3.216 \(\int \frac{\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

Optimal. Leaf size=473 \[ \frac{b \left (-170 a^2 b^2+61 a^4+105 b^4\right ) \sin (c+d x)}{15 a^7 d}+\frac{\left (-20 a^2 b^2+5 a^4+14 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{10 a^3 b^2 d (a \cos (c+d x)+b)}-\frac{\left (-61 a^2 b^2+16 a^4+42 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 d}+\frac{\left (-52 a^2 b^2+15 a^4+35 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}-\frac{\left (-86 a^2 b^2+27 a^4+56 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 a^6 d}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2-7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^8 d}+\frac{x \left (-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right )}{16 a^8}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}+\frac{a \sin (c+d x) \cos ^4(c+d x)}{6 b^2 d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^6(c+d x)}{6 a d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 b d (a \cos (c+d x)+b)} \]

[Out]

((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*x)/(16*a^8) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*(2*a^2 - 7*b^2)*A
rcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Sin[c + d*x]
)/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 +
35*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cos[c + d*x]^3*Sin[c + d*
x])/(24*a^4*b^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*b*d*(b + a*Cos[c + d*x])) + (a*Cos[c + d*x]^4*Sin[c + d*
x])/(6*b^2*d*(b + a*Cos[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(10*a^3*b^2*d
*(b + a*Cos[c + d*x])) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(30*a^2*d*(b + a*Cos[c + d*x])) - (Cos[c + d*x]^6*S
in[c + d*x])/(6*a*d*(b + a*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.71296, antiderivative size = 473, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {3872, 2896, 3047, 3049, 3023, 2735, 2659, 208} \[ \frac{b \left (-170 a^2 b^2+61 a^4+105 b^4\right ) \sin (c+d x)}{15 a^7 d}+\frac{\left (-20 a^2 b^2+5 a^4+14 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{10 a^3 b^2 d (a \cos (c+d x)+b)}-\frac{\left (-61 a^2 b^2+16 a^4+42 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 d}+\frac{\left (-52 a^2 b^2+15 a^4+35 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}-\frac{\left (-86 a^2 b^2+27 a^4+56 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 a^6 d}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2-7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^8 d}+\frac{x \left (-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right )}{16 a^8}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}+\frac{a \sin (c+d x) \cos ^4(c+d x)}{6 b^2 d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^6(c+d x)}{6 a d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 b d (a \cos (c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]

[Out]

((5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*x)/(16*a^8) - (2*(a - b)^(3/2)*b*(a + b)^(3/2)*(2*a^2 - 7*b^2)*A
rcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^8*d) + (b*(61*a^4 - 170*a^2*b^2 + 105*b^4)*Sin[c + d*x]
)/(15*a^7*d) - ((27*a^4 - 86*a^2*b^2 + 56*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^6*d) + ((15*a^4 - 52*a^2*b^2 +
35*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a^5*b*d) - ((16*a^4 - 61*a^2*b^2 + 42*b^4)*Cos[c + d*x]^3*Sin[c + d*
x])/(24*a^4*b^2*d) - (Cos[c + d*x]^3*Sin[c + d*x])/(3*b*d*(b + a*Cos[c + d*x])) + (a*Cos[c + d*x]^4*Sin[c + d*
x])/(6*b^2*d*(b + a*Cos[c + d*x])) + ((5*a^4 - 20*a^2*b^2 + 14*b^4)*Cos[c + d*x]^4*Sin[c + d*x])/(10*a^3*b^2*d
*(b + a*Cos[c + d*x])) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(30*a^2*d*(b + a*Cos[c + d*x])) - (Cos[c + d*x]^6*S
in[c + d*x])/(6*a*d*(b + a*Cos[c + d*x]))

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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]

Rule 2896

Int[cos[(e_.) + (f_.)*(x_)]^6*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Simp[(Cos[e + f*x]*(d*Sin[e + f*x])^(n + 1)*(a + b*Sin[e + f*x])^(m + 1))/(a*d*f*(n + 1)), x] +
(Dist[1/(a^2*b^2*d^2*(n + 1)*(n + 2)*(m + n + 5)*(m + n + 6)), Int[(d*Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*
x])^m*Simp[a^4*(n + 1)*(n + 2)*(n + 3)*(n + 5) - a^2*b^2*(n + 2)*(2*n + 1)*(m + n + 5)*(m + n + 6) + b^4*(m +
n + 2)*(m + n + 3)*(m + n + 5)*(m + n + 6) + a*b*m*(a^2*(n + 1)*(n + 2) - b^2*(m + n + 5)*(m + n + 6))*Sin[e +
f*x] - (a^4*(n + 1)*(n + 2)*(4 + n)*(n + 5) + b^4*(m + n + 2)*(m + n + 4)*(m + n + 5)*(m + n + 6) - a^2*b^2*(
n + 1)*(n + 2)*(m + n + 5)*(2*n + 2*m + 13))*Sin[e + f*x]^2, x], x], x] - Simp[(b*(m + n + 2)*Cos[e + f*x]*(d*
Sin[e + f*x])^(n + 2)*(a + b*Sin[e + f*x])^(m + 1))/(a^2*d^2*f*(n + 1)*(n + 2)), x] - Simp[(a*(n + 5)*Cos[e +
f*x]*(d*Sin[e + f*x])^(n + 3)*(a + b*Sin[e + f*x])^(m + 1))/(b^2*d^3*f*(m + n + 5)*(m + n + 6)), x] + Simp[(Co
s[e + f*x]*(d*Sin[e + f*x])^(n + 4)*(a + b*Sin[e + f*x])^(m + 1))/(b*d^4*f*(m + n + 6)), x]) /; FreeQ[{a, b, d
, e, f, m, n}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*m, 2*n] && NeQ[n, -1] && NeQ[n, -2] && NeQ[m + n + 5, 0]
&& NeQ[m + n + 6, 0] &&  !IGtQ[m, 0]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C - B*c*d + A*d^2)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(
c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c
*C - B*d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*
d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)
))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2,
0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

Rule 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)
*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e +
f*x])^m*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x],
x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2
, 0] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) \sin ^6(c+d x)}{(-b-a \cos (c+d x))^2} \, dx\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac{\int \frac{\cos ^4(c+d x) \left (60 \left (3 a^4-10 a^2 b^2+7 b^4\right )+12 a b \left (5 a^2-2 b^2\right ) \cos (c+d x)-12 \left (20 a^4-65 a^2 b^2+42 b^4\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^2} \, dx}{360 a^2 b^2}\\ &=-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}-\frac{\int \frac{\cos ^3(c+d x) \left (144 \left (5 a^6-25 a^4 b^2+34 a^2 b^4-14 b^6\right )+12 a b \left (10 a^4-17 a^2 b^2+7 b^4\right ) \cos (c+d x)-60 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{360 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac{\int \frac{\cos ^2(c+d x) \left (180 b \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right )+36 a b^2 \left (15 a^4-29 a^2 b^2+14 b^4\right ) \cos (c+d x)-288 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{1440 a^4 b^2 \left (a^2-b^2\right )}\\ &=\frac{\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}-\frac{\int \frac{\cos (c+d x) \left (576 b^2 \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right )+36 a b^3 \left (83 a^4-153 a^2 b^2+70 b^4\right ) \cos (c+d x)-540 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{4320 a^5 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac{\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac{\int \frac{540 b^3 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right )-36 a b^2 \left (75 a^6-449 a^4 b^2+654 a^2 b^4-280 b^6\right ) \cos (c+d x)-576 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{8640 a^6 b^2 \left (a^2-b^2\right )}\\ &=\frac{b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac{\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac{\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}-\frac{\int \frac{-540 a b^3 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right )+540 b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{8640 a^7 b^2 \left (a^2-b^2\right )}\\ &=\frac{\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}+\frac{b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac{\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac{\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac{\left (b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac{1}{-b-a \cos (c+d x)} \, dx}{a^8}\\ &=\frac{\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}+\frac{b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac{\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac{\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac{\left (2 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\right ) \operatorname{Subst}\left (\int \frac{1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^8 d}\\ &=\frac{\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}-\frac{2 (a-b)^{3/2} b (a+b)^{3/2} \left (2 a^2-7 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^8 d}+\frac{b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac{\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac{\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac{\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac{\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac{a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac{\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac{\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.90907, size = 402, normalized size = 0.85 \[ \frac{\frac{790 a^5 b^2 \sin (3 (c+d x))-42 a^5 b^2 \sin (5 (c+d x))-5440 a^4 b^3 \sin (2 (c+d x))+140 a^4 b^3 \sin (4 (c+d x))-560 a^3 b^4 \sin (3 (c+d x))+3360 a^2 b^5 \sin (2 (c+d x))-15 a \left (-576 a^4 b^2+1488 a^2 b^4+15 a^6-896 b^6\right ) \sin (c+d x)+120 a \left (-90 a^4 b^2+200 a^2 b^4+5 a^6-112 b^6\right ) (c+d x) \cos (c+d x)-10800 a^4 b^3 c+24000 a^2 b^5 c-10800 a^4 b^3 d x+24000 a^2 b^5 d x+1910 a^6 b \sin (2 (c+d x))-166 a^6 b \sin (4 (c+d x))+14 a^6 b \sin (6 (c+d x))+600 a^6 b c+600 a^6 b d x-180 a^7 \sin (3 (c+d x))+40 a^7 \sin (5 (c+d x))-5 a^7 \sin (7 (c+d x))-13440 b^7 c-13440 b^7 d x}{a \cos (c+d x)+b}+3840 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{1920 a^8 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]

[Out]

(3840*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + (600*a^6*b*c
- 10800*a^4*b^3*c + 24000*a^2*b^5*c - 13440*b^7*c + 600*a^6*b*d*x - 10800*a^4*b^3*d*x + 24000*a^2*b^5*d*x - 13
440*b^7*d*x + 120*a*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*(c + d*x)*Cos[c + d*x] - 15*a*(15*a^6 - 576*a
^4*b^2 + 1488*a^2*b^4 - 896*b^6)*Sin[c + d*x] + 1910*a^6*b*Sin[2*(c + d*x)] - 5440*a^4*b^3*Sin[2*(c + d*x)] +
3360*a^2*b^5*Sin[2*(c + d*x)] - 180*a^7*Sin[3*(c + d*x)] + 790*a^5*b^2*Sin[3*(c + d*x)] - 560*a^3*b^4*Sin[3*(c
+ d*x)] - 166*a^6*b*Sin[4*(c + d*x)] + 140*a^4*b^3*Sin[4*(c + d*x)] + 40*a^7*Sin[5*(c + d*x)] - 42*a^5*b^2*Si
n[5*(c + d*x)] + 14*a^6*b*Sin[6*(c + d*x)] - 5*a^7*Sin[7*(c + d*x)])/(b + a*Cos[c + d*x]))/(1920*a^8*d)

________________________________________________________________________________________

Maple [B]  time = 0.091, size = 1735, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^6/(a+b*sec(d*x+c))^2,x)

[Out]

-32/d*b^5/a^6/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+4/d/a^3/(1+tan(1/2*d*x
+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11*b+344/5/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*b+60/d/a^7/(1
+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*b^5-16/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*b^3+4
/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*b-272/3/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*
c)^9*b^3-33/2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*b^2-192/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^6*t
an(1/2*d*x+1/2*c)^7*b^3+10/d/a^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*b^4+14/d*b^7/a^8/((a+b)*(a-b)
)^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+21/4/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*
x+1/2*c)*b^2+120/d/a^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^7*b^5-21/4/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)
^6*tan(1/2*d*x+1/2*c)^11*b^2+33/2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*b^2-10/d/a^6/(1+tan(1/
2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*b^4+344/5/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*b+60/d/
a^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*b^5-15/d/a^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)
^3*b^4-4/d*b/a^2/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))+76/3/d/a^3/(1+tan(1
/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*b+12/d/a^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*b^5-272/3/d/a
^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*b^3+22/d*b^3/a^4/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*
d*x+1/2*c)/((a+b)*(a-b))^(1/2))-5/d/a^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)*b^4+87/4/d/a^4/(1+tan(1/
2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^3*b^2-16/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11*b^3+120/d
/a^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*b^5-87/4/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2
*c)^9*b^2+12/d/a^7/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11*b^5+76/3/d/a^3/(1+tan(1/2*d*x+1/2*c)^2)^6*
tan(1/2*d*x+1/2*c)^9*b-2/d*b^2/a^3*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)+4/d*
b^4/a^5*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)-2/d*b^6/a^7*tan(1/2*d*x+1/2*c)/
(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)+15/d/a^6/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9*b
^4-192/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5*b^3+33/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2
*d*x+1/2*c)^7-33/4/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^5-85/24/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^
6*tan(1/2*d*x+1/2*c)^3-5/8/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)-14/d/a^8*arctan(tan(1/2*d*x+1/2
*c))*b^6-45/4/d/a^4*arctan(tan(1/2*d*x+1/2*c))*b^2+25/d/a^6*arctan(tan(1/2*d*x+1/2*c))*b^4+5/8/d/a^2/(1+tan(1/
2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11+85/24/d/a^2/(1+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^9+5/d/a^6/(1
+tan(1/2*d*x+1/2*c)^2)^6*tan(1/2*d*x+1/2*c)^11*b^4+5/8/d/a^2*arctan(tan(1/2*d*x+1/2*c))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.53648, size = 1871, normalized size = 3.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

[1/240*(15*(5*a^7 - 90*a^5*b^2 + 200*a^3*b^4 - 112*a*b^6)*d*x*cos(d*x + c) + 15*(5*a^6*b - 90*a^4*b^3 + 200*a^
2*b^5 - 112*b^7)*d*x + 120*(2*a^4*b^2 - 9*a^2*b^4 + 7*b^6 + (2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c))*sqrt
(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 - 2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*si
n(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) + b^2)) - (40*a^7*cos(d*x + c)^6 - 56*a^6*b
*cos(d*x + c)^5 - 976*a^5*b^2 + 2720*a^3*b^4 - 1680*a*b^6 - 2*(65*a^7 - 42*a^5*b^2)*cos(d*x + c)^4 + 2*(111*a^
6*b - 70*a^4*b^3)*cos(d*x + c)^3 + (165*a^7 - 458*a^5*b^2 + 280*a^3*b^4)*cos(d*x + c)^2 - (571*a^6*b - 1430*a^
4*b^3 + 840*a^2*b^5)*cos(d*x + c))*sin(d*x + c))/(a^9*d*cos(d*x + c) + a^8*b*d), 1/240*(15*(5*a^7 - 90*a^5*b^2
+ 200*a^3*b^4 - 112*a*b^6)*d*x*cos(d*x + c) + 15*(5*a^6*b - 90*a^4*b^3 + 200*a^2*b^5 - 112*b^7)*d*x - 240*(2*
a^4*b^2 - 9*a^2*b^4 + 7*b^6 + (2*a^5*b - 9*a^3*b^3 + 7*a*b^5)*cos(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2
+ b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) - (40*a^7*cos(d*x + c)^6 - 56*a^6*b*cos(d*x + c)^5 -
976*a^5*b^2 + 2720*a^3*b^4 - 1680*a*b^6 - 2*(65*a^7 - 42*a^5*b^2)*cos(d*x + c)^4 + 2*(111*a^6*b - 70*a^4*b^3)*
cos(d*x + c)^3 + (165*a^7 - 458*a^5*b^2 + 280*a^3*b^4)*cos(d*x + c)^2 - (571*a^6*b - 1430*a^4*b^3 + 840*a^2*b^
5)*cos(d*x + c))*sin(d*x + c))/(a^9*d*cos(d*x + c) + a^8*b*d)]

________________________________________________________________________________________

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(sin(d*x+c)**6/(a+b*sec(d*x+c))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.38277, size = 1175, normalized size = 2.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*sec(d*x+c))^2,x, algorithm="giac")

[Out]

1/240*(15*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*(d*x + c)/a^8 - 480*(2*a^6*b - 11*a^4*b^3 + 16*a^2*b^5
- 7*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x +
1/2*c))/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^8) - 480*(a^4*b^2*tan(1/2*d*x + 1/2*c) - 2*a^2*b^4*tan(1/2*d*x
+ 1/2*c) + b^6*tan(1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)*a^7) + 2*
(75*a^5*tan(1/2*d*x + 1/2*c)^11 + 480*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 630*a^3*b^2*tan(1/2*d*x + 1/2*c)^11 - 19
20*a^2*b^3*tan(1/2*d*x + 1/2*c)^11 + 600*a*b^4*tan(1/2*d*x + 1/2*c)^11 + 1440*b^5*tan(1/2*d*x + 1/2*c)^11 + 42
5*a^5*tan(1/2*d*x + 1/2*c)^9 + 3040*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 2610*a^3*b^2*tan(1/2*d*x + 1/2*c)^9 - 10880
*a^2*b^3*tan(1/2*d*x + 1/2*c)^9 + 1800*a*b^4*tan(1/2*d*x + 1/2*c)^9 + 7200*b^5*tan(1/2*d*x + 1/2*c)^9 + 990*a^
5*tan(1/2*d*x + 1/2*c)^7 + 8256*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 1980*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 23040*a^2
*b^3*tan(1/2*d*x + 1/2*c)^7 + 1200*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 14400*b^5*tan(1/2*d*x + 1/2*c)^7 - 990*a^5*t
an(1/2*d*x + 1/2*c)^5 + 8256*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 1980*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 23040*a^2*b^
3*tan(1/2*d*x + 1/2*c)^5 - 1200*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 14400*b^5*tan(1/2*d*x + 1/2*c)^5 - 425*a^5*tan(
1/2*d*x + 1/2*c)^3 + 3040*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 2610*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 10880*a^2*b^3*t
an(1/2*d*x + 1/2*c)^3 - 1800*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 7200*b^5*tan(1/2*d*x + 1/2*c)^3 - 75*a^5*tan(1/2*d
*x + 1/2*c) + 480*a^4*b*tan(1/2*d*x + 1/2*c) + 630*a^3*b^2*tan(1/2*d*x + 1/2*c) - 1920*a^2*b^3*tan(1/2*d*x + 1
/2*c) - 600*a*b^4*tan(1/2*d*x + 1/2*c) + 1440*b^5*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^7))/
d