\(\int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx\) [412]

Optimal result

Integrand size = 21, antiderivative size = 192 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan (c+d x)}{21 d}+\frac {4 a \left (8 a^2-3 b^2\right ) \tan ^3(c+d x)}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan ^5(c+d x)}{105 d} \]

[Out]

2/63*b*(4*a^2-b^2)*sec(d*x+c)^5/d+1/9*sec(d*x+c)^9*(b+a*sin(d*x+c))*(a+b*sin(d*x+c))^2/d+2/63*sec(d*x+c)^7*(a+
b*sin(d*x+c))*(3*a*b+(4*a^2-b^2)*sin(d*x+c))/d+2/21*a*(8*a^2-3*b^2)*tan(d*x+c)/d+4/63*a*(8*a^2-3*b^2)*tan(d*x+
c)^3/d+2/105*a*(8*a^2-3*b^2)*tan(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2770, 2940, 2748, 3852} \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {2 a \left (8 a^2-3 b^2\right ) \tan ^5(c+d x)}{105 d}+\frac {4 a \left (8 a^2-3 b^2\right ) \tan ^3(c+d x)}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan (c+d x)}{21 d}+\frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (\left (4 a^2-b^2\right ) \sin (c+d x)+3 a b\right )}{63 d}+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{9 d} \]

[In]

Int[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^3,x]

[Out]

(2*b*(4*a^2 - b^2)*Sec[c + d*x]^5)/(63*d) + (Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(9*d)
+ (2*Sec[c + d*x]^7*(a + b*Sin[c + d*x])*(3*a*b + (4*a^2 - b^2)*Sin[c + d*x]))/(63*d) + (2*a*(8*a^2 - 3*b^2)*
Tan[c + d*x])/(21*d) + (4*a*(8*a^2 - 3*b^2)*Tan[c + d*x]^3)/(63*d) + (2*a*(8*a^2 - 3*b^2)*Tan[c + d*x]^5)/(105
*d)

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2770

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-(g*C
os[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Dist[1/(g^2*(p +
1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*S
in[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (Integers
Q[2*m, 2*p] || IntegerQ[m])

Rule 2940

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f
*g*(p + 1))), x] + Dist[1/(g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1)*Simp[a*c*(p
+ 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2,
0] && GtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*m] &&  !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b
*x])

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}-\frac {1}{9} \int \sec ^8(c+d x) (a+b \sin (c+d x)) \left (-8 a^2+2 b^2-6 a b \sin (c+d x)\right ) \, dx \\ & = \frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{63} \int \sec ^6(c+d x) \left (6 a \left (8 a^2-3 b^2\right )+10 b \left (4 a^2-b^2\right ) \sin (c+d x)\right ) \, dx \\ & = \frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {1}{21} \left (2 a \left (8 a^2-3 b^2\right )\right ) \int \sec ^6(c+d x) \, dx \\ & = \frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}-\frac {\left (2 a \left (8 a^2-3 b^2\right )\right ) \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{21 d} \\ & = \frac {2 b \left (4 a^2-b^2\right ) \sec ^5(c+d x)}{63 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{9 d}+\frac {2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan (c+d x)}{21 d}+\frac {4 a \left (8 a^2-3 b^2\right ) \tan ^3(c+d x)}{63 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \tan ^5(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.29 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.56 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {\sec ^9(c+d x) \left (3440640 a^2 b+409600 b^3+3150 b \left (-147 a^2+23 b^2\right ) \cos (c+d x)-737280 b^3 \cos (2 (c+d x))-308700 a^2 b \cos (3 (c+d x))+48300 b^3 \cos (3 (c+d x))-132300 a^2 b \cos (5 (c+d x))+20700 b^3 \cos (5 (c+d x))-33075 a^2 b \cos (7 (c+d x))+5175 b^3 \cos (7 (c+d x))-3675 a^2 b \cos (9 (c+d x))+575 b^3 \cos (9 (c+d x))+2064384 a^3 \sin (c+d x)+3096576 a b^2 \sin (c+d x)+1376256 a^3 \sin (3 (c+d x))-516096 a b^2 \sin (3 (c+d x))+589824 a^3 \sin (5 (c+d x))-221184 a b^2 \sin (5 (c+d x))+147456 a^3 \sin (7 (c+d x))-55296 a b^2 \sin (7 (c+d x))+16384 a^3 \sin (9 (c+d x))-6144 a b^2 \sin (9 (c+d x))\right )}{10321920 d} \]

[In]

Integrate[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^3,x]

[Out]

(Sec[c + d*x]^9*(3440640*a^2*b + 409600*b^3 + 3150*b*(-147*a^2 + 23*b^2)*Cos[c + d*x] - 737280*b^3*Cos[2*(c +
d*x)] - 308700*a^2*b*Cos[3*(c + d*x)] + 48300*b^3*Cos[3*(c + d*x)] - 132300*a^2*b*Cos[5*(c + d*x)] + 20700*b^3
*Cos[5*(c + d*x)] - 33075*a^2*b*Cos[7*(c + d*x)] + 5175*b^3*Cos[7*(c + d*x)] - 3675*a^2*b*Cos[9*(c + d*x)] + 5
75*b^3*Cos[9*(c + d*x)] + 2064384*a^3*Sin[c + d*x] + 3096576*a*b^2*Sin[c + d*x] + 1376256*a^3*Sin[3*(c + d*x)]
- 516096*a*b^2*Sin[3*(c + d*x)] + 589824*a^3*Sin[5*(c + d*x)] - 221184*a*b^2*Sin[5*(c + d*x)] + 147456*a^3*Si
n[7*(c + d*x)] - 55296*a*b^2*Sin[7*(c + d*x)] + 16384*a^3*Sin[9*(c + d*x)] - 6144*a*b^2*Sin[9*(c + d*x)]))/(10
321920*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.43 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.20

method	result	size
risch	\(-\frac {32 \left (945 i a \,b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+180 b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-1008 i a^{3} {\mathrm e}^{8 i \left (d x +c \right )}-567 i a \,b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-1680 a^{2} b \,{\mathrm e}^{9 i \left (d x +c \right )}-200 b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-672 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+252 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+180 b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-288 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+108 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-72 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+27 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-8 i a^{3}+3 i a \,b^{2}\right )}{315 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{9}}\)	\(230\)
derivativedivides	\(\frac {-a^{3} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{3 \cos \left (d x +c \right )^{9}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )}{d}\)	\(265\)
default	\(\frac {-a^{3} \left (-\frac {128}{315}-\frac {\left (\sec ^{8}\left (d x +c \right )\right )}{9}-\frac {8 \left (\sec ^{6}\left (d x +c \right )\right )}{63}-\frac {16 \left (\sec ^{4}\left (d x +c \right )\right )}{105}-\frac {64 \left (\sec ^{2}\left (d x +c \right )\right )}{315}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{3 \cos \left (d x +c \right )^{9}}+3 a \,b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \left (\sin ^{3}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{3}}\right )+b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \left (\sin ^{4}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{4}\left (d x +c \right )}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{63}\right )}{d}\)	\(265\)
parallelrisch	\(-\frac {2 \left (315 a^{3} \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1890 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+105 a^{2} b +3780 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +13230 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +8820 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +8532 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+270 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+1890 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+3150 \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+315 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+4788 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-10 b^{3}+4788 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+90 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+1512 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+1260 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+945 \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +4272 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+630 \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+1512 \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+8532 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-5112 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-5112 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+10658 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-840 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+1260 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-840 \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+1050 \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}\right )}{315 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{9}}\)	\(474\)

[In]

int(sec(d*x+c)^10*(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-32/315*(945*I*a*b^2*exp(10*I*(d*x+c))+180*b^3*exp(11*I*(d*x+c))-1008*I*a^3*exp(8*I*(d*x+c))-567*I*a*b^2*exp(8
*I*(d*x+c))-1680*a^2*b*exp(9*I*(d*x+c))-200*b^3*exp(9*I*(d*x+c))-672*I*a^3*exp(6*I*(d*x+c))+252*I*a*b^2*exp(6*
I*(d*x+c))+180*b^3*exp(7*I*(d*x+c))-288*I*a^3*exp(4*I*(d*x+c))+108*I*a*b^2*exp(4*I*(d*x+c))-72*I*a^3*exp(2*I*(
d*x+c))+27*I*a*b^2*exp(2*I*(d*x+c))-8*I*a^3+3*I*a*b^2)/d/(1+exp(2*I*(d*x+c)))^9

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.76 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {45 \, b^{3} \cos \left (d x + c\right )^{2} - 105 \, a^{2} b - 35 \, b^{3} - {\left (16 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{8} + 8 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 35 \, a^{3} + 105 \, a b^{2} + 5 \, {\left (8 \, a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \]

[In]

integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/315*(45*b^3*cos(d*x + c)^2 - 105*a^2*b - 35*b^3 - (16*(8*a^3 - 3*a*b^2)*cos(d*x + c)^8 + 8*(8*a^3 - 3*a*b^2
)*cos(d*x + c)^6 + 6*(8*a^3 - 3*a*b^2)*cos(d*x + c)^4 + 35*a^3 + 105*a*b^2 + 5*(8*a^3 - 3*a*b^2)*cos(d*x + c)^
2)*sin(d*x + c))/(d*cos(d*x + c)^9)

Sympy [F(-1)]

Timed out. \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**10*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.76 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {{\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a b^{2} - \frac {5 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} b^{3}}{\cos \left (d x + c\right )^{9}} + \frac {105 \, a^{2} b}{\cos \left (d x + c\right )^{9}}}{315 \, d} \]

[In]

integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/315*((35*tan(d*x + c)^9 + 180*tan(d*x + c)^7 + 378*tan(d*x + c)^5 + 420*tan(d*x + c)^3 + 315*tan(d*x + c))*a
^3 + 3*(35*tan(d*x + c)^9 + 135*tan(d*x + c)^7 + 189*tan(d*x + c)^5 + 105*tan(d*x + c)^3)*a*b^2 - 5*(9*cos(d*x
+ c)^2 - 7)*b^3/cos(d*x + c)^9 + 105*a^2*b/cos(d*x + c)^9)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (180) = 360\).

Time = 0.37 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.46 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {2 \, {\left (315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} + 945 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{16} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 1260 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} + 630 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} + 4788 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1512 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 8820 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 1050 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 5112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 8532 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3150 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 10658 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4272 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 13230 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 1890 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 5112 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8532 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1890 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 4788 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1512 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3780 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 270 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 840 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1260 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 90 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 315 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, a^{2} b - 10 \, b^{3}\right )}}{315 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{9} d} \]

[In]

integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

-2/315*(315*a^3*tan(1/2*d*x + 1/2*c)^17 + 945*a^2*b*tan(1/2*d*x + 1/2*c)^16 - 840*a^3*tan(1/2*d*x + 1/2*c)^15
+ 1260*a*b^2*tan(1/2*d*x + 1/2*c)^15 + 630*b^3*tan(1/2*d*x + 1/2*c)^14 + 4788*a^3*tan(1/2*d*x + 1/2*c)^13 + 15
12*a*b^2*tan(1/2*d*x + 1/2*c)^13 + 8820*a^2*b*tan(1/2*d*x + 1/2*c)^12 + 1050*b^3*tan(1/2*d*x + 1/2*c)^12 - 511
2*a^3*tan(1/2*d*x + 1/2*c)^11 + 8532*a*b^2*tan(1/2*d*x + 1/2*c)^11 + 3150*b^3*tan(1/2*d*x + 1/2*c)^10 + 10658*
a^3*tan(1/2*d*x + 1/2*c)^9 + 4272*a*b^2*tan(1/2*d*x + 1/2*c)^9 + 13230*a^2*b*tan(1/2*d*x + 1/2*c)^8 + 1890*b^3
*tan(1/2*d*x + 1/2*c)^8 - 5112*a^3*tan(1/2*d*x + 1/2*c)^7 + 8532*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 1890*b^3*tan(1
/2*d*x + 1/2*c)^6 + 4788*a^3*tan(1/2*d*x + 1/2*c)^5 + 1512*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 3780*a^2*b*tan(1/2*d
*x + 1/2*c)^4 + 270*b^3*tan(1/2*d*x + 1/2*c)^4 - 840*a^3*tan(1/2*d*x + 1/2*c)^3 + 1260*a*b^2*tan(1/2*d*x + 1/2
*c)^3 + 90*b^3*tan(1/2*d*x + 1/2*c)^2 + 315*a^3*tan(1/2*d*x + 1/2*c) + 105*a^2*b - 10*b^3)/((tan(1/2*d*x + 1/2
*c)^2 - 1)^9*d)

Mupad [B] (verification not implemented)

Time = 5.55 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.43 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b^3}{9\,d\,{\cos \left (c+d\,x\right )}^9}-\frac {b^3}{7\,d\,{\cos \left (c+d\,x\right )}^7}+\frac {a^2\,b}{3\,d\,{\cos \left (c+d\,x\right )}^9}+\frac {128\,a^3\,\sin \left (c+d\,x\right )}{315\,d\,\cos \left (c+d\,x\right )}+\frac {64\,a^3\,\sin \left (c+d\,x\right )}{315\,d\,{\cos \left (c+d\,x\right )}^3}+\frac {16\,a^3\,\sin \left (c+d\,x\right )}{105\,d\,{\cos \left (c+d\,x\right )}^5}+\frac {8\,a^3\,\sin \left (c+d\,x\right )}{63\,d\,{\cos \left (c+d\,x\right )}^7}+\frac {a^3\,\sin \left (c+d\,x\right )}{9\,d\,{\cos \left (c+d\,x\right )}^9}-\frac {16\,a\,b^2\,\sin \left (c+d\,x\right )}{105\,d\,\cos \left (c+d\,x\right )}-\frac {8\,a\,b^2\,\sin \left (c+d\,x\right )}{105\,d\,{\cos \left (c+d\,x\right )}^3}-\frac {2\,a\,b^2\,\sin \left (c+d\,x\right )}{35\,d\,{\cos \left (c+d\,x\right )}^5}-\frac {a\,b^2\,\sin \left (c+d\,x\right )}{21\,d\,{\cos \left (c+d\,x\right )}^7}+\frac {a\,b^2\,\sin \left (c+d\,x\right )}{3\,d\,{\cos \left (c+d\,x\right )}^9} \]

[In]

int((a + b*sin(c + d*x))^3/cos(c + d*x)^10,x)

[Out]

b^3/(9*d*cos(c + d*x)^9) - b^3/(7*d*cos(c + d*x)^7) + (a^2*b)/(3*d*cos(c + d*x)^9) + (128*a^3*sin(c + d*x))/(3
15*d*cos(c + d*x)) + (64*a^3*sin(c + d*x))/(315*d*cos(c + d*x)^3) + (16*a^3*sin(c + d*x))/(105*d*cos(c + d*x)^
5) + (8*a^3*sin(c + d*x))/(63*d*cos(c + d*x)^7) + (a^3*sin(c + d*x))/(9*d*cos(c + d*x)^9) - (16*a*b^2*sin(c +
d*x))/(105*d*cos(c + d*x)) - (8*a*b^2*sin(c + d*x))/(105*d*cos(c + d*x)^3) - (2*a*b^2*sin(c + d*x))/(35*d*cos(
c + d*x)^5) - (a*b^2*sin(c + d*x))/(21*d*cos(c + d*x)^7) + (a*b^2*sin(c + d*x))/(3*d*cos(c + d*x)^9)