3.3.28 \(\int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx\) [228]

Optimal. Leaf size=539 \[ \frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}-\frac {\sqrt {a-b} b \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))} \]

[Out]

1/16*(5*a^6-180*a^4*b^2+600*a^2*b^4-448*b^6)*x/a^9+1/30*b*(213*a^4-985*a^2*b^2+840*b^4)*sin(d*x+c)/a^8/d-1/16*
(43*a^4-244*a^2*b^2+224*b^4)*cos(d*x+c)*sin(d*x+c)/a^7/d+1/30*(45*a^4-291*a^2*b^2+280*b^4)*cos(d*x+c)^2*sin(d*
x+c)/a^6/b/d-1/24*(24*a^4-169*a^2*b^2+168*b^4)*cos(d*x+c)^3*sin(d*x+c)/a^5/b^2/d-1/4*cos(d*x+c)^4*sin(d*x+c)/b
/d/(b+a*cos(d*x+c))^2+1/10*a*cos(d*x+c)^5*sin(d*x+c)/b^2/d/(b+a*cos(d*x+c))^2+1/60*(9*a^4-60*a^2*b^2+56*b^4)*c
os(d*x+c)^5*sin(d*x+c)/a^3/b^2/d/(b+a*cos(d*x+c))^2+4/15*b*cos(d*x+c)^6*sin(d*x+c)/a^2/d/(b+a*cos(d*x+c))^2-1/
6*cos(d*x+c)^7*sin(d*x+c)/a/d/(b+a*cos(d*x+c))^2+1/20*(15*a^4-110*a^2*b^2+112*b^4)*cos(d*x+c)^4*sin(d*x+c)/a^4
/b^2/d/(b+a*cos(d*x+c))-b*(6*a^4-47*a^2*b^2+56*b^4)*arctanh((a-b)^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))*(a-b)^
(1/2)*(a+b)^(1/2)/a^9/d

________________________________________________________________________________________

Rubi [A]
time = 1.61, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 11, 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 {4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{20 a^4 b^2 d (a \cos (c+d x)+b)}-\frac {b \sqrt {a-b} \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^5 b^2 d}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos ^5(c+d x)}{60 a^3 b^2 d (a \cos (c+d x)+b)^2}+\frac {x \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right )}{16 a^9}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac {\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^3,x]

[Out]

((5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*x)/(16*a^9) - (Sqrt[a - b]*b*Sqrt[a + b]*(6*a^4 - 47*a^2*b^2 +
56*b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^9*d) + (b*(213*a^4 - 985*a^2*b^2 + 840*b^4)*Si
n[c + d*x])/(30*a^8*d) - ((43*a^4 - 244*a^2*b^2 + 224*b^4)*Cos[c + d*x]*Sin[c + d*x])/(16*a^7*d) + ((45*a^4 -
291*a^2*b^2 + 280*b^4)*Cos[c + d*x]^2*Sin[c + d*x])/(30*a^6*b*d) - ((24*a^4 - 169*a^2*b^2 + 168*b^4)*Cos[c + d
*x]^3*Sin[c + d*x])/(24*a^5*b^2*d) - (Cos[c + d*x]^4*Sin[c + d*x])/(4*b*d*(b + a*Cos[c + d*x])^2) + (a*Cos[c +
d*x]^5*Sin[c + d*x])/(10*b^2*d*(b + a*Cos[c + d*x])^2) + ((9*a^4 - 60*a^2*b^2 + 56*b^4)*Cos[c + d*x]^5*Sin[c
+ d*x])/(60*a^3*b^2*d*(b + a*Cos[c + d*x])^2) + (4*b*Cos[c + d*x]^6*Sin[c + d*x])/(15*a^2*d*(b + a*Cos[c + d*x
])^2) - (Cos[c + d*x]^7*Sin[c + d*x])/(6*a*d*(b + a*Cos[c + d*x])^2) + ((15*a^4 - 110*a^2*b^2 + 112*b^4)*Cos[c
+ d*x]^4*Sin[c + d*x])/(20*a^4*b^2*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))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^6(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}-\frac {\int \frac {\cos ^5(c+d x) \left (30 \left (6 a^4-35 a^2 b^2+32 b^4\right )+30 a b \left (3 a^2-2 b^2\right ) \cos (c+d x)-20 \left (12 a^4-65 a^2 b^2+56 b^4\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^3} \, dx}{600 a^2 b^2}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^4(c+d x) \left (100 \left (9 a^6-69 a^4 b^2+116 a^2 b^4-56 b^6\right )+20 a b \left (15 a^4-31 a^2 b^2+16 b^4\right ) \cos (c+d x)-40 \left (30 a^6-215 a^4 b^2+353 a^2 b^4-168 b^6\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^2} \, dx}{1200 a^3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (240 \left (a^2-b^2\right )^2 \left (15 a^4-110 a^2 b^2+112 b^4\right )+40 a b \left (15 a^2-28 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-200 \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{1200 a^4 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (600 b \left (a^2-b^2\right )^2 \left (24 a^4-169 a^2 b^2+168 b^4\right )+840 a b^2 \left (5 a^2-8 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-480 b \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{4800 a^5 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (960 b^2 \left (a^2-b^2\right )^2 \left (45 a^4-291 a^2 b^2+280 b^4\right )+120 a b^3 \left (207 a^2-280 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-1800 b^2 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{14400 a^6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\int \frac {1800 b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right )-120 a b^2 \left (a^2-b^2\right )^2 \left (75 a^4-996 a^2 b^2+1120 b^4\right ) \cos (c+d x)-960 b^3 \left (a^2-b^2\right )^2 \left (213 a^4-985 a^2 b^2+840 b^4\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{28800 a^7 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}-\frac {\int \frac {-1800 a b^3 \left (a^2-b^2\right )^2 \left (43 a^4-244 a^2 b^2+224 b^4\right )+1800 b^2 \left (a^2-b^2\right )^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{28800 a^8 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\left (b \left (a^2-b^2\right ) \left (6 a^4-47 a^2 b^2+56 b^4\right )\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{2 a^9}\\ &=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}+\frac {\left (b \left (a^2-b^2\right ) \left (6 a^4-47 a^2 b^2+56 b^4\right )\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^9 d}\\ &=\frac {\left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) x}{16 a^9}-\frac {\sqrt {a-b} b \sqrt {a+b} \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^9 d}+\frac {b \left (213 a^4-985 a^2 b^2+840 b^4\right ) \sin (c+d x)}{30 a^8 d}-\frac {\left (43 a^4-244 a^2 b^2+224 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^7 d}+\frac {\left (45 a^4-291 a^2 b^2+280 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{30 a^6 b d}-\frac {\left (24 a^4-169 a^2 b^2+168 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^5 b^2 d}-\frac {\cos ^4(c+d x) \sin (c+d x)}{4 b d (b+a \cos (c+d x))^2}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{10 b^2 d (b+a \cos (c+d x))^2}+\frac {\left (9 a^4-60 a^2 b^2+56 b^4\right ) \cos ^5(c+d x) \sin (c+d x)}{60 a^3 b^2 d (b+a \cos (c+d x))^2}+\frac {4 b \cos ^6(c+d x) \sin (c+d x)}{15 a^2 d (b+a \cos (c+d x))^2}-\frac {\cos ^7(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))^2}+\frac {\left (15 a^4-110 a^2 b^2+112 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{20 a^4 b^2 d (b+a \cos (c+d x))}\\ \end {align*}

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Mathematica [A]
time = 7.86, size = 599, normalized size = 1.11 \begin {gather*} \frac {-7680 b \left (-a^2+b^2\right )^3 \left (6 a^4-47 a^2 b^2+56 b^4\right ) \tanh ^{-1}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right ) (b+a \cos (c+d x))^2+2 \left (a^2-b^2\right )^{5/2} \left (600 a^8 c-20400 a^6 b^2 c+28800 a^4 b^4 c+90240 a^2 b^6 c-107520 b^8 c+600 a^8 d x-20400 a^6 b^2 d x+28800 a^4 b^4 d x+90240 a^2 b^6 d x-107520 b^8 d x+480 a b \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (c+d x)+120 a^2 \left (5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right ) (c+d x) \cos (2 (c+d x))+2640 a^7 b \sin (c+d x)+16160 a^5 b^3 \sin (c+d x)-117120 a^3 b^5 \sin (c+d x)+107520 a b^7 \sin (c+d x)-405 a^8 \sin (2 (c+d x))+24600 a^6 b^2 \sin (2 (c+d x))-99040 a^4 b^4 \sin (2 (c+d x))+80640 a^2 b^6 \sin (2 (c+d x))+2436 a^7 b \sin (3 (c+d x))-10880 a^5 b^3 \sin (3 (c+d x))+8960 a^3 b^5 \sin (3 (c+d x))-140 a^8 \sin (4 (c+d x))+1164 a^6 b^2 \sin (4 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))-188 a^7 b \sin (5 (c+d x))+224 a^5 b^3 \sin (5 (c+d x))+35 a^8 \sin (6 (c+d x))-56 a^6 b^2 \sin (6 (c+d x))+16 a^7 b \sin (7 (c+d x))-5 a^8 \sin (8 (c+d x))\right )}{7680 a^9 (a-b)^2 (a+b)^2 \sqrt {a^2-b^2} d (b+a \cos (c+d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^3,x]

[Out]

(-7680*b*(-a^2 + b^2)^3*(6*a^4 - 47*a^2*b^2 + 56*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b
+ a*Cos[c + d*x])^2 + 2*(a^2 - b^2)^(5/2)*(600*a^8*c - 20400*a^6*b^2*c + 28800*a^4*b^4*c + 90240*a^2*b^6*c - 1
07520*b^8*c + 600*a^8*d*x - 20400*a^6*b^2*d*x + 28800*a^4*b^4*d*x + 90240*a^2*b^6*d*x - 107520*b^8*d*x + 480*a
*b*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*(c + d*x)*Cos[c + d*x] + 120*a^2*(5*a^6 - 180*a^4*b^2 + 600*a
^2*b^4 - 448*b^6)*(c + d*x)*Cos[2*(c + d*x)] + 2640*a^7*b*Sin[c + d*x] + 16160*a^5*b^3*Sin[c + d*x] - 117120*a
^3*b^5*Sin[c + d*x] + 107520*a*b^7*Sin[c + d*x] - 405*a^8*Sin[2*(c + d*x)] + 24600*a^6*b^2*Sin[2*(c + d*x)] -
99040*a^4*b^4*Sin[2*(c + d*x)] + 80640*a^2*b^6*Sin[2*(c + d*x)] + 2436*a^7*b*Sin[3*(c + d*x)] - 10880*a^5*b^3*
Sin[3*(c + d*x)] + 8960*a^3*b^5*Sin[3*(c + d*x)] - 140*a^8*Sin[4*(c + d*x)] + 1164*a^6*b^2*Sin[4*(c + d*x)] -
1120*a^4*b^4*Sin[4*(c + d*x)] - 188*a^7*b*Sin[5*(c + d*x)] + 224*a^5*b^3*Sin[5*(c + d*x)] + 35*a^8*Sin[6*(c +
d*x)] - 56*a^6*b^2*Sin[6*(c + d*x)] + 16*a^7*b*Sin[7*(c + d*x)] - 5*a^8*Sin[8*(c + d*x)]))/(7680*a^9*(a - b)^2
*(a + b)^2*Sqrt[a^2 - b^2]*d*(b + a*Cos[c + d*x])^2)

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Maple [A]
time = 0.48, size = 582, 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))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(2*(a-b)*(a+b)*b/a^9*(((5/2*a^3*b^2-7*a*b^4-3*a^4*b+15/2*a^2*b^3)*tan(1/2*d*x+1/2*c)^3+(5/2*a^3*b^2-7*a*b^
4+3*a^4*b-15/2*a^2*b^3)*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)^2-1/2*(6*a^4-4
7*a^2*b^2+56*b^4)/((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^9*(((5/16*a^6
+3*a^5*b-21/4*a^4*b^2-20*a^3*b^3+15/2*a^2*b^4+21*a*b^5)*tan(1/2*d*x+1/2*c)^11+(19*a^5*b-87/4*a^4*b^2+45/2*a^2*
b^4+105*a*b^5+85/48*a^6-340/3*a^3*b^3)*tan(1/2*d*x+1/2*c)^9+(258/5*a^5*b-33/2*a^4*b^2-240*a^3*b^3+15*a^2*b^4+2
10*a*b^5+33/8*a^6)*tan(1/2*d*x+1/2*c)^7+(-33/8*a^6+33/2*a^4*b^2-15*a^2*b^4+258/5*a^5*b-240*a^3*b^3+210*a*b^5)*
tan(1/2*d*x+1/2*c)^5+(19*a^5*b+87/4*a^4*b^2-340/3*a^3*b^3-45/2*a^2*b^4+105*a*b^5-85/48*a^6)*tan(1/2*d*x+1/2*c)
^3+(3*a^5*b-20*a^3*b^3+21*a*b^5-5/16*a^6+21/4*a^4*b^2-15/2*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^6-180*a^4*b^2+600*a^2*b^4-448*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))^3,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 = 4.38, size = 1057, normalized size = 1.96 \begin {gather*} \left [\frac {15 \, {\left (5 \, a^{8} - 180 \, a^{6} b^{2} + 600 \, a^{4} b^{4} - 448 \, a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 30 \, {\left (5 \, a^{7} b - 180 \, a^{5} b^{3} + 600 \, a^{3} b^{5} - 448 \, a b^{7}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b^{2} - 180 \, a^{4} b^{4} + 600 \, a^{2} b^{6} - 448 \, b^{8}\right )} d x + 60 \, {\left (6 \, a^{4} b^{3} - 47 \, a^{2} b^{5} + 56 \, b^{7} + {\left (6 \, a^{6} b - 47 \, a^{4} b^{3} + 56 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 47 \, a^{3} b^{4} + 56 \, a b^{6}\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^{8} \cos \left (d x + c\right )^{7} - 64 \, a^{7} b \cos \left (d x + c\right )^{6} - 1704 \, a^{5} b^{3} + 7880 \, a^{3} b^{5} - 6720 \, a b^{7} - 2 \, {\left (65 \, a^{8} - 56 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (67 \, a^{7} b - 56 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (165 \, a^{8} - 694 \, a^{6} b^{2} + 560 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (387 \, a^{7} b - 1444 \, a^{5} b^{3} + 1120 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (2763 \, a^{6} b^{2} - 12100 \, a^{4} b^{4} + 10080 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{11} d \cos \left (d x + c\right )^{2} + 2 \, a^{10} b d \cos \left (d x + c\right ) + a^{9} b^{2} d\right )}}, \frac {15 \, {\left (5 \, a^{8} - 180 \, a^{6} b^{2} + 600 \, a^{4} b^{4} - 448 \, a^{2} b^{6}\right )} d x \cos \left (d x + c\right )^{2} + 30 \, {\left (5 \, a^{7} b - 180 \, a^{5} b^{3} + 600 \, a^{3} b^{5} - 448 \, a b^{7}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b^{2} - 180 \, a^{4} b^{4} + 600 \, a^{2} b^{6} - 448 \, b^{8}\right )} d x - 120 \, {\left (6 \, a^{4} b^{3} - 47 \, a^{2} b^{5} + 56 \, b^{7} + {\left (6 \, a^{6} b - 47 \, a^{4} b^{3} + 56 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 47 \, a^{3} b^{4} + 56 \, a b^{6}\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^{8} \cos \left (d x + c\right )^{7} - 64 \, a^{7} b \cos \left (d x + c\right )^{6} - 1704 \, a^{5} b^{3} + 7880 \, a^{3} b^{5} - 6720 \, a b^{7} - 2 \, {\left (65 \, a^{8} - 56 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (67 \, a^{7} b - 56 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{4} + {\left (165 \, a^{8} - 694 \, a^{6} b^{2} + 560 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (387 \, a^{7} b - 1444 \, a^{5} b^{3} + 1120 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (2763 \, a^{6} b^{2} - 12100 \, a^{4} b^{4} + 10080 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{11} d \cos \left (d x + c\right )^{2} + 2 \, a^{10} b d \cos \left (d x + c\right ) + a^{9} b^{2} 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))^3,x, algorithm="fricas")

[Out]

[1/240*(15*(5*a^8 - 180*a^6*b^2 + 600*a^4*b^4 - 448*a^2*b^6)*d*x*cos(d*x + c)^2 + 30*(5*a^7*b - 180*a^5*b^3 +
600*a^3*b^5 - 448*a*b^7)*d*x*cos(d*x + c) + 15*(5*a^6*b^2 - 180*a^4*b^4 + 600*a^2*b^6 - 448*b^8)*d*x + 60*(6*a
^4*b^3 - 47*a^2*b^5 + 56*b^7 + (6*a^6*b - 47*a^4*b^3 + 56*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 47*a^3*b^4
+ 56*a*b^6)*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)*sin(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^8*cos(d*x + c)^7 - 64*a^7*b*cos(d*x + c)^6 - 1704*a^5*b^3 + 7880*a^3*b^5 - 6720*a*b^7 - 2*(65*a^8 - 56*a^6*
b^2)*cos(d*x + c)^5 + 4*(67*a^7*b - 56*a^5*b^3)*cos(d*x + c)^4 + (165*a^8 - 694*a^6*b^2 + 560*a^4*b^4)*cos(d*x
+ c)^3 - 2*(387*a^7*b - 1444*a^5*b^3 + 1120*a^3*b^5)*cos(d*x + c)^2 - (2763*a^6*b^2 - 12100*a^4*b^4 + 10080*a
^2*b^6)*cos(d*x + c))*sin(d*x + c))/(a^11*d*cos(d*x + c)^2 + 2*a^10*b*d*cos(d*x + c) + a^9*b^2*d), 1/240*(15*(
5*a^8 - 180*a^6*b^2 + 600*a^4*b^4 - 448*a^2*b^6)*d*x*cos(d*x + c)^2 + 30*(5*a^7*b - 180*a^5*b^3 + 600*a^3*b^5
- 448*a*b^7)*d*x*cos(d*x + c) + 15*(5*a^6*b^2 - 180*a^4*b^4 + 600*a^2*b^6 - 448*b^8)*d*x - 120*(6*a^4*b^3 - 47
*a^2*b^5 + 56*b^7 + (6*a^6*b - 47*a^4*b^3 + 56*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 47*a^3*b^4 + 56*a*b^6)
*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))) - (4
0*a^8*cos(d*x + c)^7 - 64*a^7*b*cos(d*x + c)^6 - 1704*a^5*b^3 + 7880*a^3*b^5 - 6720*a*b^7 - 2*(65*a^8 - 56*a^6
*b^2)*cos(d*x + c)^5 + 4*(67*a^7*b - 56*a^5*b^3)*cos(d*x + c)^4 + (165*a^8 - 694*a^6*b^2 + 560*a^4*b^4)*cos(d*
x + c)^3 - 2*(387*a^7*b - 1444*a^5*b^3 + 1120*a^3*b^5)*cos(d*x + c)^2 - (2763*a^6*b^2 - 12100*a^4*b^4 + 10080*
a^2*b^6)*cos(d*x + c))*sin(d*x + c))/(a^11*d*cos(d*x + c)^2 + 2*a^10*b*d*cos(d*x + c) + a^9*b^2*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 )^{3}}\, 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))**3,x)

[Out]

Integral(sin(c + d*x)**6/(a + b*sec(c + d*x))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (508) = 1016\).
time = 0.64, size = 1030, normalized size = 1.91 \begin {gather*} \frac {\frac {15 \, {\left (5 \, a^{6} - 180 \, a^{4} b^{2} + 600 \, a^{2} b^{4} - 448 \, b^{6}\right )} {\left (d x + c\right )}}{a^{9}} - \frac {240 \, {\left (6 \, a^{6} b - 53 \, a^{4} b^{3} + 103 \, a^{2} b^{5} - 56 \, 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^{9}} - \frac {240 \, {\left (6 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 19 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 14 \, b^{7} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 19 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 15 \, a b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, b^{7} \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 )}^{2} a^{8}} + \frac {2 \, {\left (75 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 720 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 1260 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4800 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1800 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 5040 \, 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} + 4560 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 5220 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 27200 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5400 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 25200 \, 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} + 12384 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3960 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 57600 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3600 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 50400 \, 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} + 12384 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3960 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 57600 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3600 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 50400 \, 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} + 4560 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5220 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 27200 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5400 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 25200 \, 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 ) + 720 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1260 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4800 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1800 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5040 \, 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^{8}}}{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))^3,x, algorithm="giac")

[Out]

1/240*(15*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*(d*x + c)/a^9 - 240*(6*a^6*b - 53*a^4*b^3 + 103*a^2*b^
5 - 56*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^9) - 240*(6*a^5*b^2*tan(1/2*d*x + 1/2*c)^3 - 5*a^4*b^3*tan(
1/2*d*x + 1/2*c)^3 - 21*a^3*b^4*tan(1/2*d*x + 1/2*c)^3 + 19*a^2*b^5*tan(1/2*d*x + 1/2*c)^3 + 15*a*b^6*tan(1/2*
d*x + 1/2*c)^3 - 14*b^7*tan(1/2*d*x + 1/2*c)^3 - 6*a^5*b^2*tan(1/2*d*x + 1/2*c) - 5*a^4*b^3*tan(1/2*d*x + 1/2*
c) + 21*a^3*b^4*tan(1/2*d*x + 1/2*c) + 19*a^2*b^5*tan(1/2*d*x + 1/2*c) - 15*a*b^6*tan(1/2*d*x + 1/2*c) - 14*b^
7*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^8) + 2*(75*a^5*tan(
1/2*d*x + 1/2*c)^11 + 720*a^4*b*tan(1/2*d*x + 1/2*c)^11 - 1260*a^3*b^2*tan(1/2*d*x + 1/2*c)^11 - 4800*a^2*b^3*
tan(1/2*d*x + 1/2*c)^11 + 1800*a*b^4*tan(1/2*d*x + 1/2*c)^11 + 5040*b^5*tan(1/2*d*x + 1/2*c)^11 + 425*a^5*tan(
1/2*d*x + 1/2*c)^9 + 4560*a^4*b*tan(1/2*d*x + 1/2*c)^9 - 5220*a^3*b^2*tan(1/2*d*x + 1/2*c)^9 - 27200*a^2*b^3*t
an(1/2*d*x + 1/2*c)^9 + 5400*a*b^4*tan(1/2*d*x + 1/2*c)^9 + 25200*b^5*tan(1/2*d*x + 1/2*c)^9 + 990*a^5*tan(1/2
*d*x + 1/2*c)^7 + 12384*a^4*b*tan(1/2*d*x + 1/2*c)^7 - 3960*a^3*b^2*tan(1/2*d*x + 1/2*c)^7 - 57600*a^2*b^3*tan
(1/2*d*x + 1/2*c)^7 + 3600*a*b^4*tan(1/2*d*x + 1/2*c)^7 + 50400*b^5*tan(1/2*d*x + 1/2*c)^7 - 990*a^5*tan(1/2*d
*x + 1/2*c)^5 + 12384*a^4*b*tan(1/2*d*x + 1/2*c)^5 + 3960*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 57600*a^2*b^3*tan(1
/2*d*x + 1/2*c)^5 - 3600*a*b^4*tan(1/2*d*x + 1/2*c)^5 + 50400*b^5*tan(1/2*d*x + 1/2*c)^5 - 425*a^5*tan(1/2*d*x
+ 1/2*c)^3 + 4560*a^4*b*tan(1/2*d*x + 1/2*c)^3 + 5220*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - 27200*a^2*b^3*tan(1/2*
d*x + 1/2*c)^3 - 5400*a*b^4*tan(1/2*d*x + 1/2*c)^3 + 25200*b^5*tan(1/2*d*x + 1/2*c)^3 - 75*a^5*tan(1/2*d*x + 1
/2*c) + 720*a^4*b*tan(1/2*d*x + 1/2*c) + 1260*a^3*b^2*tan(1/2*d*x + 1/2*c) - 4800*a^2*b^3*tan(1/2*d*x + 1/2*c)
- 1800*a*b^4*tan(1/2*d*x + 1/2*c) + 5040*b^5*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^8))/d

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Mupad [B]
time = 5.71, size = 2500, normalized size = 4.64 \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))^3,x)

[Out]

((tan(c/2 + (d*x)/2)^3*(10080*a*b^6 + 454*a^6*b - 55*a^7 + 9408*b^7 - 9688*a^2*b^5 - 12212*a^3*b^4 + 608*a^4*b
^3 + 2969*a^5*b^2))/(24*a^8) + (tan(c/2 + (d*x)/2)^13*(454*a^6*b - 10080*a*b^6 + 55*a^7 + 9408*b^7 - 9688*a^2*
b^5 + 12212*a^3*b^4 + 608*a^4*b^3 - 2969*a^5*b^2))/(24*a^8) + (tan(c/2 + (d*x)/2)^5*(90720*a*b^6 + 2154*a^6*b
- 215*a^7 + 141120*b^7 - 163240*a^2*b^5 - 107220*a^3*b^4 + 32224*a^4*b^3 + 22673*a^5*b^2))/(120*a^8) + (tan(c/
2 + (d*x)/2)^11*(2154*a^6*b - 90720*a*b^6 + 215*a^7 + 141120*b^7 - 163240*a^2*b^5 + 107220*a^3*b^4 + 32224*a^4
*b^3 - 22673*a^5*b^2))/(120*a^8) + (tan(c/2 + (d*x)/2)^7*(50400*a*b^6 - 4994*a^6*b + 2545*a^7 + 235200*b^7 - 2
87000*a^2*b^5 - 58820*a^3*b^4 + 74752*a^4*b^3 + 11173*a^5*b^2))/(120*a^8) - (tan(c/2 + (d*x)/2)^9*(50400*a*b^6
+ 4994*a^6*b + 2545*a^7 - 235200*b^7 + 287000*a^2*b^5 - 58820*a^3*b^4 - 74752*a^4*b^3 + 11173*a^5*b^2))/(120*
a^8) + (tan(c/2 + (d*x)/2)^15*(a - b)*(224*a*b^5 + 43*a^5*b + 5*a^6 - 448*b^6 + 600*a^2*b^4 - 244*a^3*b^3 - 18
0*a^4*b^2))/(8*a^8) - (tan(c/2 + (d*x)/2)*(2*a*b + a^2 + b^2)*(224*a*b^4 - 48*a^4*b + 5*a^5 - 448*b^5 + 376*a^
2*b^3 - 132*a^3*b^2))/(8*a^8))/(d*(2*a*b - tan(c/2 + (d*x)/2)^8*(10*a^2 - 70*b^2) + tan(c/2 + (d*x)/2)^2*(12*a
*b + 4*a^2 + 8*b^2) + tan(c/2 + (d*x)/2)^14*(4*a^2 - 12*a*b + 8*b^2) + tan(c/2 + (d*x)/2)^4*(28*a*b + 4*a^2 +
28*b^2) + tan(c/2 + (d*x)/2)^12*(4*a^2 - 28*a*b + 28*b^2) + tan(c/2 + (d*x)/2)^6*(28*a*b - 4*a^2 + 56*b^2) - t
an(c/2 + (d*x)/2)^10*(28*a*b + 4*a^2 - 56*b^2) + tan(c/2 + (d*x)/2)^16*(a^2 - 2*a*b + b^2) + a^2 + b^2)) + (at
an(((((((106*a^25*b - 10*a^26 + 896*a^18*b^8 - 1344*a^19*b^7 - 1200*a^20*b^6 + 2360*a^21*b^5 + 136*a^22*b^4 -
1122*a^23*b^3 + 178*a^24*b^2)/a^24 - (tan(c/2 + (d*x)/2)*(512*a^20*b + 512*a^18*b^3 - 1024*a^19*b^2)*(a^6*5i -
b^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(128*a^25))*(a^6*5i - b^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(16*a
^9) + (tan(c/2 + (d*x)/2)*(802816*a*b^14 - 75*a^14*b + 25*a^15 - 401408*b^15 + 673792*a^2*b^13 - 2150400*a^3*b
^12 + 32640*a^4*b^11 + 2085120*a^5*b^10 - 601600*a^6*b^9 - 881920*a^7*b^8 + 364160*a^8*b^7 + 153600*a^9*b^6 -
72696*a^10*b^5 - 7704*a^11*b^4 + 3071*a^12*b^3 + 579*a^13*b^2))/(8*a^16))*(a^6*5i - b^6*448i + a^2*b^4*600i -
a^4*b^2*180i)*1i)/(16*a^9) - (((((106*a^25*b - 10*a^26 + 896*a^18*b^8 - 1344*a^19*b^7 - 1200*a^20*b^6 + 2360*a
^21*b^5 + 136*a^22*b^4 - 1122*a^23*b^3 + 178*a^24*b^2)/a^24 + (tan(c/2 + (d*x)/2)*(512*a^20*b + 512*a^18*b^3 -
1024*a^19*b^2)*(a^6*5i - b^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(128*a^25))*(a^6*5i - b^6*448i + a^2*b^4*60
0i - a^4*b^2*180i))/(16*a^9) - (tan(c/2 + (d*x)/2)*(802816*a*b^14 - 75*a^14*b + 25*a^15 - 401408*b^15 + 673792
*a^2*b^13 - 2150400*a^3*b^12 + 32640*a^4*b^11 + 2085120*a^5*b^10 - 601600*a^6*b^9 - 881920*a^7*b^8 + 364160*a^
8*b^7 + 153600*a^9*b^6 - 72696*a^10*b^5 - 7704*a^11*b^4 + 3071*a^12*b^3 + 579*a^13*b^2))/(8*a^16))*(a^6*5i - b
^6*448i + a^2*b^4*600i - a^4*b^2*180i)*1i)/(16*a^9))/(((75*a^19*b)/4 - 2107392*a*b^19 + 1404928*b^20 - 5644800
*a^2*b^18 + 9345280*a^3*b^17 + 8902208*a^4*b^16 - 17144736*a^5*b^15 - 6722456*a^6*b^14 + 16804748*a^7*b^13 + 2
126380*a^8*b^12 - 9486373*a^9*b^11 + 163573*a^10*b^10 + 3099308*a^11*b^9 - 297558*a^12*b^8 - (4466945*a^13*b^7
)/8 + (296845*a^14*b^6)/4 + (196765*a^15*b^5)/4 - (26515*a^16*b^4)/4 - (13415*a^17*b^3)/8 + (285*a^18*b^2)/2)/
a^24 + (((((106*a^25*b - 10*a^26 + 896*a^18*b^8 - 1344*a^19*b^7 - 1200*a^20*b^6 + 2360*a^21*b^5 + 136*a^22*b^4
- 1122*a^23*b^3 + 178*a^24*b^2)/a^24 - (tan(c/2 + (d*x)/2)*(512*a^20*b + 512*a^18*b^3 - 1024*a^19*b^2)*(a^6*5
i - b^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(128*a^25))*(a^6*5i - b^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(1
6*a^9) + (tan(c/2 + (d*x)/2)*(802816*a*b^14 - 75*a^14*b + 25*a^15 - 401408*b^15 + 673792*a^2*b^13 - 2150400*a^
3*b^12 + 32640*a^4*b^11 + 2085120*a^5*b^10 - 601600*a^6*b^9 - 881920*a^7*b^8 + 364160*a^8*b^7 + 153600*a^9*b^6
- 72696*a^10*b^5 - 7704*a^11*b^4 + 3071*a^12*b^3 + 579*a^13*b^2))/(8*a^16))*(a^6*5i - b^6*448i + a^2*b^4*600i
- a^4*b^2*180i))/(16*a^9) + (((((106*a^25*b - 10*a^26 + 896*a^18*b^8 - 1344*a^19*b^7 - 1200*a^20*b^6 + 2360*a
^21*b^5 + 136*a^22*b^4 - 1122*a^23*b^3 + 178*a^24*b^2)/a^24 + (tan(c/2 + (d*x)/2)*(512*a^20*b + 512*a^18*b^3 -
1024*a^19*b^2)*(a^6*5i - b^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(128*a^25))*(a^6*5i - b^6*448i + a^2*b^4*60
0i - a^4*b^2*180i))/(16*a^9) - (tan(c/2 + (d*x)/2)*(802816*a*b^14 - 75*a^14*b + 25*a^15 - 401408*b^15 + 673792
*a^2*b^13 - 2150400*a^3*b^12 + 32640*a^4*b^11 + 2085120*a^5*b^10 - 601600*a^6*b^9 - 881920*a^7*b^8 + 364160*a^
8*b^7 + 153600*a^9*b^6 - 72696*a^10*b^5 - 7704*a^11*b^4 + 3071*a^12*b^3 + 579*a^13*b^2))/(8*a^16))*(a^6*5i - b
^6*448i + a^2*b^4*600i - a^4*b^2*180i))/(16*a^9)))*(a^6*5i - b^6*448i + a^2*b^4*600i - a^4*b^2*180i)*1i)/(8*a^
9*d) + (b*atan(((b*((a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(802816*a*b^14 - 75*a^14*b + 25*a^15 - 401408*
b^15 + 673792*a^2*b^13 - 2150400*a^3*b^12 + 32640*a^4*b^11 + 2085120*a^5*b^10 - 601600*a^6*b^9 - 881920*a^7*b^
8 + 364160*a^8*b^7 + 153600*a^9*b^6 - 72696*a^1...

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