3.3.16 \(\int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) [216]

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

[Out]

1/16*(5*a^6-90*a^4*b^2+200*a^2*b^4-112*b^6)*x/a^8-2*(a-b)^(3/2)*b*(a+b)^(3/2)*(2*a^2-7*b^2)*arctanh((a-b)^(1/2
)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/a^8/d+1/15*b*(61*a^4-170*a^2*b^2+105*b^4)*sin(d*x+c)/a^7/d-1/16*(27*a^4-86*a
^2*b^2+56*b^4)*cos(d*x+c)*sin(d*x+c)/a^6/d+1/15*(15*a^4-52*a^2*b^2+35*b^4)*cos(d*x+c)^2*sin(d*x+c)/a^5/b/d-1/2
4*(16*a^4-61*a^2*b^2+42*b^4)*cos(d*x+c)^3*sin(d*x+c)/a^4/b^2/d-1/3*cos(d*x+c)^3*sin(d*x+c)/b/d/(b+a*cos(d*x+c)
)+1/6*a*cos(d*x+c)^4*sin(d*x+c)/b^2/d/(b+a*cos(d*x+c))+1/10*(5*a^4-20*a^2*b^2+14*b^4)*cos(d*x+c)^4*sin(d*x+c)/
a^3/b^2/d/(b+a*cos(d*x+c))+7/30*b*cos(d*x+c)^5*sin(d*x+c)/a^2/d/(b+a*cos(d*x+c))-1/6*cos(d*x+c)^6*sin(d*x+c)/a
/d/(b+a*cos(d*x+c))

________________________________________________________________________________________

Rubi [A]
time = 1.17, antiderivative size = 473, normalized size of antiderivative = 1.00, 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 = {3957, 2975, 3126, 3128, 3102, 2814, 2738, 214} \begin {gather*} \frac {7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}-\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 {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 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 ) \sin (c+d x) \cos (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}+\frac {\left (5 a^4-20 a^2 b^2+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 {x \left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right )}{16 a^8}+\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)} \end {gather*}

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 214

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

Rule 2738

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 2814

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 2975

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*S
in[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[Cos
[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 3102

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 3126

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 3128

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 3957

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

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 ) \text {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*}

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Mathematica [A]
time = 4.58, size = 402, normalized size = 0.85 \begin {gather*} \frac {3840 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {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-13440 b^7 d x+120 a \left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) (c+d x) \cos (c+d x)-15 a \left (15 a^6-576 a^4 b^2+1488 a^2 b^4-896 b^6\right ) \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 \sin (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} \end {gather*}

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)

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Maple [A]
time = 0.26, size = 509, normalized size = 1.08 Too large to display

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,method=_RETURNVERBOSE)

[Out]

1/d*(2*(a-b)^2*(a+b)^2*b/a^8*(-a*b*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)-(2*a
^2-7*b^2)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)))+2/a^8*(((5/16*a^6+2*a^5*b
-21/8*a^4*b^2-8*a^3*b^3+5/2*a^2*b^4+6*a*b^5)*tan(1/2*d*x+1/2*c)^11+(38/3*a^5*b-87/8*a^4*b^2+15/2*a^2*b^4+30*a*
b^5+85/48*a^6-136/3*a^3*b^3)*tan(1/2*d*x+1/2*c)^9+(172/5*a^5*b-33/4*a^4*b^2-96*a^3*b^3+5*a^2*b^4+60*a*b^5+33/8
*a^6)*tan(1/2*d*x+1/2*c)^7+(-33/8*a^6+33/4*a^4*b^2-5*a^2*b^4+172/5*a^5*b-96*a^3*b^3+60*a*b^5)*tan(1/2*d*x+1/2*
c)^5+(38/3*a^5*b+87/8*a^4*b^2-136/3*a^3*b^3-15/2*a^2*b^4+30*a*b^5-85/48*a^6)*tan(1/2*d*x+1/2*c)^3+(2*a^5*b-8*a
^3*b^3+6*a*b^5-5/16*a^6+21/8*a^4*b^2-5/2*a^2*b^4)*tan(1/2*d*x+1/2*c))/(1+tan(1/2*d*x+1/2*c)^2)^6+1/16*(5*a^6-9
0*a^4*b^2+200*a^2*b^4-112*b^6)*arctan(tan(1/2*d*x+1/2*c))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?`
for more de

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Fricas [A]
time = 5.24, size = 793, normalized size = 1.68 \begin {gather*} \left [\frac {15 \, {\left (5 \, a^{7} - 90 \, a^{5} b^{2} + 200 \, a^{3} b^{4} - 112 \, a b^{6}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b - 90 \, a^{4} b^{3} + 200 \, a^{2} b^{5} - 112 \, b^{7}\right )} d x + 120 \, {\left (2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 7 \, b^{6} + {\left (2 \, a^{5} b - 9 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (40 \, a^{7} \cos \left (d x + c\right )^{6} - 56 \, a^{6} b \cos \left (d x + c\right )^{5} - 976 \, a^{5} b^{2} + 2720 \, a^{3} b^{4} - 1680 \, a b^{6} - 2 \, {\left (65 \, a^{7} - 42 \, a^{5} b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (111 \, a^{6} b - 70 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (165 \, a^{7} - 458 \, a^{5} b^{2} + 280 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (571 \, a^{6} b - 1430 \, a^{4} b^{3} + 840 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{9} d \cos \left (d x + c\right ) + a^{8} b d\right )}}, \frac {15 \, {\left (5 \, a^{7} - 90 \, a^{5} b^{2} + 200 \, a^{3} b^{4} - 112 \, a b^{6}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b - 90 \, a^{4} b^{3} + 200 \, a^{2} b^{5} - 112 \, b^{7}\right )} d x - 240 \, {\left (2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 7 \, b^{6} + {\left (2 \, a^{5} b - 9 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (40 \, a^{7} \cos \left (d x + c\right )^{6} - 56 \, a^{6} b \cos \left (d x + c\right )^{5} - 976 \, a^{5} b^{2} + 2720 \, a^{3} b^{4} - 1680 \, a b^{6} - 2 \, {\left (65 \, a^{7} - 42 \, a^{5} b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (111 \, a^{6} b - 70 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (165 \, a^{7} - 458 \, a^{5} b^{2} + 280 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (571 \, a^{6} b - 1430 \, a^{4} b^{3} + 840 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{9} d \cos \left (d x + c\right ) + a^{8} b d\right )}}\right ] \end {gather*}

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)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{6}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.49, size = 870, normalized size = 1.84 \begin {gather*} \frac {\frac {15 \, {\left (5 \, a^{6} - 90 \, a^{4} b^{2} + 200 \, a^{2} b^{4} - 112 \, b^{6}\right )} {\left (d x + c\right )}}{a^{8}} - \frac {480 \, {\left (2 \, a^{6} b - 11 \, a^{4} b^{3} + 16 \, a^{2} b^{5} - 7 \, b^{7}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} a^{8}} - \frac {480 \, {\left (a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a - b\right )} a^{7}} + \frac {2 \, {\left (75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 480 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 630 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1920 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 600 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1440 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 425 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3040 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 2610 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 10880 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1800 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 7200 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 990 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 8256 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1980 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 23040 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1200 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 14400 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 990 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 8256 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1980 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 23040 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1200 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14400 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 425 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3040 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2610 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10880 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1800 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 7200 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 630 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1920 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1440 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{7}}}{240 \, d} \end {gather*}

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

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Mupad [B]
time = 4.87, size = 2500, normalized size = 5.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((((74*a^23*b - 10*a^24 + 224*a^16*b^8 - 336*a^17*b^7 - 400*a^18*b^6 + 740*a^19*b^5 + 124*a^20*b^4 - 4
78*a^21*b^3 + 62*a^22*b^2)/a^21 - (tan(c/2 + (d*x)/2)*(512*a^18*b + 512*a^16*b^3 - 1024*a^17*b^2)*(a^6*5i - b^
6*112i + a^2*b^4*200i - a^4*b^2*90i))/(128*a^22))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(16*a^8) +
(tan(c/2 + (d*x)/2)*(50176*a*b^14 - 75*a^14*b + 25*a^15 - 25088*b^15 + 64512*a^2*b^13 - 179200*a^3*b^12 - 307
20*a^4*b^11 + 240640*a^5*b^10 - 46080*a^6*b^9 - 148480*a^7*b^8 + 53900*a^8*b^7 + 40540*a^9*b^6 - 18136*a^10*b^
5 - 3864*a^11*b^4 + 1651*a^12*b^3 + 199*a^13*b^2))/(8*a^14))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i)*
1i)/(16*a^8) - (((((74*a^23*b - 10*a^24 + 224*a^16*b^8 - 336*a^17*b^7 - 400*a^18*b^6 + 740*a^19*b^5 + 124*a^20
*b^4 - 478*a^21*b^3 + 62*a^22*b^2)/a^21 + (tan(c/2 + (d*x)/2)*(512*a^18*b + 512*a^16*b^3 - 1024*a^17*b^2)*(a^6
*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(128*a^22))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(1
6*a^8) - (tan(c/2 + (d*x)/2)*(50176*a*b^14 - 75*a^14*b + 25*a^15 - 25088*b^15 + 64512*a^2*b^13 - 179200*a^3*b^
12 - 30720*a^4*b^11 + 240640*a^5*b^10 - 46080*a^6*b^9 - 148480*a^7*b^8 + 53900*a^8*b^7 + 40540*a^9*b^6 - 18136
*a^10*b^5 - 3864*a^11*b^4 + 1651*a^12*b^3 + 199*a^13*b^2))/(8*a^14))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b
^2*90i)*1i)/(16*a^8))/((((((74*a^23*b - 10*a^24 + 224*a^16*b^8 - 336*a^17*b^7 - 400*a^18*b^6 + 740*a^19*b^5 +
124*a^20*b^4 - 478*a^21*b^3 + 62*a^22*b^2)/a^21 - (tan(c/2 + (d*x)/2)*(512*a^18*b + 512*a^16*b^3 - 1024*a^17*b
^2)*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(128*a^22))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*
90i))/(16*a^8) + (tan(c/2 + (d*x)/2)*(50176*a*b^14 - 75*a^14*b + 25*a^15 - 25088*b^15 + 64512*a^2*b^13 - 17920
0*a^3*b^12 - 30720*a^4*b^11 + 240640*a^5*b^10 - 46080*a^6*b^9 - 148480*a^7*b^8 + 53900*a^8*b^7 + 40540*a^9*b^6
- 18136*a^10*b^5 - 3864*a^11*b^4 + 1651*a^12*b^3 + 199*a^13*b^2))/(8*a^14))*(a^6*5i - b^6*112i + a^2*b^4*200i
- a^4*b^2*90i))/(16*a^8) - (32928*a*b^19 - (25*a^19*b)/2 - 21952*b^20 + 117600*a^2*b^18 - 190120*a^3*b^17 - 2
57432*a^4*b^16 + 463764*a^5*b^15 + 290284*a^6*b^14 - 620037*a^7*b^13 - 169030*a^8*b^12 + 492572*a^9*b^11 + 355
58*a^10*b^10 - (941393*a^11*b^9)/4 + (22469*a^12*b^8)/2 + (260375*a^13*b^7)/4 - 7490*a^14*b^6 - (37705*a^15*b^
5)/4 + (2565*a^16*b^4)/2 + (2345*a^17*b^3)/4 - 55*a^18*b^2)/a^21 + (((((74*a^23*b - 10*a^24 + 224*a^16*b^8 - 3
36*a^17*b^7 - 400*a^18*b^6 + 740*a^19*b^5 + 124*a^20*b^4 - 478*a^21*b^3 + 62*a^22*b^2)/a^21 + (tan(c/2 + (d*x)
/2)*(512*a^18*b + 512*a^16*b^3 - 1024*a^17*b^2)*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(128*a^22))*
(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(16*a^8) - (tan(c/2 + (d*x)/2)*(50176*a*b^14 - 75*a^14*b + 2
5*a^15 - 25088*b^15 + 64512*a^2*b^13 - 179200*a^3*b^12 - 30720*a^4*b^11 + 240640*a^5*b^10 - 46080*a^6*b^9 - 14
8480*a^7*b^8 + 53900*a^8*b^7 + 40540*a^9*b^6 - 18136*a^10*b^5 - 3864*a^11*b^4 + 1651*a^12*b^3 + 199*a^13*b^2))
/(8*a^14))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^4*b^2*90i))/(16*a^8)))*(a^6*5i - b^6*112i + a^2*b^4*200i - a^
4*b^2*90i)*1i)/(8*a^8*d) - ((tan(c/2 + (d*x)/2)^11*(336*a*b^5 + 206*a^5*b + 35*a^6 - 1008*b^6 + 1688*a^2*b^4 -
572*a^3*b^3 - 694*a^4*b^2))/(12*a^7) - (tan(c/2 + (d*x)/2)^3*(336*a*b^5 + 206*a^5*b - 35*a^6 + 1008*b^6 - 168
8*a^2*b^4 - 572*a^3*b^3 + 694*a^4*b^2))/(12*a^7) - (tan(c/2 + (d*x)/2)^5*(4200*a*b^5 + 3801*a^5*b - 565*a^6 +
25200*b^6 - 40520*a^2*b^4 - 7570*a^3*b^3 + 14266*a^4*b^2))/(120*a^7) + (tan(c/2 + (d*x)/2)^9*(4200*a*b^5 + 380
1*a^5*b + 565*a^6 - 25200*b^6 + 40520*a^2*b^4 - 7570*a^3*b^3 - 14266*a^4*b^2))/(120*a^7) - (tan(c/2 + (d*x)/2)
^7*(165*a^6 + 2800*b^6 - 4440*a^2*b^4 + 1446*a^4*b^2))/(10*a^7) + (tan(c/2 + (d*x)/2)^13*(a - b)*(56*a*b^4 + 3
2*a^4*b + 5*a^5 + 112*b^5 - 144*a^2*b^3 - 58*a^3*b^2))/(8*a^7) + (tan(c/2 + (d*x)/2)*(a + b)*(56*a*b^4 - 32*a^
4*b + 5*a^5 - 112*b^5 + 144*a^2*b^3 - 58*a^3*b^2))/(8*a^7))/(d*(a + b - tan(c/2 + (d*x)/2)^14*(a - b) + tan(c/
2 + (d*x)/2)^2*(5*a + 7*b) - tan(c/2 + (d*x)/2)^12*(5*a - 7*b) + tan(c/2 + (d*x)/2)^4*(9*a + 21*b) - tan(c/2 +
(d*x)/2)^10*(9*a - 21*b) + tan(c/2 + (d*x)/2)^6*(5*a + 35*b) - tan(c/2 + (d*x)/2)^8*(5*a - 35*b))) - (b*atan(
((b*((tan(c/2 + (d*x)/2)*(50176*a*b^14 - 75*a^14*b + 25*a^15 - 25088*b^15 + 64512*a^2*b^13 - 179200*a^3*b^12 -
30720*a^4*b^11 + 240640*a^5*b^10 - 46080*a^6*b^9 - 148480*a^7*b^8 + 53900*a^8*b^7 + 40540*a^9*b^6 - 18136*a^1
0*b^5 - 3864*a^11*b^4 + 1651*a^12*b^3 + 199*a^13*b^2))/(8*a^14) + (b*((74*a^23*b - 10*a^24 + 224*a^16*b^8 - 33
6*a^17*b^7 - 400*a^18*b^6 + 740*a^19*b^5 + 124*a^20*b^4 - 478*a^21*b^3 + 62*a^22*b^2)/a^21 - (b*tan(c/2 + (d*x
)/2)*(2*a^2 - 7*b^2)*((a + b)^3*(a - b)^3)^(1/2)*(512*a^18*b + 512*a^16*b^3 - 1024*a^17*b^2))/(8*a^22))*(2*a^2
- 7*b^2)*((a + b)^3*(a - b)^3)^(1/2))/a^8)*(2*a^2 - 7*b^2)*((a + b)^3*(a - b)^3)^(1/2)*1i)/a^8 + (b*((tan(c/2
+ (d*x)/2)*(50176*a*b^14 - 75*a^14*b + 25*a^15 - 25088*b^15 + 64512*a^2*b^13 - 179200*a^3*b^12 - 30720*a^4*b^
11 + 240640*a^5*b^10 - 46080*a^6*b^9 - 148480*a...

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