3.968 \(\int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=119 \[ \frac{a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac{a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac{a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac{a (8 A-B) \tan (c+d x)}{9 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d} \]

[Out]

((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x]))/(9*d) + (a*(8*A - B)*Tan[c + d*x])/(9*d) + (a*(8*A - B)*Tan[c +
d*x]^3)/(9*d) + (a*(8*A - B)*Tan[c + d*x]^5)/(15*d) + (a*(8*A - B)*Tan[c + d*x]^7)/(63*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0865134, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2855, 3767} \[ \frac{a (8 A-B) \tan ^7(c+d x)}{63 d}+\frac{a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac{a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac{a (8 A-B) \tan (c+d x)}{9 d}+\frac{(A+B) \sec ^9(c+d x) (a \sin (c+d x)+a)}{9 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A + B)*Sec[c + d*x]^9*(a + a*Sin[c + d*x]))/(9*d) + (a*(8*A - B)*Tan[c + d*x])/(9*d) + (a*(8*A - B)*Tan[c +
d*x]^3)/(9*d) + (a*(8*A - B)*Tan[c + d*x]^5)/(15*d) + (a*(8*A - B)*Tan[c + d*x]^7)/(63*d)

Rule 2855

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

Rule 3767

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*} \int \sec ^{10}(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac{(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))}{9 d}+\frac{1}{9} (a (8 A-B)) \int \sec ^8(c+d x) \, dx\\ &=\frac{(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))}{9 d}-\frac{(a (8 A-B)) \operatorname{Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{9 d}\\ &=\frac{(A+B) \sec ^9(c+d x) (a+a \sin (c+d x))}{9 d}+\frac{a (8 A-B) \tan (c+d x)}{9 d}+\frac{a (8 A-B) \tan ^3(c+d x)}{9 d}+\frac{a (8 A-B) \tan ^5(c+d x)}{15 d}+\frac{a (8 A-B) \tan ^7(c+d x)}{63 d}\\ \end{align*}

Mathematica [B]  time = 4.27944, size = 407, normalized size = 3.42 \[ \frac{a \sec (c) (-85750 (A+B) \cos (c+d x)+17150 A \sin (2 (c+d x))+17150 A \sin (4 (c+d x))+7350 A \sin (6 (c+d x))+1225 A \sin (8 (c+d x))+688128 A \sin (2 c+3 d x)+229376 A \sin (4 c+5 d x)+32768 A \sin (6 c+7 d x)-51450 A \cos (3 (c+d x))-17150 A \cos (5 (c+d x))-2450 A \cos (7 (c+d x))+229376 A \cos (c+2 d x)+229376 A \cos (3 c+4 d x)+98304 A \cos (5 c+6 d x)+16384 A \cos (7 c+8 d x)+1146880 A \sin (d x)+17150 B \sin (2 (c+d x))+17150 B \sin (4 (c+d x))+7350 B \sin (6 (c+d x))+1225 B \sin (8 (c+d x))-86016 B \sin (2 c+3 d x)-28672 B \sin (4 c+5 d x)-4096 B \sin (6 c+7 d x)-51450 B \cos (3 (c+d x))-17150 B \cos (5 (c+d x))-2450 B \cos (7 (c+d x))-28672 B \cos (c+2 d x)-28672 B \cos (3 c+4 d x)-12288 B \cos (5 c+6 d x)-2048 B \cos (7 c+8 d x)+645120 B \cos (c)-143360 B \sin (d x))}{5160960 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^9 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sec[c]*(645120*B*Cos[c] - 85750*(A + B)*Cos[c + d*x] - 51450*A*Cos[3*(c + d*x)] - 51450*B*Cos[3*(c + d*x)]
- 17150*A*Cos[5*(c + d*x)] - 17150*B*Cos[5*(c + d*x)] - 2450*A*Cos[7*(c + d*x)] - 2450*B*Cos[7*(c + d*x)] + 22
9376*A*Cos[c + 2*d*x] - 28672*B*Cos[c + 2*d*x] + 229376*A*Cos[3*c + 4*d*x] - 28672*B*Cos[3*c + 4*d*x] + 98304*
A*Cos[5*c + 6*d*x] - 12288*B*Cos[5*c + 6*d*x] + 16384*A*Cos[7*c + 8*d*x] - 2048*B*Cos[7*c + 8*d*x] + 1146880*A
*Sin[d*x] - 143360*B*Sin[d*x] + 17150*A*Sin[2*(c + d*x)] + 17150*B*Sin[2*(c + d*x)] + 17150*A*Sin[4*(c + d*x)]
+ 17150*B*Sin[4*(c + d*x)] + 7350*A*Sin[6*(c + d*x)] + 7350*B*Sin[6*(c + d*x)] + 1225*A*Sin[8*(c + d*x)] + 12
25*B*Sin[8*(c + d*x)] + 688128*A*Sin[2*c + 3*d*x] - 86016*B*Sin[2*c + 3*d*x] + 229376*A*Sin[4*c + 5*d*x] - 286
72*B*Sin[4*c + 5*d*x] + 32768*A*Sin[6*c + 7*d*x] - 4096*B*Sin[6*c + 7*d*x]))/(5160960*d*(Cos[(c + d*x)/2] - Si
n[(c + d*x)/2])^9*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)

________________________________________________________________________________________

Maple [A]  time = 0.119, size = 158, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ({\frac{aA}{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}+aB \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}}+{\frac{2\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{21\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{16\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{315\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) -aA \left ( -{\frac{128}{315}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{8}}{9}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{63}}-{\frac{16\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{315}} \right ) \tan \left ( dx+c \right ) +{\frac{aB}{9\, \left ( \cos \left ( dx+c \right ) \right ) ^{9}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

1/d*(1/9*a*A/cos(d*x+c)^9+a*B*(1/9*sin(d*x+c)^3/cos(d*x+c)^9+2/21*sin(d*x+c)^3/cos(d*x+c)^7+8/105*sin(d*x+c)^3
/cos(d*x+c)^5+16/315*sin(d*x+c)^3/cos(d*x+c)^3)-a*A*(-128/315-1/9*sec(d*x+c)^8-8/63*sec(d*x+c)^6-16/105*sec(d*
x+c)^4-64/315*sec(d*x+c)^2)*tan(d*x+c)+1/9*a*B/cos(d*x+c)^9)

________________________________________________________________________________________

Maxima [A]  time = 1.02318, size = 170, normalized size = 1.43 \begin{align*} \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 a +{\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 )} B a + \frac{35 \, A a}{\cos \left (d x + c\right )^{9}} + \frac{35 \, B a}{\cos \left (d x + c\right )^{9}}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),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
*a + (35*tan(d*x + c)^9 + 135*tan(d*x + c)^7 + 189*tan(d*x + c)^5 + 105*tan(d*x + c)^3)*B*a + 35*A*a/cos(d*x +
c)^9 + 35*B*a/cos(d*x + c)^9)/d

________________________________________________________________________________________

Fricas [A]  time = 1.91598, size = 436, normalized size = 3.66 \begin{align*} -\frac{16 \,{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{8} - 8 \,{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 2 \,{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} -{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 5 \,{\left (A - 8 \, B\right )} a +{\left (16 \,{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{6} + 8 \,{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 6 \,{\left (8 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 5 \,{\left (8 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{315 \,{\left (d \cos \left (d x + c\right )^{7} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(16*(8*A - B)*a*cos(d*x + c)^8 - 8*(8*A - B)*a*cos(d*x + c)^6 - 2*(8*A - B)*a*cos(d*x + c)^4 - (8*A - B
)*a*cos(d*x + c)^2 - 5*(A - 8*B)*a + (16*(8*A - B)*a*cos(d*x + c)^6 + 8*(8*A - B)*a*cos(d*x + c)^4 + 6*(8*A -
B)*a*cos(d*x + c)^2 + 5*(8*A - B)*a)*sin(d*x + c))/(d*cos(d*x + c)^7*sin(d*x + c) - d*cos(d*x + c)^7)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**10*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.36597, size = 628, normalized size = 5.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/40320*(3*(9765*A*a*tan(1/2*d*x + 1/2*c)^6 - 3675*B*a*tan(1/2*d*x + 1/2*c)^6 + 48720*A*a*tan(1/2*d*x + 1/2*c
)^5 - 15960*B*a*tan(1/2*d*x + 1/2*c)^5 + 109865*A*a*tan(1/2*d*x + 1/2*c)^4 - 33775*B*a*tan(1/2*d*x + 1/2*c)^4
+ 136640*A*a*tan(1/2*d*x + 1/2*c)^3 - 39760*B*a*tan(1/2*d*x + 1/2*c)^3 + 99183*A*a*tan(1/2*d*x + 1/2*c)^2 - 28
161*B*a*tan(1/2*d*x + 1/2*c)^2 + 39536*A*a*tan(1/2*d*x + 1/2*c) - 11032*B*a*tan(1/2*d*x + 1/2*c) + 7043*A*a -
2101*B*a)/(tan(1/2*d*x + 1/2*c) + 1)^7 + (51345*A*a*tan(1/2*d*x + 1/2*c)^8 + 11025*B*a*tan(1/2*d*x + 1/2*c)^8
- 322560*A*a*tan(1/2*d*x + 1/2*c)^7 - 47880*B*a*tan(1/2*d*x + 1/2*c)^7 + 976500*A*a*tan(1/2*d*x + 1/2*c)^6 + 1
17180*B*a*tan(1/2*d*x + 1/2*c)^6 - 1753920*A*a*tan(1/2*d*x + 1/2*c)^5 - 168840*B*a*tan(1/2*d*x + 1/2*c)^5 + 20
37294*A*a*tan(1/2*d*x + 1/2*c)^4 + 165942*B*a*tan(1/2*d*x + 1/2*c)^4 - 1550976*A*a*tan(1/2*d*x + 1/2*c)^3 - 10
6008*B*a*tan(1/2*d*x + 1/2*c)^3 + 760644*A*a*tan(1/2*d*x + 1/2*c)^2 + 47772*B*a*tan(1/2*d*x + 1/2*c)^2 - 21945
6*A*a*tan(1/2*d*x + 1/2*c) - 12888*B*a*tan(1/2*d*x + 1/2*c) + 30089*A*a + 2657*B*a)/(tan(1/2*d*x + 1/2*c) - 1)
^9)/d