3.967 \(\int \sec ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx\)

Optimal. Leaf size=96 \[ \frac{a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac{2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac{a (6 A-B) \tan (c+d x)}{7 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d} \]

[Out]

((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x]))/(7*d) + (a*(6*A - B)*Tan[c + d*x])/(7*d) + (2*a*(6*A - B)*Tan[c
+ d*x]^3)/(21*d) + (a*(6*A - B)*Tan[c + d*x]^5)/(35*d)

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Rubi [A]  time = 0.0797963, antiderivative size = 96, 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} \[ \frac{a (6 A-B) \tan ^5(c+d x)}{35 d}+\frac{2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac{a (6 A-B) \tan (c+d x)}{7 d}+\frac{(A+B) \sec ^7(c+d x) (a \sin (c+d x)+a)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

((A + B)*Sec[c + d*x]^7*(a + a*Sin[c + d*x]))/(7*d) + (a*(6*A - B)*Tan[c + d*x])/(7*d) + (2*a*(6*A - B)*Tan[c
+ d*x]^3)/(21*d) + (a*(6*A - B)*Tan[c + d*x]^5)/(35*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 ^8(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac{(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac{1}{7} (a (6 A-B)) \int \sec ^6(c+d x) \, dx\\ &=\frac{(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}-\frac{(a (6 A-B)) \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{7 d}\\ &=\frac{(A+B) \sec ^7(c+d x) (a+a \sin (c+d x))}{7 d}+\frac{a (6 A-B) \tan (c+d x)}{7 d}+\frac{2 a (6 A-B) \tan ^3(c+d x)}{21 d}+\frac{a (6 A-B) \tan ^5(c+d x)}{35 d}\\ \end{align*}

Mathematica [B]  time = 2.02256, size = 315, normalized size = 3.28 \[ \frac{a \sec (c) (-1500 (A+B) \cos (c+d x)+375 A \sin (2 (c+d x))+300 A \sin (4 (c+d x))+75 A \sin (6 (c+d x))+7680 A \sin (2 c+3 d x)+1536 A \sin (4 c+5 d x)-750 A \cos (3 (c+d x))-150 A \cos (5 (c+d x))+3840 A \cos (c+2 d x)+3072 A \cos (3 c+4 d x)+768 A \cos (5 c+6 d x)+15360 A \sin (d x)+375 B \sin (2 (c+d x))+300 B \sin (4 (c+d x))+75 B \sin (6 (c+d x))-1280 B \sin (2 c+3 d x)-256 B \sin (4 c+5 d x)-750 B \cos (3 (c+d x))-150 B \cos (5 (c+d x))-640 B \cos (c+2 d x)-512 B \cos (3 c+4 d x)-128 B \cos (5 c+6 d x)+8960 B \cos (c)-2560 B \sin (d x))}{53760 d \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^7 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])*(A + B*Sin[c + d*x]),x]

[Out]

(a*Sec[c]*(8960*B*Cos[c] - 1500*(A + B)*Cos[c + d*x] - 750*A*Cos[3*(c + d*x)] - 750*B*Cos[3*(c + d*x)] - 150*A
*Cos[5*(c + d*x)] - 150*B*Cos[5*(c + d*x)] + 3840*A*Cos[c + 2*d*x] - 640*B*Cos[c + 2*d*x] + 3072*A*Cos[3*c + 4
*d*x] - 512*B*Cos[3*c + 4*d*x] + 768*A*Cos[5*c + 6*d*x] - 128*B*Cos[5*c + 6*d*x] + 15360*A*Sin[d*x] - 2560*B*S
in[d*x] + 375*A*Sin[2*(c + d*x)] + 375*B*Sin[2*(c + d*x)] + 300*A*Sin[4*(c + d*x)] + 300*B*Sin[4*(c + d*x)] +
75*A*Sin[6*(c + d*x)] + 75*B*Sin[6*(c + d*x)] + 7680*A*Sin[2*c + 3*d*x] - 1280*B*Sin[2*c + 3*d*x] + 1536*A*Sin
[4*c + 5*d*x] - 256*B*Sin[4*c + 5*d*x]))/(53760*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*(Cos[(c + d*x)/2] +
Sin[(c + d*x)/2])^5)

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Maple [A]  time = 0.105, size = 130, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({\frac{aA}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+aB \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) -aA \left ( -{\frac{16}{35}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{35}}-{\frac{8\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \tan \left ( dx+c \right ) +{\frac{aB}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

1/d*(1/7*a*A/cos(d*x+c)^7+a*B*(1/7*sin(d*x+c)^3/cos(d*x+c)^7+4/35*sin(d*x+c)^3/cos(d*x+c)^5+8/105*sin(d*x+c)^3
/cos(d*x+c)^3)-a*A*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan(d*x+c)+1/7*a*B/cos(d*x+c)
^7)

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Maxima [A]  time = 1.01267, size = 144, normalized size = 1.5 \begin{align*} \frac{3 \,{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a +{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac{15 \, A a}{\cos \left (d x + c\right )^{7}} + \frac{15 \, B a}{\cos \left (d x + c\right )^{7}}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/105*(3*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*A*a + (15*tan(d*x + c)^7
+ 42*tan(d*x + c)^5 + 35*tan(d*x + c)^3)*B*a + 15*A*a/cos(d*x + c)^7 + 15*B*a/cos(d*x + c)^7)/d

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Fricas [A]  time = 1.78687, size = 350, normalized size = 3.65 \begin{align*} -\frac{8 \,{\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{6} - 4 \,{\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{4} -{\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{2} - 3 \,{\left (A - 6 \, B\right )} a +{\left (8 \,{\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{4} + 4 \,{\left (6 \, A - B\right )} a \cos \left (d x + c\right )^{2} + 3 \,{\left (6 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/105*(8*(6*A - B)*a*cos(d*x + c)^6 - 4*(6*A - B)*a*cos(d*x + c)^4 - (6*A - B)*a*cos(d*x + c)^2 - 3*(A - 6*B)
*a + (8*(6*A - B)*a*cos(d*x + c)^4 + 4*(6*A - B)*a*cos(d*x + c)^2 + 3*(6*A - B)*a)*sin(d*x + c))/(d*cos(d*x +
c)^5*sin(d*x + c) - d*cos(d*x + c)^5)

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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)**8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x)

[Out]

Timed out

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Giac [B]  time = 1.26275, size = 466, normalized size = 4.85 \begin{align*} -\frac{\frac{7 \,{\left (165 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 75 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 540 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 210 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 750 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 280 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 480 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 170 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 129 \, A a - 49 \, B a\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}} + \frac{2205 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 525 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 10080 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 1470 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 21945 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2555 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 26460 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2240 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 18963 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1407 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7476 \, A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 434 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1383 \, A a + 137 \, B a}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{7}}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^8*(a+a*sin(d*x+c))*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/1680*(7*(165*A*a*tan(1/2*d*x + 1/2*c)^4 - 75*B*a*tan(1/2*d*x + 1/2*c)^4 + 540*A*a*tan(1/2*d*x + 1/2*c)^3 -
210*B*a*tan(1/2*d*x + 1/2*c)^3 + 750*A*a*tan(1/2*d*x + 1/2*c)^2 - 280*B*a*tan(1/2*d*x + 1/2*c)^2 + 480*A*a*tan
(1/2*d*x + 1/2*c) - 170*B*a*tan(1/2*d*x + 1/2*c) + 129*A*a - 49*B*a)/(tan(1/2*d*x + 1/2*c) + 1)^5 + (2205*A*a*
tan(1/2*d*x + 1/2*c)^6 + 525*B*a*tan(1/2*d*x + 1/2*c)^6 - 10080*A*a*tan(1/2*d*x + 1/2*c)^5 - 1470*B*a*tan(1/2*
d*x + 1/2*c)^5 + 21945*A*a*tan(1/2*d*x + 1/2*c)^4 + 2555*B*a*tan(1/2*d*x + 1/2*c)^4 - 26460*A*a*tan(1/2*d*x +
1/2*c)^3 - 2240*B*a*tan(1/2*d*x + 1/2*c)^3 + 18963*A*a*tan(1/2*d*x + 1/2*c)^2 + 1407*B*a*tan(1/2*d*x + 1/2*c)^
2 - 7476*A*a*tan(1/2*d*x + 1/2*c) - 434*B*a*tan(1/2*d*x + 1/2*c) + 1383*A*a + 137*B*a)/(tan(1/2*d*x + 1/2*c) -
1)^7)/d