\(\int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx\) [67]

Optimal result

Integrand size = 21, antiderivative size = 382 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {a \left (5 a^8-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6-3 b^8\right ) x}{16 \left (a^2+b^2\right )^6}+\frac {a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^6 b}{2 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {2 a^5 b \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d} \]

[Out]

1/16*a*(5*a^8-180*a^6*b^2+390*a^4*b^4-68*a^2*b^6-3*b^8)*x/(a^2+b^2)^6+a^4*b*(3*a^4-22*a^2*b^2+15*b^4)*ln(a*cos
(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^6/d-1/2*a^6*b/(a^2+b^2)^4/d/(a+b*tan(d*x+c))^2-2*a^5*b*(a^2-3*b^2)/(a^2+b^2)^5
/d/(a+b*tan(d*x+c))-1/6*cos(d*x+c)^6*(b*(3*a^2-b^2)+a*(a^2-3*b^2)*tan(d*x+c))/(a^2+b^2)^3/d+1/24*cos(d*x+c)^4*
(6*b*(9*a^4-4*a^2*b^2-b^4)+a*(13*a^4-62*a^2*b^2-3*b^4)*tan(d*x+c))/(a^2+b^2)^4/d-1/16*a*cos(d*x+c)^2*(24*a^3*b
*(3*a^2-5*b^2)+(11*a^6-119*a^4*b^2+65*a^2*b^4+3*b^6)*tan(d*x+c))/(a^2+b^2)^5/d

Rubi [A] (verified)

Time = 2.12 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3597, 1661, 1643, 649, 209, 266} \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=-\frac {\cos ^6(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{6 d \left (a^2+b^2\right )^3}-\frac {a^6 b}{2 d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}-\frac {2 a^5 b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^5 (a+b \tan (c+d x))}+\frac {\cos ^4(c+d x) \left (a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)+6 b \left (9 a^4-4 a^2 b^2-b^4\right )\right )}{24 d \left (a^2+b^2\right )^4}+\frac {a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^6}+\frac {a x \left (5 a^8-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6-3 b^8\right )}{16 \left (a^2+b^2\right )^6}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 d \left (a^2+b^2\right )^5} \]

[In]

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

[Out]

(a*(5*a^8 - 180*a^6*b^2 + 390*a^4*b^4 - 68*a^2*b^6 - 3*b^8)*x)/(16*(a^2 + b^2)^6) + (a^4*b*(3*a^4 - 22*a^2*b^2
+ 15*b^4)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^6*d) - (a^6*b)/(2*(a^2 + b^2)^4*d*(a + b*Tan[c +
d*x])^2) - (2*a^5*b*(a^2 - 3*b^2))/((a^2 + b^2)^5*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]^6*(b*(3*a^2 - b^2)
+ a*(a^2 - 3*b^2)*Tan[c + d*x]))/(6*(a^2 + b^2)^3*d) + (Cos[c + d*x]^4*(6*b*(9*a^4 - 4*a^2*b^2 - b^4) + a*(13*
a^4 - 62*a^2*b^2 - 3*b^4)*Tan[c + d*x]))/(24*(a^2 + b^2)^4*d) - (a*Cos[c + d*x]^2*(24*a^3*b*(3*a^2 - 5*b^2) +
(11*a^6 - 119*a^4*b^2 + 65*a^2*b^4 + 3*b^6)*Tan[c + d*x]))/(16*(a^2 + b^2)^5*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 3597

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[x^m*((a + x)^n/(b^2 + x^2)^(m/2 + 1)), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rubi steps \begin{align*} \text {integral}& = \frac {b \text {Subst}\left (\int \frac {x^6}{(a+x)^3 \left (b^2+x^2\right )^4} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}-\frac {\text {Subst}\left (\int \frac {-\frac {a^4 b^6 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac {3 a^3 b^6 \left (5 a^2+b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {3 a^2 b^4 \left (2 a^4+11 a^2 b^2-3 b^4\right ) x^2}{\left (a^2+b^2\right )^3}+\frac {5 a b^6 \left (a^2-3 b^2\right ) x^3}{\left (a^2+b^2\right )^3}-6 b^2 x^4}{(a+x)^3 \left (b^2+x^2\right )^3} \, dx,x,b \tan (c+d x)\right )}{6 b d} \\ & = -\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}+\frac {\text {Subst}\left (\int \frac {-\frac {9 a^4 b^6 \left (a^4-6 a^2 b^2+b^4\right )}{\left (a^2+b^2\right )^4}+\frac {9 a^3 b^6 \left (13 a^4+2 a^2 b^2-3 b^4\right ) x}{\left (a^2+b^2\right )^4}+\frac {3 a^2 b^4 \left (8 a^6+71 a^4 b^2-66 a^2 b^4-9 b^6\right ) x^2}{\left (a^2+b^2\right )^4}+\frac {3 a b^6 \left (13 a^4-62 a^2 b^2-3 b^4\right ) x^3}{\left (a^2+b^2\right )^4}}{(a+x)^3 \left (b^2+x^2\right )^2} \, dx,x,b \tan (c+d x)\right )}{24 b^3 d} \\ & = -\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d}-\frac {\text {Subst}\left (\int \frac {-\frac {3 a^4 b^6 \left (5 a^6-89 a^4 b^2+95 a^2 b^4-3 b^6\right )}{\left (a^2+b^2\right )^5}+\frac {9 a^3 b^6 \left (11 a^6+9 a^4 b^2-63 a^2 b^4+3 b^6\right ) x}{\left (a^2+b^2\right )^5}+\frac {9 a^2 b^6 \left (11 a^6-71 a^4 b^2-15 a^2 b^4+3 b^6\right ) x^2}{\left (a^2+b^2\right )^5}+\frac {3 a b^6 \left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) x^3}{\left (a^2+b^2\right )^5}}{(a+x)^3 \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{48 b^5 d} \\ & = -\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d}-\frac {\text {Subst}\left (\int \left (-\frac {48 a^6 b^6}{\left (a^2+b^2\right )^4 (a+x)^3}-\frac {96 a^5 b^6 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 (a+x)^2}-\frac {48 a^4 b^6 \left (3 a^4-22 a^2 b^2+15 b^4\right )}{\left (a^2+b^2\right )^6 (a+x)}+\frac {3 a b^6 \left (-5 a^8+180 a^6 b^2-390 a^4 b^4+68 a^2 b^6+3 b^8+16 a^3 \left (3 a^4-22 a^2 b^2+15 b^4\right ) x\right )}{\left (a^2+b^2\right )^6 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{48 b^5 d} \\ & = \frac {a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^6 b}{2 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {2 a^5 b \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d}-\frac {(a b) \text {Subst}\left (\int \frac {-5 a^8+180 a^6 b^2-390 a^4 b^4+68 a^2 b^6+3 b^8+16 a^3 \left (3 a^4-22 a^2 b^2+15 b^4\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^6 d} \\ & = \frac {a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^6 b}{2 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {2 a^5 b \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d}-\frac {\left (a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^6 d}-\frac {\left (a b \left (-5 a^8+180 a^6 b^2-390 a^4 b^4+68 a^2 b^6+3 b^8\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^6 d} \\ & = \frac {a \left (5 a^8-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6-3 b^8\right ) x}{16 \left (a^2+b^2\right )^6}+\frac {a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^6 d}+\frac {a^4 b \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6 d}-\frac {a^6 b}{2 \left (a^2+b^2\right )^4 d (a+b \tan (c+d x))^2}-\frac {2 a^5 b \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (b \left (3 a^2-b^2\right )+a \left (a^2-3 b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) \left (6 b \left (9 a^4-4 a^2 b^2-b^4\right )+a \left (13 a^4-62 a^2 b^2-3 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^4 d}-\frac {a \cos ^2(c+d x) \left (24 a^3 b \left (3 a^2-5 b^2\right )+\left (11 a^6-119 a^4 b^2+65 a^2 b^4+3 b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.72 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.95 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {b \left (-\frac {3 a^5 \left (a^2-7 b^2\right ) \arctan (\tan (c+d x))}{2 b \left (a^2+b^2\right )^5}-\frac {5 a \left (a^2-3 b^2\right ) \arctan (\tan (c+d x))}{16 b \left (a^2+b^2\right )^3}+\frac {9 a \left (a^4-4 a^2 b^2-b^4\right ) \arctan (\tan (c+d x))}{8 b \left (a^2+b^2\right )^4}-\frac {3 a^4 \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^5}+\frac {\left (9 a^4-4 a^2 b^2-b^4\right ) \cos ^4(c+d x)}{4 \left (a^2+b^2\right )^4}-\frac {\left (3 a^2-b^2\right ) \cos ^6(c+d x)}{6 \left (a^2+b^2\right )^3}-\frac {a^4 \left (3 a^4-22 a^2 b^2+15 b^4-\frac {a^5-18 a^3 b^2+21 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^6}+\frac {a^4 \left (3 a^4-22 a^2 b^2+15 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6}-\frac {a^4 \left (3 a^4-22 a^2 b^2+15 b^4+\frac {a^5-18 a^3 b^2+21 a b^4}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^6}-\frac {3 a^5 \left (a^2-7 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b \left (a^2+b^2\right )^5}-\frac {5 a \left (a^2-3 b^2\right ) \cos (c+d x) \sin (c+d x)}{16 b \left (a^2+b^2\right )^3}+\frac {9 a \left (a^4-4 a^2 b^2-b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b \left (a^2+b^2\right )^4}-\frac {5 a \left (a^2-3 b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 b \left (a^2+b^2\right )^3}+\frac {3 a \left (a^4-4 a^2 b^2-b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{4 b \left (a^2+b^2\right )^4}-\frac {a \left (a^2-3 b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{6 b \left (a^2+b^2\right )^3}-\frac {a^6}{2 \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}-\frac {2 a^5 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 (a+b \tan (c+d x))}\right )}{d} \]

[In]

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

[Out]

(b*((-3*a^5*(a^2 - 7*b^2)*ArcTan[Tan[c + d*x]])/(2*b*(a^2 + b^2)^5) - (5*a*(a^2 - 3*b^2)*ArcTan[Tan[c + d*x]])
/(16*b*(a^2 + b^2)^3) + (9*a*(a^4 - 4*a^2*b^2 - b^4)*ArcTan[Tan[c + d*x]])/(8*b*(a^2 + b^2)^4) - (3*a^4*(3*a^2
- 5*b^2)*Cos[c + d*x]^2)/(2*(a^2 + b^2)^5) + ((9*a^4 - 4*a^2*b^2 - b^4)*Cos[c + d*x]^4)/(4*(a^2 + b^2)^4) - (
(3*a^2 - b^2)*Cos[c + d*x]^6)/(6*(a^2 + b^2)^3) - (a^4*(3*a^4 - 22*a^2*b^2 + 15*b^4 - (a^5 - 18*a^3*b^2 + 21*a
*b^4)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Tan[c + d*x]])/(2*(a^2 + b^2)^6) + (a^4*(3*a^4 - 22*a^2*b^2 + 15*b^4)*Log
[a + b*Tan[c + d*x]])/(a^2 + b^2)^6 - (a^4*(3*a^4 - 22*a^2*b^2 + 15*b^4 + (a^5 - 18*a^3*b^2 + 21*a*b^4)/Sqrt[-
b^2])*Log[Sqrt[-b^2] + b*Tan[c + d*x]])/(2*(a^2 + b^2)^6) - (3*a^5*(a^2 - 7*b^2)*Cos[c + d*x]*Sin[c + d*x])/(2
*b*(a^2 + b^2)^5) - (5*a*(a^2 - 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*b*(a^2 + b^2)^3) + (9*a*(a^4 - 4*a^2*b^2
- b^4)*Cos[c + d*x]*Sin[c + d*x])/(8*b*(a^2 + b^2)^4) - (5*a*(a^2 - 3*b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*b
*(a^2 + b^2)^3) + (3*a*(a^4 - 4*a^2*b^2 - b^4)*Cos[c + d*x]^3*Sin[c + d*x])/(4*b*(a^2 + b^2)^4) - (a*(a^2 - 3*
b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(6*b*(a^2 + b^2)^3) - a^6/(2*(a^2 + b^2)^4*(a + b*Tan[c + d*x])^2) - (2*a^5*
(a^2 - 3*b^2))/((a^2 + b^2)^5*(a + b*Tan[c + d*x]))))/d

Maple [A] (verified)

Time = 70.94 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.22

method	result	size
derivativedivides	\(\frac {-\frac {b \,a^{6}}{2 \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{4} b \left (3 a^{4}-22 a^{2} b^{2}+15 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{6}}-\frac {2 b \,a^{5} \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{5} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-\frac {11}{16} a^{9}+\frac {27}{4} b^{2} a^{7}+\frac {27}{8} b^{4} a^{5}-\frac {17}{4} b^{6} a^{3}-\frac {3}{16} a \,b^{8}\right ) \left (\tan ^{5}\left (d x +c \right )\right )+\left (-\frac {9}{2} a^{8} b +3 a^{6} b^{3}+\frac {15}{2} a^{4} b^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )+\left (-\frac {5}{6} a^{9}+12 b^{2} a^{7}+2 b^{4} a^{5}-\frac {34}{3} b^{6} a^{3}-\frac {1}{2} a \,b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-\frac {27}{4} a^{8} b +\frac {19}{2} a^{6} b^{3}+15 a^{4} b^{5}-\frac {3}{2} a^{2} b^{7}-\frac {1}{4} b^{9}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {5}{16} a^{9}+\frac {21}{4} b^{2} a^{7}-\frac {3}{8} b^{4} a^{5}-\frac {23}{4} b^{6} a^{3}+\frac {3}{16} a \,b^{8}\right ) \tan \left (d x +c \right )-\frac {11 a^{8} b}{4}+\frac {31 a^{6} b^{3}}{6}+\frac {13 a^{4} b^{5}}{2}-\frac {3 a^{2} b^{7}}{2}-\frac {b^{9}}{12}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {a \left (\frac {\left (-48 b \,a^{7}+352 b^{3} a^{5}-240 a^{3} b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (5 a^{8}-180 a^{6} b^{2}+390 a^{4} b^{4}-68 b^{6} a^{2}-3 b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{16}}{\left (a^{2}+b^{2}\right )^{6}}}{d}\)	\(466\)
default	\(\frac {-\frac {b \,a^{6}}{2 \left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{4} b \left (3 a^{4}-22 a^{2} b^{2}+15 b^{4}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{6}}-\frac {2 b \,a^{5} \left (a^{2}-3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{5} \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-\frac {11}{16} a^{9}+\frac {27}{4} b^{2} a^{7}+\frac {27}{8} b^{4} a^{5}-\frac {17}{4} b^{6} a^{3}-\frac {3}{16} a \,b^{8}\right ) \left (\tan ^{5}\left (d x +c \right )\right )+\left (-\frac {9}{2} a^{8} b +3 a^{6} b^{3}+\frac {15}{2} a^{4} b^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )+\left (-\frac {5}{6} a^{9}+12 b^{2} a^{7}+2 b^{4} a^{5}-\frac {34}{3} b^{6} a^{3}-\frac {1}{2} a \,b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-\frac {27}{4} a^{8} b +\frac {19}{2} a^{6} b^{3}+15 a^{4} b^{5}-\frac {3}{2} a^{2} b^{7}-\frac {1}{4} b^{9}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {5}{16} a^{9}+\frac {21}{4} b^{2} a^{7}-\frac {3}{8} b^{4} a^{5}-\frac {23}{4} b^{6} a^{3}+\frac {3}{16} a \,b^{8}\right ) \tan \left (d x +c \right )-\frac {11 a^{8} b}{4}+\frac {31 a^{6} b^{3}}{6}+\frac {13 a^{4} b^{5}}{2}-\frac {3 a^{2} b^{7}}{2}-\frac {b^{9}}{12}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {a \left (\frac {\left (-48 b \,a^{7}+352 b^{3} a^{5}-240 a^{3} b^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (5 a^{8}-180 a^{6} b^{2}+390 a^{4} b^{4}-68 b^{6} a^{2}-3 b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )\right )}{16}}{\left (a^{2}+b^{2}\right )^{6}}}{d}\)	\(466\)
risch	\(\text {Expression too large to display}\)	\(1412\)

[In]

int(sin(d*x+c)^6/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/2*b*a^6/(a^2+b^2)^4/(a+b*tan(d*x+c))^2+a^4*b*(3*a^4-22*a^2*b^2+15*b^4)/(a^2+b^2)^6*ln(a+b*tan(d*x+c))-
2*b*a^5*(a^2-3*b^2)/(a^2+b^2)^5/(a+b*tan(d*x+c))+1/(a^2+b^2)^6*(((-11/16*a^9+27/4*b^2*a^7+27/8*b^4*a^5-17/4*b^
6*a^3-3/16*a*b^8)*tan(d*x+c)^5+(-9/2*a^8*b+3*a^6*b^3+15/2*a^4*b^5)*tan(d*x+c)^4+(-5/6*a^9+12*b^2*a^7+2*b^4*a^5
-34/3*b^6*a^3-1/2*a*b^8)*tan(d*x+c)^3+(-27/4*a^8*b+19/2*a^6*b^3+15*a^4*b^5-3/2*a^2*b^7-1/4*b^9)*tan(d*x+c)^2+(
-5/16*a^9+21/4*b^2*a^7-3/8*b^4*a^5-23/4*b^6*a^3+3/16*a*b^8)*tan(d*x+c)-11/4*a^8*b+31/6*a^6*b^3+13/2*a^4*b^5-3/
2*a^2*b^7-1/12*b^9)/(1+tan(d*x+c)^2)^3+1/16*a*(1/2*(-48*a^7*b+352*a^5*b^3-240*a^3*b^5)*ln(1+tan(d*x+c)^2)+(5*a
^8-180*a^6*b^2+390*a^4*b^4-68*a^2*b^6-3*b^8)*arctan(tan(d*x+c)))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (372) = 744\).

Time = 0.38 (sec) , antiderivative size = 932, normalized size of antiderivative = 2.44 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {195 \, a^{8} b^{3} - 427 \, a^{6} b^{5} - 165 \, a^{4} b^{7} + 27 \, a^{2} b^{9} + 2 \, b^{11} - 8 \, {\left (a^{10} b + 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} + 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} + b^{11}\right )} \cos \left (d x + c\right )^{8} + 20 \, {\left (2 \, a^{10} b + 9 \, a^{8} b^{3} + 16 \, a^{6} b^{5} + 14 \, a^{4} b^{7} + 6 \, a^{2} b^{9} + b^{11}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (49 \, a^{10} b + 162 \, a^{8} b^{3} + 198 \, a^{6} b^{5} + 112 \, a^{4} b^{7} + 33 \, a^{2} b^{9} + 6 \, b^{11}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{9} b^{2} - 180 \, a^{7} b^{4} + 390 \, a^{5} b^{6} - 68 \, a^{3} b^{8} - 3 \, a b^{10}\right )} d x + {\left (9 \, a^{10} b - 46 \, a^{8} b^{3} + 994 \, a^{6} b^{5} + 144 \, a^{4} b^{7} - 43 \, a^{2} b^{9} - 2 \, b^{11} + 3 \, {\left (5 \, a^{11} - 185 \, a^{9} b^{2} + 570 \, a^{7} b^{4} - 458 \, a^{5} b^{6} + 65 \, a^{3} b^{8} + 3 \, a b^{10}\right )} d x\right )} \cos \left (d x + c\right )^{2} + 24 \, {\left (3 \, a^{8} b^{3} - 22 \, a^{6} b^{5} + 15 \, a^{4} b^{7} + {\left (3 \, a^{10} b - 25 \, a^{8} b^{3} + 37 \, a^{6} b^{5} - 15 \, a^{4} b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{9} b^{2} - 22 \, a^{7} b^{4} + 15 \, a^{5} b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - {\left (8 \, {\left (a^{11} + 5 \, a^{9} b^{2} + 10 \, a^{7} b^{4} + 10 \, a^{5} b^{6} + 5 \, a^{3} b^{8} + a b^{10}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (13 \, a^{11} + 55 \, a^{9} b^{2} + 90 \, a^{7} b^{4} + 70 \, a^{5} b^{6} + 25 \, a^{3} b^{8} + 3 \, a b^{10}\right )} \cos \left (d x + c\right )^{5} + {\left (33 \, a^{11} + 49 \, a^{9} b^{2} - 54 \, a^{7} b^{4} - 126 \, a^{5} b^{6} - 59 \, a^{3} b^{8} - 3 \, a b^{10}\right )} \cos \left (d x + c\right )^{3} - {\left (261 \, a^{9} b^{2} - 338 \, a^{7} b^{4} + 120 \, a^{5} b^{6} - 150 \, a^{3} b^{8} - 5 \, a b^{10} + 6 \, {\left (5 \, a^{10} b - 180 \, a^{8} b^{3} + 390 \, a^{6} b^{5} - 68 \, a^{4} b^{7} - 3 \, a^{2} b^{9}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left ({\left (a^{14} + 5 \, a^{12} b^{2} + 9 \, a^{10} b^{4} + 5 \, a^{8} b^{6} - 5 \, a^{6} b^{8} - 9 \, a^{4} b^{10} - 5 \, a^{2} b^{12} - b^{14}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{13} b + 6 \, a^{11} b^{3} + 15 \, a^{9} b^{5} + 20 \, a^{7} b^{7} + 15 \, a^{5} b^{9} + 6 \, a^{3} b^{11} + a b^{13}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{12} b^{2} + 6 \, a^{10} b^{4} + 15 \, a^{8} b^{6} + 20 \, a^{6} b^{8} + 15 \, a^{4} b^{10} + 6 \, a^{2} b^{12} + b^{14}\right )} d\right )}} \]

[In]

integrate(sin(d*x+c)^6/(a+b*tan(d*x+c))^3,x, algorithm="fricas")

[Out]

1/48*(195*a^8*b^3 - 427*a^6*b^5 - 165*a^4*b^7 + 27*a^2*b^9 + 2*b^11 - 8*(a^10*b + 5*a^8*b^3 + 10*a^6*b^5 + 10*
a^4*b^7 + 5*a^2*b^9 + b^11)*cos(d*x + c)^8 + 20*(2*a^10*b + 9*a^8*b^3 + 16*a^6*b^5 + 14*a^4*b^7 + 6*a^2*b^9 +
b^11)*cos(d*x + c)^6 - 2*(49*a^10*b + 162*a^8*b^3 + 198*a^6*b^5 + 112*a^4*b^7 + 33*a^2*b^9 + 6*b^11)*cos(d*x +
c)^4 + 3*(5*a^9*b^2 - 180*a^7*b^4 + 390*a^5*b^6 - 68*a^3*b^8 - 3*a*b^10)*d*x + (9*a^10*b - 46*a^8*b^3 + 994*a
^6*b^5 + 144*a^4*b^7 - 43*a^2*b^9 - 2*b^11 + 3*(5*a^11 - 185*a^9*b^2 + 570*a^7*b^4 - 458*a^5*b^6 + 65*a^3*b^8
+ 3*a*b^10)*d*x)*cos(d*x + c)^2 + 24*(3*a^8*b^3 - 22*a^6*b^5 + 15*a^4*b^7 + (3*a^10*b - 25*a^8*b^3 + 37*a^6*b^
5 - 15*a^4*b^7)*cos(d*x + c)^2 + 2*(3*a^9*b^2 - 22*a^7*b^4 + 15*a^5*b^6)*cos(d*x + c)*sin(d*x + c))*log(2*a*b*
cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) - (8*(a^11 + 5*a^9*b^2 + 10*a^7*b^4 + 10*a^5*b^6
+ 5*a^3*b^8 + a*b^10)*cos(d*x + c)^7 - 2*(13*a^11 + 55*a^9*b^2 + 90*a^7*b^4 + 70*a^5*b^6 + 25*a^3*b^8 + 3*a*b
^10)*cos(d*x + c)^5 + (33*a^11 + 49*a^9*b^2 - 54*a^7*b^4 - 126*a^5*b^6 - 59*a^3*b^8 - 3*a*b^10)*cos(d*x + c)^3
- (261*a^9*b^2 - 338*a^7*b^4 + 120*a^5*b^6 - 150*a^3*b^8 - 5*a*b^10 + 6*(5*a^10*b - 180*a^8*b^3 + 390*a^6*b^5
- 68*a^4*b^7 - 3*a^2*b^9)*d*x)*cos(d*x + c))*sin(d*x + c))/((a^14 + 5*a^12*b^2 + 9*a^10*b^4 + 5*a^8*b^6 - 5*a
^6*b^8 - 9*a^4*b^10 - 5*a^2*b^12 - b^14)*d*cos(d*x + c)^2 + 2*(a^13*b + 6*a^11*b^3 + 15*a^9*b^5 + 20*a^7*b^7 +
15*a^5*b^9 + 6*a^3*b^11 + a*b^13)*d*cos(d*x + c)*sin(d*x + c) + (a^12*b^2 + 6*a^10*b^4 + 15*a^8*b^6 + 20*a^6*
b^8 + 15*a^4*b^10 + 6*a^2*b^12 + b^14)*d)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

integrate(sin(d*x+c)**6/(a+b*tan(d*x+c))**3,x)

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1088 vs. \(2 (372) = 744\).

Time = 0.34 (sec) , antiderivative size = 1088, normalized size of antiderivative = 2.85 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]

[In]

integrate(sin(d*x+c)^6/(a+b*tan(d*x+c))^3,x, algorithm="maxima")

[Out]

1/48*(3*(5*a^9 - 180*a^7*b^2 + 390*a^5*b^4 - 68*a^3*b^6 - 3*a*b^8)*(d*x + c)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) + 48*(3*a^8*b - 22*a^6*b^3 + 15*a^4*b^5)*log(b*tan(d*x + c) + a)
/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) - 24*(3*a^8*b - 22*a^6*b^3 + 1
5*a^4*b^5)*log(tan(d*x + c)^2 + 1)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^
12) - (252*a^8*b - 644*a^6*b^3 + 68*a^4*b^5 + 4*a^2*b^7 + 3*(43*a^7*b^2 - 215*a^5*b^4 + 65*a^3*b^6 + 3*a*b^8)*
tan(d*x + c)^7 + 6*(31*a^8*b - 127*a^6*b^3 + 5*a^4*b^5 + 3*a^2*b^7)*tan(d*x + c)^6 + (33*a^9 + 403*a^7*b^2 - 2
005*a^5*b^4 + 529*a^3*b^6 + 24*a*b^8)*tan(d*x + c)^5 + 4*(164*a^8*b - 515*a^6*b^3 + 65*a^4*b^5 + 27*a^2*b^7 +
3*b^9)*tan(d*x + c)^4 + (40*a^9 + 335*a^7*b^2 - 2171*a^5*b^4 + 429*a^3*b^6 + 15*a*b^8)*tan(d*x + c)^3 + 2*(357
*a^8*b - 987*a^6*b^3 + 125*a^4*b^5 + 31*a^2*b^7 + 2*b^9)*tan(d*x + c)^2 + (15*a^9 + 93*a^7*b^2 - 763*a^5*b^4 +
127*a^3*b^6 + 8*a*b^8)*tan(d*x + c))/(a^12 + 5*a^10*b^2 + 10*a^8*b^4 + 10*a^6*b^6 + 5*a^4*b^8 + a^2*b^10 + (a
^10*b^2 + 5*a^8*b^4 + 10*a^6*b^6 + 10*a^4*b^8 + 5*a^2*b^10 + b^12)*tan(d*x + c)^8 + 2*(a^11*b + 5*a^9*b^3 + 10
*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*tan(d*x + c)^7 + (a^12 + 8*a^10*b^2 + 25*a^8*b^4 + 40*a^6*b^6 + 35
*a^4*b^8 + 16*a^2*b^10 + 3*b^12)*tan(d*x + c)^6 + 6*(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9
+ a*b^11)*tan(d*x + c)^5 + 3*(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*ta
n(d*x + c)^4 + 6*(a^11*b + 5*a^9*b^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*tan(d*x + c)^3 + (3*a^12
+ 16*a^10*b^2 + 35*a^8*b^4 + 40*a^6*b^6 + 25*a^4*b^8 + 8*a^2*b^10 + b^12)*tan(d*x + c)^2 + 2*(a^11*b + 5*a^9*b
^3 + 10*a^7*b^5 + 10*a^5*b^7 + 5*a^3*b^9 + a*b^11)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 923 vs. \(2 (372) = 744\).

Time = 0.69 (sec) , antiderivative size = 923, normalized size of antiderivative = 2.42 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {3 \, {\left (5 \, a^{9} - 180 \, a^{7} b^{2} + 390 \, a^{5} b^{4} - 68 \, a^{3} b^{6} - 3 \, a b^{8}\right )} {\left (d x + c\right )}}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} - \frac {24 \, {\left (3 \, a^{8} b - 22 \, a^{6} b^{3} + 15 \, a^{4} b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}} + \frac {48 \, {\left (3 \, a^{8} b^{2} - 22 \, a^{6} b^{4} + 15 \, a^{4} b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{12} b + 6 \, a^{10} b^{3} + 15 \, a^{8} b^{5} + 20 \, a^{6} b^{7} + 15 \, a^{4} b^{9} + 6 \, a^{2} b^{11} + b^{13}} - \frac {24 \, {\left (9 \, a^{8} b^{3} \tan \left (d x + c\right )^{2} - 66 \, a^{6} b^{5} \tan \left (d x + c\right )^{2} + 45 \, a^{4} b^{7} \tan \left (d x + c\right )^{2} + 22 \, a^{9} b^{2} \tan \left (d x + c\right ) - 140 \, a^{7} b^{4} \tan \left (d x + c\right ) + 78 \, a^{5} b^{6} \tan \left (d x + c\right ) + 14 \, a^{10} b - 72 \, a^{8} b^{3} + 34 \, a^{6} b^{5}\right )}}{{\left (a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} + \frac {132 \, a^{8} b \tan \left (d x + c\right )^{6} - 968 \, a^{6} b^{3} \tan \left (d x + c\right )^{6} + 660 \, a^{4} b^{5} \tan \left (d x + c\right )^{6} - 33 \, a^{9} \tan \left (d x + c\right )^{5} + 324 \, a^{7} b^{2} \tan \left (d x + c\right )^{5} + 162 \, a^{5} b^{4} \tan \left (d x + c\right )^{5} - 204 \, a^{3} b^{6} \tan \left (d x + c\right )^{5} - 9 \, a b^{8} \tan \left (d x + c\right )^{5} + 180 \, a^{8} b \tan \left (d x + c\right )^{4} - 2760 \, a^{6} b^{3} \tan \left (d x + c\right )^{4} + 2340 \, a^{4} b^{5} \tan \left (d x + c\right )^{4} - 40 \, a^{9} \tan \left (d x + c\right )^{3} + 576 \, a^{7} b^{2} \tan \left (d x + c\right )^{3} + 96 \, a^{5} b^{4} \tan \left (d x + c\right )^{3} - 544 \, a^{3} b^{6} \tan \left (d x + c\right )^{3} - 24 \, a b^{8} \tan \left (d x + c\right )^{3} + 72 \, a^{8} b \tan \left (d x + c\right )^{2} - 2448 \, a^{6} b^{3} \tan \left (d x + c\right )^{2} + 2700 \, a^{4} b^{5} \tan \left (d x + c\right )^{2} - 72 \, a^{2} b^{7} \tan \left (d x + c\right )^{2} - 12 \, b^{9} \tan \left (d x + c\right )^{2} - 15 \, a^{9} \tan \left (d x + c\right ) + 252 \, a^{7} b^{2} \tan \left (d x + c\right ) - 18 \, a^{5} b^{4} \tan \left (d x + c\right ) - 276 \, a^{3} b^{6} \tan \left (d x + c\right ) + 9 \, a b^{8} \tan \left (d x + c\right ) - 720 \, a^{6} b^{3} + 972 \, a^{4} b^{5} - 72 \, a^{2} b^{7} - 4 \, b^{9}}{{\left (a^{12} + 6 \, a^{10} b^{2} + 15 \, a^{8} b^{4} + 20 \, a^{6} b^{6} + 15 \, a^{4} b^{8} + 6 \, a^{2} b^{10} + b^{12}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3}}}{48 \, d} \]

[In]

integrate(sin(d*x+c)^6/(a+b*tan(d*x+c))^3,x, algorithm="giac")

[Out]

1/48*(3*(5*a^9 - 180*a^7*b^2 + 390*a^5*b^4 - 68*a^3*b^6 - 3*a*b^8)*(d*x + c)/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 +
20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) - 24*(3*a^8*b - 22*a^6*b^3 + 15*a^4*b^5)*log(tan(d*x + c)^2 + 1)
/(a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12) + 48*(3*a^8*b^2 - 22*a^6*b^4 +
15*a^4*b^6)*log(abs(b*tan(d*x + c) + a))/(a^12*b + 6*a^10*b^3 + 15*a^8*b^5 + 20*a^6*b^7 + 15*a^4*b^9 + 6*a^2*
b^11 + b^13) - 24*(9*a^8*b^3*tan(d*x + c)^2 - 66*a^6*b^5*tan(d*x + c)^2 + 45*a^4*b^7*tan(d*x + c)^2 + 22*a^9*b
^2*tan(d*x + c) - 140*a^7*b^4*tan(d*x + c) + 78*a^5*b^6*tan(d*x + c) + 14*a^10*b - 72*a^8*b^3 + 34*a^6*b^5)/((
a^12 + 6*a^10*b^2 + 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*(b*tan(d*x + c) + a)^2) + (132*a
^8*b*tan(d*x + c)^6 - 968*a^6*b^3*tan(d*x + c)^6 + 660*a^4*b^5*tan(d*x + c)^6 - 33*a^9*tan(d*x + c)^5 + 324*a^
7*b^2*tan(d*x + c)^5 + 162*a^5*b^4*tan(d*x + c)^5 - 204*a^3*b^6*tan(d*x + c)^5 - 9*a*b^8*tan(d*x + c)^5 + 180*
a^8*b*tan(d*x + c)^4 - 2760*a^6*b^3*tan(d*x + c)^4 + 2340*a^4*b^5*tan(d*x + c)^4 - 40*a^9*tan(d*x + c)^3 + 576
*a^7*b^2*tan(d*x + c)^3 + 96*a^5*b^4*tan(d*x + c)^3 - 544*a^3*b^6*tan(d*x + c)^3 - 24*a*b^8*tan(d*x + c)^3 + 7
2*a^8*b*tan(d*x + c)^2 - 2448*a^6*b^3*tan(d*x + c)^2 + 2700*a^4*b^5*tan(d*x + c)^2 - 72*a^2*b^7*tan(d*x + c)^2
- 12*b^9*tan(d*x + c)^2 - 15*a^9*tan(d*x + c) + 252*a^7*b^2*tan(d*x + c) - 18*a^5*b^4*tan(d*x + c) - 276*a^3*
b^6*tan(d*x + c) + 9*a*b^8*tan(d*x + c) - 720*a^6*b^3 + 972*a^4*b^5 - 72*a^2*b^7 - 4*b^9)/((a^12 + 6*a^10*b^2
+ 15*a^8*b^4 + 20*a^6*b^6 + 15*a^4*b^8 + 6*a^2*b^10 + b^12)*(tan(d*x + c)^2 + 1)^3))/d

Mupad [B] (verification not implemented)

Time = 6.86 (sec) , antiderivative size = 1068, normalized size of antiderivative = 2.80 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {3\,b}{{\left (a^2+b^2\right )}^2}-\frac {34\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {99\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {108\,b^7}{{\left (a^2+b^2\right )}^5}+\frac {40\,b^9}{{\left (a^2+b^2\right )}^6}\right )}{d}-\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (31\,a^8\,b-127\,a^6\,b^3+5\,a^4\,b^5+3\,a^2\,b^7\right )}{8\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (43\,a^7\,b^2-215\,a^5\,b^4+65\,a^3\,b^6+3\,a\,b^8\right )}{16\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (33\,a^9+403\,a^7\,b^2-2005\,a^5\,b^4+529\,a^3\,b^6+24\,a\,b^8\right )}{48\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (164\,a^8\,b-515\,a^6\,b^3+65\,a^4\,b^5+27\,a^2\,b^7+3\,b^9\right )}{12\,\left (a^{10}+5\,a^8\,b^2+10\,a^6\,b^4+10\,a^4\,b^6+5\,a^2\,b^8+b^{10}\right )}+\frac {a^2\,\left (63\,a^6\,b-161\,a^4\,b^3+17\,a^2\,b^5+b^7\right )}{12\,\left (a^2+b^2\right )\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (40\,a^9+335\,a^7\,b^2-2171\,a^5\,b^4+429\,a^3\,b^6+15\,a\,b^8\right )}{48\,\left (a^2+b^2\right )\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (357\,a^8\,b-987\,a^6\,b^3+125\,a^4\,b^5+31\,a^2\,b^7+2\,b^9\right )}{24\,\left (a^2+b^2\right )\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (15\,a^8+93\,a^6\,b^2-763\,a^4\,b^4+127\,a^2\,b^6+8\,b^8\right )}{48\,\left (a^2+b^2\right )\,\left (a^8+4\,a^6\,b^2+6\,a^4\,b^4+4\,a^2\,b^6+b^8\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (3\,a^2+b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (a^2+3\,b^2\right )+a^2+{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (3\,a^2+3\,b^2\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^8+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+6\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+6\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^7\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (a^3\,5{}\mathrm {i}-18\,a^2\,b+a\,b^2\,3{}\mathrm {i}\right )}{32\,d\,\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (5\,a^3-a^2\,b\,18{}\mathrm {i}+3\,a\,b^2\right )}{32\,d\,\left (-a^6\,1{}\mathrm {i}+6\,a^5\,b+a^4\,b^2\,15{}\mathrm {i}-20\,a^3\,b^3-a^2\,b^4\,15{}\mathrm {i}+6\,a\,b^5+b^6\,1{}\mathrm {i}\right )} \]

[In]

int(sin(c + d*x)^6/(a + b*tan(c + d*x))^3,x)

[Out]

(log(a + b*tan(c + d*x))*((3*b)/(a^2 + b^2)^2 - (34*b^3)/(a^2 + b^2)^3 + (99*b^5)/(a^2 + b^2)^4 - (108*b^7)/(a
^2 + b^2)^5 + (40*b^9)/(a^2 + b^2)^6))/d - ((tan(c + d*x)^6*(31*a^8*b + 3*a^2*b^7 + 5*a^4*b^5 - 127*a^6*b^3))/
(8*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^7*(3*a*b^8 + 65*a^3*b^6 -
215*a^5*b^4 + 43*a^7*b^2))/(16*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x
)^5*(24*a*b^8 + 33*a^9 + 529*a^3*b^6 - 2005*a^5*b^4 + 403*a^7*b^2))/(48*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6
+ 10*a^6*b^4 + 5*a^8*b^2)) + (tan(c + d*x)^4*(164*a^8*b + 3*b^9 + 27*a^2*b^7 + 65*a^4*b^5 - 515*a^6*b^3))/(12*
(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2)) + (a^2*(63*a^6*b + b^7 + 17*a^2*b^5 - 161*a^4
*b^3))/(12*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)^3*(15*a*b^8 + 40*a^9 +
429*a^3*b^6 - 2171*a^5*b^4 + 335*a^7*b^2))/(48*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) +
(tan(c + d*x)^2*(357*a^8*b + 2*b^9 + 31*a^2*b^7 + 125*a^4*b^5 - 987*a^6*b^3))/(24*(a^2 + b^2)*(a^8 + b^8 + 4*
a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (a*tan(c + d*x)*(15*a^8 + 8*b^8 + 127*a^2*b^6 - 763*a^4*b^4 + 93*a^6*b^2))
/(48*(a^2 + b^2)*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)))/(d*(tan(c + d*x)^2*(3*a^2 + b^2) + tan(c +
d*x)^6*(a^2 + 3*b^2) + a^2 + tan(c + d*x)^4*(3*a^2 + 3*b^2) + b^2*tan(c + d*x)^8 + 2*a*b*tan(c + d*x) + 6*a*b*
tan(c + d*x)^3 + 6*a*b*tan(c + d*x)^5 + 2*a*b*tan(c + d*x)^7)) - (log(tan(c + d*x) + 1i)*(a*b^2*3i - 18*a^2*b
+ a^3*5i))/(32*d*(a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)) - (log(tan(c + d*x
) - 1i)*(3*a*b^2 - a^2*b*18i + 5*a^3))/(32*d*(6*a*b^5 + 6*a^5*b - a^6*1i + b^6*1i - a^2*b^4*15i - 20*a^3*b^3 +
a^4*b^2*15i))