*x+2*c)+6/65*(9*A+B)*cos(4*d*
x+4*c)+4/65*(59/3*A+9*B)*sin(3*d*x+3*c)-32/195*A*sin(5*d*x+5*c)+4/13*(-3*B+187/15*A)*sin(d*x+c)-102/65*A+74/65
*B)*a^3/d/(-sin(5*d*x+5*c)+14*sin(d*x+c)+6*cos(4*d*x+4*c)-14+13*sin(3*d*x+3*c)-8*cos(2*d*x+2*c))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (154) = 308\).

Time = 0.29 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.18 \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {6 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 26 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 12 \, {\left (3 \, A - 5 \, B\right )} a^{3} + 3 \, {\left (3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 4 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - {\left ({\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 4 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 4 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - {\left ({\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{4} - 4 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 6 \, {\left (3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} - 2 \, {\left (5 \, A - 3 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{192 \, {\left (3 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )}} \]

[In]

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

[Out]

1/192*(6*(5*A - 3*B)*a^3*cos(d*x + c)^4 - 26*(5*A - 3*B)*a^3*cos(d*x + c)^2 + 12*(3*A - 5*B)*a^3 + 3*(3*(5*A -
3*B)*a^3*cos(d*x + c)^4 - 4*(5*A - 3*B)*a^3*cos(d*x + c)^2 - ((5*A - 3*B)*a^3*cos(d*x + c)^4 - 4*(5*A - 3*B)*
a^3*cos(d*x + c)^2)*sin(d*x + c))*log(sin(d*x + c) + 1) - 3*(3*(5*A - 3*B)*a^3*cos(d*x + c)^4 - 4*(5*A - 3*B)*
a^3*cos(d*x + c)^2 - ((5*A - 3*B)*a^3*cos(d*x + c)^4 - 4*(5*A - 3*B)*a^3*cos(d*x + c)^2)*sin(d*x + c))*log(-si
n(d*x + c) + 1) + 6*(3*(5*A - 3*B)*a^3*cos(d*x + c)^2 - 2*(5*A - 3*B)*a^3)*sin(d*x + c))/(3*d*cos(d*x + c)^4 -
4*d*cos(d*x + c)^2 - (d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{4} - 9 \, {\left (5 \, A - 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{3} + 7 \, {\left (5 \, A - 3 \, B\right )} a^{3} \sin \left (d x + c\right )^{2} + 3 \, {\left (5 \, A - 3 \, B\right )} a^{3} \sin \left (d x + c\right ) - 32 \, A a^{3}\right )}}{\sin \left (d x + c\right )^{5} - 3 \, \sin \left (d x + c\right )^{4} + 2 \, \sin \left (d x + c\right )^{3} + 2 \, \sin \left (d x + c\right )^{2} - 3 \, \sin \left (d x + c\right ) + 1}}{192 \, d} \]

[In]

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

[Out]

1/192*(3*(5*A - 3*B)*a^3*log(sin(d*x + c) + 1) - 3*(5*A - 3*B)*a^3*log(sin(d*x + c) - 1) - 2*(3*(5*A - 3*B)*a^
3*sin(d*x + c)^4 - 9*(5*A - 3*B)*a^3*sin(d*x + c)^3 + 7*(5*A - 3*B)*a^3*sin(d*x + c)^2 + 3*(5*A - 3*B)*a^3*sin
(d*x + c) - 32*A*a^3)/(sin(d*x + c)^5 - 3*sin(d*x + c)^4 + 2*sin(d*x + c)^3 + 2*sin(d*x + c)^2 - 3*sin(d*x + c
) + 1))/d

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.46 \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {12 \, {\left (5 \, A a^{3} - 3 \, B a^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 12 \, {\left (5 \, A a^{3} - 3 \, B a^{3}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {12 \, {\left (5 \, A a^{3} \sin \left (d x + c\right ) - 3 \, B a^{3} \sin \left (d x + c\right ) + 7 \, A a^{3} - 5 \, B a^{3}\right )}}{\sin \left (d x + c\right ) + 1} + \frac {125 \, A a^{3} \sin \left (d x + c\right )^{4} - 75 \, B a^{3} \sin \left (d x + c\right )^{4} - 596 \, A a^{3} \sin \left (d x + c\right )^{3} + 348 \, B a^{3} \sin \left (d x + c\right )^{3} + 1110 \, A a^{3} \sin \left (d x + c\right )^{2} - 618 \, B a^{3} \sin \left (d x + c\right )^{2} - 996 \, A a^{3} \sin \left (d x + c\right ) + 492 \, B a^{3} \sin \left (d x + c\right ) + 405 \, A a^{3} - 99 \, B a^{3}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{4}}}{768 \, d} \]

[In]

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

[Out]

1/768*(12*(5*A*a^3 - 3*B*a^3)*log(abs(sin(d*x + c) + 1)) - 12*(5*A*a^3 - 3*B*a^3)*log(abs(sin(d*x + c) - 1)) -
12*(5*A*a^3*sin(d*x + c) - 3*B*a^3*sin(d*x + c) + 7*A*a^3 - 5*B*a^3)/(sin(d*x + c) + 1) + (125*A*a^3*sin(d*x
+ c)^4 - 75*B*a^3*sin(d*x + c)^4 - 596*A*a^3*sin(d*x + c)^3 + 348*B*a^3*sin(d*x + c)^3 + 1110*A*a^3*sin(d*x +
c)^2 - 618*B*a^3*sin(d*x + c)^2 - 996*A*a^3*sin(d*x + c) + 492*B*a^3*sin(d*x + c) + 405*A*a^3 - 99*B*a^3)/(sin
(d*x + c) - 1)^4)/d

Mupad [B] (verification not implemented)

Time = 9.84 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.06 \[ \int \sec ^9(c+d x) (a+a \sin (c+d x))^3 (A+B \sin (c+d x)) \, dx=\frac {a^3\,\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (5\,A-3\,B\right )}{32\,d}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {5\,A\,a^3}{32}-\frac {3\,B\,a^3}{32}\right )-{\sin \left (c+d\,x\right )}^3\,\left (\frac {15\,A\,a^3}{32}-\frac {9\,B\,a^3}{32}\right )+{\sin \left (c+d\,x\right )}^2\,\left (\frac {35\,A\,a^3}{96}-\frac {7\,B\,a^3}{32}\right )-\frac {A\,a^3}{3}+\sin \left (c+d\,x\right )\,\left (\frac {5\,A\,a^3}{32}-\frac {3\,B\,a^3}{32}\right )}{d\,\left ({\sin \left (c+d\,x\right )}^5-3\,{\sin \left (c+d\,x\right )}^4+2\,{\sin \left (c+d\,x\right )}^3+2\,{\sin \left (c+d\,x\right )}^2-3\,\sin \left (c+d\,x\right )+1\right )} \]

[In]

int(((A + B*sin(c + d*x))*(a + a*sin(c + d*x))^3)/cos(c + d*x)^9,x)

[Out]

(a^3*atanh(sin(c + d*x))*(5*A - 3*B))/(32*d) - (sin(c + d*x)^4*((5*A*a^3)/32 - (3*B*a^3)/32) - sin(c + d*x)^3*
((15*A*a^3)/32 - (9*B*a^3)/32) + sin(c + d*x)^2*((35*A*a^3)/96 - (7*B*a^3)/32) - (A*a^3)/3 + sin(c + d*x)*((5*
A*a^3)/32 - (3*B*a^3)/32))/(d*(2*sin(c + d*x)^2 - 3*sin(c + d*x) + 2*sin(c + d*x)^3 - 3*sin(c + d*x)^4 + sin(c
+ d*x)^5 + 1))