\(\int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [61]

Optimal result

Integrand size = 21, antiderivative size = 297 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right ) x}{16 \left (a^2+b^2\right )^5}+\frac {2 a^5 b \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^6 b}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d} \]

[Out]

1/16*(5*a^8-80*a^6*b^2+50*a^4*b^4+8*a^2*b^6+b^8)*x/(a^2+b^2)^5+2*a^5*b*(a^2-3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+c
))/(a^2+b^2)^5/d-a^6*b/(a^2+b^2)^4/d/(a+b*tan(d*x+c))-1/6*cos(d*x+c)^6*(2*a*b+(a^2-b^2)*tan(d*x+c))/(a^2+b^2)^
2/d+1/24*cos(d*x+c)^4*(12*a*b*(3*a^2+b^2)+(13*a^4-18*a^2*b^2-7*b^4)*tan(d*x+c))/(a^2+b^2)^3/d-1/16*cos(d*x+c)^
2*(48*a^5*b+(11*a^6-43*a^4*b^2-7*a^2*b^4-b^6)*tan(d*x+c))/(a^2+b^2)^4/d

Rubi [A] (verified)

Time = 1.27 (sec) , antiderivative size = 297, 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} \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {\cos ^6(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{6 d \left (a^2+b^2\right )^2}-\frac {a^6 b}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}+\frac {2 a^5 b \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 d \left (a^2+b^2\right )^3}+\frac {x \left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right )}{16 \left (a^2+b^2\right )^5}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 d \left (a^2+b^2\right )^4} \]

[In]

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

[Out]

((5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*x)/(16*(a^2 + b^2)^5) + (2*a^5*b*(a^2 - 3*b^2)*Log[a*Cos[
c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)^5*d) - (a^6*b)/((a^2 + b^2)^4*d*(a + b*Tan[c + d*x])) - (Cos[c + d*x]
^6*(2*a*b + (a^2 - b^2)*Tan[c + d*x]))/(6*(a^2 + b^2)^2*d) + (Cos[c + d*x]^4*(12*a*b*(3*a^2 + b^2) + (13*a^4 -
18*a^2*b^2 - 7*b^4)*Tan[c + d*x]))/(24*(a^2 + b^2)^3*d) - (Cos[c + d*x]^2*(48*a^5*b + (11*a^6 - 43*a^4*b^2 -
7*a^2*b^4 - b^6)*Tan[c + d*x]))/(16*(a^2 + b^2)^4*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)^2 \left (b^2+x^2\right )^4} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = -\frac {\cos ^6(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}-\frac {\text {Subst}\left (\int \frac {-\frac {a^2 b^6 \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2}+\frac {2 a b^6 \left (5 a^2+b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac {b^4 \left (6 a^4+17 a^2 b^2+b^4\right ) x^2}{\left (a^2+b^2\right )^2}-6 b^2 x^4}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {-\frac {3 a^2 b^6 \left (3 a^4-6 a^2 b^2-b^4\right )}{\left (a^2+b^2\right )^3}+\frac {6 a b^6 \left (13 a^4+6 a^2 b^2+b^4\right ) x}{\left (a^2+b^2\right )^3}+\frac {3 b^4 \left (8 a^6+37 a^4 b^2+6 a^2 b^4+b^6\right ) x^2}{\left (a^2+b^2\right )^3}}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d}-\frac {\text {Subst}\left (\int \frac {-\frac {3 a^2 b^6 \left (5 a^6-37 a^4 b^2+7 a^2 b^4+b^6\right )}{\left (a^2+b^2\right )^4}+\frac {6 a b^6 \left (11 a^4-6 a^2 b^2-b^4\right ) x}{\left (a^2+b^2\right )^3}+\frac {3 b^6 \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) x^2}{\left (a^2+b^2\right )^4}}{(a+x)^2 \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 (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d}-\frac {\text {Subst}\left (\int \left (-\frac {48 a^6 b^6}{\left (a^2+b^2\right )^4 (a+x)^2}-\frac {96 a^5 b^6 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 (a+x)}+\frac {3 b^6 \left (-5 a^8+80 a^6 b^2-50 a^4 b^4-8 a^2 b^6-b^8+32 a^5 \left (a^2-3 b^2\right ) x\right )}{\left (a^2+b^2\right )^5 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{48 b^5 d} \\ & = \frac {2 a^5 b \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^6 b}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d}-\frac {b \text {Subst}\left (\int \frac {-5 a^8+80 a^6 b^2-50 a^4 b^4-8 a^2 b^6-b^8+32 a^5 \left (a^2-3 b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^5 d} \\ & = \frac {2 a^5 b \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^6 b}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d}-\frac {\left (2 a^5 b \left (a^2-3 b^2\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 )^5 d}+\frac {\left (b \left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+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 )^5 d} \\ & = \frac {\left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right ) x}{16 \left (a^2+b^2\right )^5}+\frac {2 a^5 b \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^5 d}+\frac {2 a^5 b \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5 d}-\frac {a^6 b}{\left (a^2+b^2\right )^4 d (a+b \tan (c+d x))}-\frac {\cos ^6(c+d x) \left (2 a b+\left (a^2-b^2\right ) \tan (c+d x)\right )}{6 \left (a^2+b^2\right )^2 d}+\frac {\cos ^4(c+d x) \left (12 a b \left (3 a^2+b^2\right )+\left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{24 \left (a^2+b^2\right )^3 d}-\frac {\cos ^2(c+d x) \left (48 a^5 b+\left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{16 \left (a^2+b^2\right )^4 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(664\) vs. \(2(297)=594\).

Time = 6.72 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.24 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {b \left (-\frac {5 \left (a^2-b^2\right ) \arctan (\tan (c+d x))}{16 b \left (a^2+b^2\right )^2}+\frac {3 \left (3 a^4-3 a^2 b^2-2 b^4\right ) \arctan (\tan (c+d x))}{8 b \left (a^2+b^2\right )^3}-\frac {\left (3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right ) \arctan (\tan (c+d x))}{2 b \left (a^2+b^2\right )^4}-\frac {3 a^5 \cos ^2(c+d x)}{\left (a^2+b^2\right )^4}+\frac {a \left (3 a^2+b^2\right ) \cos ^4(c+d x)}{2 \left (a^2+b^2\right )^3}-\frac {a \cos ^6(c+d x)}{3 \left (a^2+b^2\right )^2}-\frac {a^5 \left (2 a^2-6 b^2-\frac {a^3-7 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}+\frac {2 a^5 \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}-\frac {a^5 \left (2 a^2-6 b^2+\frac {a^3-7 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}-\frac {5 (a-b) (a+b) \cos (c+d x) \sin (c+d x)}{16 b \left (a^2+b^2\right )^2}+\frac {3 \left (3 a^4-3 a^2 b^2-2 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b \left (a^2+b^2\right )^3}-\frac {\left (3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right ) \cos (c+d x) \sin (c+d x)}{2 b \left (a^2+b^2\right )^4}-\frac {5 \left (a^2-b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 b \left (a^2+b^2\right )^2}+\frac {\left (3 a^4-3 a^2 b^2-2 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{4 b \left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{6 b \left (a^2+b^2\right )^2}-\frac {a^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}\right )}{d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 33.91 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(\frac {-\frac {b \,a^{6}}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \,a^{5} \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (-\frac {11}{16} a^{8}+2 a^{6} b^{2}+\frac {25}{8} a^{4} b^{4}+\frac {1}{2} b^{6} a^{2}+\frac {1}{16} b^{8}\right ) \left (\tan ^{5}\left (d x +c \right )\right )+\left (-3 b \,a^{7}-3 b^{3} a^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )+\left (-\frac {5}{6} a^{8}+\frac {13}{3} a^{6} b^{2}+5 a^{4} b^{4}-\frac {1}{3} b^{6} a^{2}-\frac {1}{6} b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-\frac {9}{2} b \,a^{7}-\frac {5}{2} b^{3} a^{5}+\frac {5}{2} a^{3} b^{5}+\frac {1}{2} a \,b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {5}{16} a^{8}+2 a^{6} b^{2}+\frac {15}{8} a^{4} b^{4}-\frac {1}{2} b^{6} a^{2}-\frac {1}{16} b^{8}\right ) \tan \left (d x +c \right )-\frac {11 b \,a^{7}}{6}-\frac {b^{3} a^{5}}{2}+\frac {3 a^{3} b^{5}}{2}+\frac {a \,b^{7}}{6}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {\left (-32 b \,a^{7}+96 b^{3} a^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{32}+\frac {\left (5 a^{8}-80 a^{6} b^{2}+50 a^{4} b^{4}+8 b^{6} a^{2}+b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{16}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\)	\(383\)
default	\(\frac {-\frac {b \,a^{6}}{\left (a^{2}+b^{2}\right )^{4} \left (a +b \tan \left (d x +c \right )\right )}+\frac {2 b \,a^{5} \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{5}}+\frac {\frac {\left (-\frac {11}{16} a^{8}+2 a^{6} b^{2}+\frac {25}{8} a^{4} b^{4}+\frac {1}{2} b^{6} a^{2}+\frac {1}{16} b^{8}\right ) \left (\tan ^{5}\left (d x +c \right )\right )+\left (-3 b \,a^{7}-3 b^{3} a^{5}\right ) \left (\tan ^{4}\left (d x +c \right )\right )+\left (-\frac {5}{6} a^{8}+\frac {13}{3} a^{6} b^{2}+5 a^{4} b^{4}-\frac {1}{3} b^{6} a^{2}-\frac {1}{6} b^{8}\right ) \left (\tan ^{3}\left (d x +c \right )\right )+\left (-\frac {9}{2} b \,a^{7}-\frac {5}{2} b^{3} a^{5}+\frac {5}{2} a^{3} b^{5}+\frac {1}{2} a \,b^{7}\right ) \left (\tan ^{2}\left (d x +c \right )\right )+\left (-\frac {5}{16} a^{8}+2 a^{6} b^{2}+\frac {15}{8} a^{4} b^{4}-\frac {1}{2} b^{6} a^{2}-\frac {1}{16} b^{8}\right ) \tan \left (d x +c \right )-\frac {11 b \,a^{7}}{6}-\frac {b^{3} a^{5}}{2}+\frac {3 a^{3} b^{5}}{2}+\frac {a \,b^{7}}{6}}{\left (1+\tan ^{2}\left (d x +c \right )\right )^{3}}+\frac {\left (-32 b \,a^{7}+96 b^{3} a^{5}\right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{32}+\frac {\left (5 a^{8}-80 a^{6} b^{2}+50 a^{4} b^{4}+8 b^{6} a^{2}+b^{8}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{16}}{\left (a^{2}+b^{2}\right )^{5}}}{d}\)	\(383\)
risch	\(\text {Expression too large to display}\)	\(1139\)

[In]

int(sin(d*x+c)^6/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(-b*a^6/(a^2+b^2)^4/(a+b*tan(d*x+c))+2*b*a^5*(a^2-3*b^2)/(a^2+b^2)^5*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^5*(((-
11/16*a^8+2*a^6*b^2+25/8*a^4*b^4+1/2*b^6*a^2+1/16*b^8)*tan(d*x+c)^5+(-3*a^7*b-3*a^5*b^3)*tan(d*x+c)^4+(-5/6*a^
8+13/3*a^6*b^2+5*a^4*b^4-1/3*b^6*a^2-1/6*b^8)*tan(d*x+c)^3+(-9/2*b*a^7-5/2*b^3*a^5+5/2*a^3*b^5+1/2*a*b^7)*tan(
d*x+c)^2+(-5/16*a^8+2*a^6*b^2+15/8*a^4*b^4-1/2*b^6*a^2-1/16*b^8)*tan(d*x+c)-11/6*b*a^7-1/2*b^3*a^5+3/2*a^3*b^5
+1/6*a*b^7)/(1+tan(d*x+c)^2)^3+1/32*(-32*a^7*b+96*a^5*b^3)*ln(1+tan(d*x+c)^2)+1/16*(5*a^8-80*a^6*b^2+50*a^4*b^
4+8*a^2*b^6+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. 619 vs. \(2 (289) = 578\).

Time = 0.34 (sec) , antiderivative size = 619, normalized size of antiderivative = 2.08 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=-\frac {8 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{7} - 2 \, {\left (19 \, a^{8} b + 68 \, a^{6} b^{3} + 90 \, a^{4} b^{5} + 52 \, a^{2} b^{7} + 11 \, b^{9}\right )} \cos \left (d x + c\right )^{5} + {\left (85 \, a^{8} b + 224 \, a^{6} b^{3} + 210 \, a^{4} b^{5} + 88 \, a^{2} b^{7} + 17 \, b^{9}\right )} \cos \left (d x + c\right )^{3} - {\left (17 \, a^{8} b + 72 \, a^{6} b^{3} + 120 \, a^{4} b^{5} + 20 \, a^{2} b^{7} + 3 \, b^{9} + 3 \, {\left (5 \, a^{9} - 80 \, a^{7} b^{2} + 50 \, a^{5} b^{4} + 8 \, a^{3} b^{6} + a b^{8}\right )} d x\right )} \cos \left (d x + c\right ) - 48 \, {\left ({\left (a^{8} b - 3 \, a^{6} b^{3}\right )} \cos \left (d x + c\right ) + {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\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 (98 \, a^{7} b^{2} + 24 \, a^{5} b^{4} - 30 \, a^{3} b^{6} - 4 \, a b^{8} - 8 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (13 \, a^{9} + 44 \, a^{7} b^{2} + 54 \, a^{5} b^{4} + 28 \, a^{3} b^{6} + 5 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (5 \, a^{8} b - 80 \, a^{6} b^{3} + 50 \, a^{4} b^{5} + 8 \, a^{2} b^{7} + b^{9}\right )} d x - 3 \, {\left (11 \, a^{9} + 16 \, a^{7} b^{2} - 2 \, a^{5} b^{4} - 8 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, {\left ({\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 )} d \cos \left (d x + c\right ) + {\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 )} d \sin \left (d x + c\right )\right )}} \]

[In]

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

[Out]

-1/48*(8*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cos(d*x + c)^7 - 2*(19*a^8*b + 68*a^6*b^3 + 90*a^4*
b^5 + 52*a^2*b^7 + 11*b^9)*cos(d*x + c)^5 + (85*a^8*b + 224*a^6*b^3 + 210*a^4*b^5 + 88*a^2*b^7 + 17*b^9)*cos(d
*x + c)^3 - (17*a^8*b + 72*a^6*b^3 + 120*a^4*b^5 + 20*a^2*b^7 + 3*b^9 + 3*(5*a^9 - 80*a^7*b^2 + 50*a^5*b^4 + 8
*a^3*b^6 + a*b^8)*d*x)*cos(d*x + c) - 48*((a^8*b - 3*a^6*b^3)*cos(d*x + c) + (a^7*b^2 - 3*a^5*b^4)*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) - (98*a^7*b^2 + 24*a^5*b^4 - 30*a^3
*b^6 - 4*a*b^8 - 8*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cos(d*x + c)^6 + 2*(13*a^9 + 44*a^7*b^2 +
54*a^5*b^4 + 28*a^3*b^6 + 5*a*b^8)*cos(d*x + c)^4 + 3*(5*a^8*b - 80*a^6*b^3 + 50*a^4*b^5 + 8*a^2*b^7 + b^9)*d
*x - 3*(11*a^9 + 16*a^7*b^2 - 2*a^5*b^4 - 8*a^3*b^6 - a*b^8)*cos(d*x + c)^2)*sin(d*x + c))/((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)*d*cos(d*x + c) + (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)*d*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (289) = 578\).

Time = 0.32 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.69 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (5 \, a^{8} - 80 \, a^{6} b^{2} + 50 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {96 \, {\left (a^{7} b - 3 \, a^{5} b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {48 \, {\left (a^{7} b - 3 \, a^{5} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {136 \, a^{6} b - 64 \, a^{4} b^{3} - 8 \, a^{2} b^{5} + 3 \, {\left (27 \, a^{6} b - 43 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - b^{7}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (11 \, a^{7} + 5 \, a^{5} b^{2} - 7 \, a^{3} b^{4} - a b^{6}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (41 \, a^{6} b - 31 \, a^{4} b^{3} + a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{4} + 8 \, {\left (5 \, a^{7} - 4 \, a^{5} b^{2} - 11 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (125 \, a^{6} b - 69 \, a^{4} b^{3} - a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + {\left (15 \, a^{7} - 23 \, a^{5} b^{2} - 43 \, a^{3} b^{4} - 5 \, a b^{6}\right )} \tan \left (d x + c\right )}{a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8} + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{7} + {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{5} + 3 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{4} + 3 \, {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (a^{9} + 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} + 4 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )}}{48 \, d} \]

[In]

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

[Out]

1/48*(3*(5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*(d*x + c)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*
b^6 + 5*a^2*b^8 + b^10) + 96*(a^7*b - 3*a^5*b^3)*log(b*tan(d*x + c) + a)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a
^4*b^6 + 5*a^2*b^8 + b^10) - 48*(a^7*b - 3*a^5*b^3)*log(tan(d*x + c)^2 + 1)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 1
0*a^4*b^6 + 5*a^2*b^8 + b^10) - (136*a^6*b - 64*a^4*b^3 - 8*a^2*b^5 + 3*(27*a^6*b - 43*a^4*b^3 - 7*a^2*b^5 - b
^7)*tan(d*x + c)^6 + 3*(11*a^7 + 5*a^5*b^2 - 7*a^3*b^4 - a*b^6)*tan(d*x + c)^5 + 8*(41*a^6*b - 31*a^4*b^3 + a^
2*b^5 + b^7)*tan(d*x + c)^4 + 8*(5*a^7 - 4*a^5*b^2 - 11*a^3*b^4 - 2*a*b^6)*tan(d*x + c)^3 + 3*(125*a^6*b - 69*
a^4*b^3 - a^2*b^5 + b^7)*tan(d*x + c)^2 + (15*a^7 - 23*a^5*b^2 - 43*a^3*b^4 - 5*a*b^6)*tan(d*x + c))/(a^9 + 4*
a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8 + (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*tan(d*x + c)^7 + (
a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*tan(d*x + c)^6 + 3*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7
+ b^9)*tan(d*x + c)^5 + 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*tan(d*x + c)^4 + 3*(a^8*b + 4*a^6
*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*tan(d*x + c)^3 + 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*tan(d
*x + c)^2 + (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*tan(d*x + c)))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (289) = 578\).

Time = 0.54 (sec) , antiderivative size = 735, normalized size of antiderivative = 2.47 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\frac {3 \, {\left (5 \, a^{8} - 80 \, a^{6} b^{2} + 50 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\right )} {\left (d x + c\right )}}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} - \frac {48 \, {\left (a^{7} b - 3 \, a^{5} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} + 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} + 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} + b^{10}} + \frac {96 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{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}} - \frac {48 \, {\left (2 \, a^{7} b^{2} \tan \left (d x + c\right ) - 6 \, a^{5} b^{4} \tan \left (d x + c\right ) + 3 \, a^{8} b - 5 \, a^{6} b^{3}\right )}}{{\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 )} {\left (b \tan \left (d x + c\right ) + a\right )}} + \frac {88 \, a^{7} b \tan \left (d x + c\right )^{6} - 264 \, a^{5} b^{3} \tan \left (d x + c\right )^{6} - 33 \, a^{8} \tan \left (d x + c\right )^{5} + 96 \, a^{6} b^{2} \tan \left (d x + c\right )^{5} + 150 \, a^{4} b^{4} \tan \left (d x + c\right )^{5} + 24 \, a^{2} b^{6} \tan \left (d x + c\right )^{5} + 3 \, b^{8} \tan \left (d x + c\right )^{5} + 120 \, a^{7} b \tan \left (d x + c\right )^{4} - 936 \, a^{5} b^{3} \tan \left (d x + c\right )^{4} - 40 \, a^{8} \tan \left (d x + c\right )^{3} + 208 \, a^{6} b^{2} \tan \left (d x + c\right )^{3} + 240 \, a^{4} b^{4} \tan \left (d x + c\right )^{3} - 16 \, a^{2} b^{6} \tan \left (d x + c\right )^{3} - 8 \, b^{8} \tan \left (d x + c\right )^{3} + 48 \, a^{7} b \tan \left (d x + c\right )^{2} - 912 \, a^{5} b^{3} \tan \left (d x + c\right )^{2} + 120 \, a^{3} b^{5} \tan \left (d x + c\right )^{2} + 24 \, a b^{7} \tan \left (d x + c\right )^{2} - 15 \, a^{8} \tan \left (d x + c\right ) + 96 \, a^{6} b^{2} \tan \left (d x + c\right ) + 90 \, a^{4} b^{4} \tan \left (d x + c\right ) - 24 \, a^{2} b^{6} \tan \left (d x + c\right ) - 3 \, b^{8} \tan \left (d x + c\right ) - 288 \, a^{5} b^{3} + 72 \, a^{3} b^{5} + 8 \, a b^{7}}{{\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 )} {\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))^2,x, algorithm="giac")

[Out]

1/48*(3*(5*a^8 - 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*(d*x + c)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*
b^6 + 5*a^2*b^8 + b^10) - 48*(a^7*b - 3*a^5*b^3)*log(tan(d*x + c)^2 + 1)/(a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a
^4*b^6 + 5*a^2*b^8 + b^10) + 96*(a^7*b^2 - 3*a^5*b^4)*log(abs(b*tan(d*x + c) + a))/(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) - 48*(2*a^7*b^2*tan(d*x + c) - 6*a^5*b^4*tan(d*x + c) + 3*a^8*b - 5*a^6
*b^3)/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*(b*tan(d*x + c) + a)) + (88*a^7*b*tan(d
*x + c)^6 - 264*a^5*b^3*tan(d*x + c)^6 - 33*a^8*tan(d*x + c)^5 + 96*a^6*b^2*tan(d*x + c)^5 + 150*a^4*b^4*tan(d
*x + c)^5 + 24*a^2*b^6*tan(d*x + c)^5 + 3*b^8*tan(d*x + c)^5 + 120*a^7*b*tan(d*x + c)^4 - 936*a^5*b^3*tan(d*x
+ c)^4 - 40*a^8*tan(d*x + c)^3 + 208*a^6*b^2*tan(d*x + c)^3 + 240*a^4*b^4*tan(d*x + c)^3 - 16*a^2*b^6*tan(d*x
+ c)^3 - 8*b^8*tan(d*x + c)^3 + 48*a^7*b*tan(d*x + c)^2 - 912*a^5*b^3*tan(d*x + c)^2 + 120*a^3*b^5*tan(d*x + c
)^2 + 24*a*b^7*tan(d*x + c)^2 - 15*a^8*tan(d*x + c) + 96*a^6*b^2*tan(d*x + c) + 90*a^4*b^4*tan(d*x + c) - 24*a
^2*b^6*tan(d*x + c) - 3*b^8*tan(d*x + c) - 288*a^5*b^3 + 72*a^3*b^5 + 8*a*b^7)/((a^10 + 5*a^8*b^2 + 10*a^6*b^4
+ 10*a^4*b^6 + 5*a^2*b^8 + b^10)*(tan(d*x + c)^2 + 1)^3))/d

Mupad [B] (verification not implemented)

Time = 6.11 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.55 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {2\,a\,b}{{\left (a^2+b^2\right )}^2}-\frac {12\,a\,b^3}{{\left (a^2+b^2\right )}^3}+\frac {18\,a\,b^5}{{\left (a^2+b^2\right )}^4}-\frac {8\,a\,b^7}{{\left (a^2+b^2\right )}^5}\right )}{d}+\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-5\,a^5+9\,a^3\,b^2+2\,a\,b^4\right )}{6\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (-11\,a^5+6\,a^3\,b^2+a\,b^4\right )}{16\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (-27\,a^6\,b+43\,a^4\,b^3+7\,a^2\,b^5+b^7\right )}{16\,\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 )\,\left (-15\,a^5+38\,a^3\,b^2+5\,a\,b^4\right )}{48\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {a\,\left (-17\,a^5\,b+8\,a^3\,b^3+a\,b^5\right )}{6\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (41\,a^6\,b-31\,a^4\,b^3+a^2\,b^5+b^7\right )}{6\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}-\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (125\,a^6\,b-69\,a^4\,b^3-a^2\,b^5+b^7\right )}{16\,\left (a^2+b^2\right )\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}}{d\,\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^7+a\,{\mathrm {tan}\left (c+d\,x\right )}^6+3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^5+3\,a\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+3\,a\,{\mathrm {tan}\left (c+d\,x\right )}^2+b\,\mathrm {tan}\left (c+d\,x\right )+a\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (a^3\,5{}\mathrm {i}-7\,a^2\,b+a\,b^2\,5{}\mathrm {i}+b^3\right )}{32\,d\,\left (a^5-a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2+a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4-b^5\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a^3\,5{}\mathrm {i}+7\,a^2\,b+a\,b^2\,5{}\mathrm {i}-b^3\right )}{32\,d\,\left (a^5+a^4\,b\,5{}\mathrm {i}-10\,a^3\,b^2-a^2\,b^3\,10{}\mathrm {i}+5\,a\,b^4+b^5\,1{}\mathrm {i}\right )} \]

[In]

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

[Out]

(log(a + b*tan(c + d*x))*((2*a*b)/(a^2 + b^2)^2 - (12*a*b^3)/(a^2 + b^2)^3 + (18*a*b^5)/(a^2 + b^2)^4 - (8*a*b
^7)/(a^2 + b^2)^5))/d + ((tan(c + d*x)^3*(2*a*b^4 - 5*a^5 + 9*a^3*b^2))/(6*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)
) + (tan(c + d*x)^5*(a*b^4 - 11*a^5 + 6*a^3*b^2))/(16*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (tan(c + d*x)^6*(
b^7 - 27*a^6*b + 7*a^2*b^5 + 43*a^4*b^3))/(16*(a^8 + b^8 + 4*a^2*b^6 + 6*a^4*b^4 + 4*a^6*b^2)) + (tan(c + d*x)
*(5*a*b^4 - 15*a^5 + 38*a^3*b^2))/(48*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*(a*b^5 - 17*a^5*b + 8*a^3*b^3)
)/(6*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^4*(41*a^6*b + b^7 + a^2*b^5 - 31*a^4*b^3
))/(6*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (tan(c + d*x)^2*(125*a^6*b + b^7 - a^2*b^5 - 69*a^4*b
^3))/(16*(a^2 + b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))/(d*(a + b*tan(c + d*x) + 3*a*tan(c + d*x)^2 + 3*a*t
an(c + d*x)^4 + a*tan(c + d*x)^6 + 3*b*tan(c + d*x)^3 + 3*b*tan(c + d*x)^5 + b*tan(c + d*x)^7)) + (log(tan(c +
d*x) + 1i)*(a*b^2*5i - 7*a^2*b + a^3*5i + b^3))/(32*d*(5*a*b^4 - a^4*b*5i + a^5 - b^5*1i + a^2*b^3*10i - 10*a
^3*b^2)) - (log(tan(c + d*x) - 1i)*(a*b^2*5i + 7*a^2*b + a^3*5i - b^3))/(32*d*(5*a*b^4 + a^4*b*5i + a^5 + b^5*
1i - a^2*b^3*10i - 10*a^3*b^2))