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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.43182, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3516, 1647, 1629, 635, 203, 260} \[ -\frac{a^6 b}{2 d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}-\frac{2 a^5 b \left (a^2-3 b^2\right )}{d \left (a^2+b^2\right )^5 (a+b \tan (c+d x))}-\frac{a \cos ^2(c+d x) \left (\left (-119 a^4 b^2+65 a^2 b^4+11 a^6+3 b^6\right ) \tan (c+d x)+24 a^3 b \left (3 a^2-5 b^2\right )\right )}{16 d \left (a^2+b^2\right )^5}-\frac{\cos ^6(c+d x) \left (a \left (a^2-3 b^2\right ) \tan (c+d x)+b \left (3 a^2-b^2\right )\right )}{6 d \left (a^2+b^2\right )^3}+\frac{\cos ^4(c+d x) \left (a \left (-62 a^2 b^2+13 a^4-3 b^4\right ) \tan (c+d x)+6 b \left (-4 a^2 b^2+9 a^4-b^4\right )\right )}{24 d \left (a^2+b^2\right )^4}+\frac{a^4 b \left (-22 a^2 b^2+3 a^4+15 b^4\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^6}+\frac{a x \left (-180 a^6 b^2+390 a^4 b^4-68 a^2 b^6+5 a^8-3 b^8\right )}{16 \left (a^2+b^2\right )^6} \]

Antiderivative was successfully verified.

[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 3516

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]

Rule 1647

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 1629

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 635

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 203

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

Rule 260

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]

Rubi steps

\begin{align*} \int \frac{\sin ^6(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac{b \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 6.60369, size = 683, normalized size = 1.79 \[ \frac{b \left (-\frac{3 a^4 \left (3 a^2-5 b^2\right ) \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^5}-\frac{\left (3 a^2-b^2\right ) \cos ^6(c+d x)}{6 \left (a^2+b^2\right )^3}+\frac{\left (-4 a^2 b^2+9 a^4-b^4\right ) \cos ^4(c+d x)}{4 \left (a^2+b^2\right )^4}-\frac{3 a^5 \left (a^2-7 b^2\right ) \tan ^{-1}(\tan (c+d x))}{2 b \left (a^2+b^2\right )^5}-\frac{a^6}{2 \left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}-\frac{2 a^5 \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^5 (a+b \tan (c+d x))}-\frac{a^4 \left (-22 a^2 b^2-\frac{-18 a^3 b^2+a^5+21 a b^4}{\sqrt{-b^2}}+3 a^4+15 b^4\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^6}+\frac{a^4 \left (-22 a^2 b^2+3 a^4+15 b^4\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^6}-\frac{a^4 \left (-22 a^2 b^2+\frac{-18 a^3 b^2+a^5+21 a b^4}{\sqrt{-b^2}}+3 a^4+15 b^4\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^6}-\frac{a \left (a^2-3 b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{6 b \left (a^2+b^2\right )^3}+\frac{3 a \left (-4 a^2 b^2+a^4-b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{4 b \left (a^2+b^2\right )^4}-\frac{3 a^5 \left (a^2-7 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b \left (a^2+b^2\right )^5}-\frac{5 a \left (a^2-3 b^2\right ) \left (3 b^2 \left (\frac{\tan ^{-1}(\tan (c+d x))}{b^3}+\frac{\sin (c+d x) \cos (c+d x)}{b^3}\right )+\frac{2 \sin (c+d x) \cos ^3(c+d x)}{b}\right )}{48 \left (a^2+b^2\right )^3}+\frac{9 a \left (-4 a^2 b^2+a^4-b^4\right ) \left (\frac{\tan ^{-1}(\tan (c+d x))}{b}+\frac{\sin (c+d x) \cos (c+d x)}{b}\right )}{8 \left (a^2+b^2\right )^4}\right )}{d} \]

Antiderivative was successfully verified.

[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) - (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*Ta
n[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) + (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) + (9*a*(a^4 - 4*a^2*b^2 - b^4)*(ArcTan[Tan[c + d*x]]/b + (Cos[c + d*x]*Sin[c + d*
x])/b))/(8*(a^2 + b^2)^4) - (5*a*(a^2 - 3*b^2)*((2*Cos[c + d*x]^3*Sin[c + d*x])/b + 3*b^2*(ArcTan[Tan[c + d*x]
]/b^3 + (Cos[c + d*x]*Sin[c + d*x])/b^3)))/(48*(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

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Maple [B]  time = 0.118, size = 1449, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/2/d/(a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a^4*b^5+13/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*a^4*b^5-11/4/d/(a^2+b^2)^
6/(1+tan(d*x+c)^2)^3*a^8*b+31/6/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*a^6*b^3-3/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*
a^2*b^7-11/16/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^9-5/6/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c
)^3*a^9-1/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*b^9-5/16/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)
*a^9+3/d*a^8*b/(a^2+b^2)^6*ln(a+b*tan(d*x+c))-22/d*a^6*b^3/(a^2+b^2)^6*ln(a+b*tan(d*x+c))+15/d*a^4*b^5/(a^2+b^
2)^6*ln(a+b*tan(d*x+c))-2/d*b*a^7/(a^2+b^2)^5/(a+b*tan(d*x+c))+6/d*b^3*a^5/(a^2+b^2)^5/(a+b*tan(d*x+c))-45/4/d
/(a^2+b^2)^6*arctan(tan(d*x+c))*a^7*b^2+195/8/d/(a^2+b^2)^6*arctan(tan(d*x+c))*a^5*b^4-17/4/d/(a^2+b^2)^6*arct
an(tan(d*x+c))*a^3*b^6-3/16/d/(a^2+b^2)^6*arctan(tan(d*x+c))*a*b^8-3/2/d/(a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a^8*b+
11/d/(a^2+b^2)^6*ln(1+tan(d*x+c)^2)*a^6*b^3-1/12/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*b^9+5/16/d/(a^2+b^2)^6*arcta
n(tan(d*x+c))*a^9+27/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^7*b^2+27/8/d/(a^2+b^2)^6/(1+tan(d*x+c)^
2)^3*tan(d*x+c)^5*a^5*b^4-17/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^3*b^6-3/16/d/(a^2+b^2)^6/(1+tan
(d*x+c)^2)^3*tan(d*x+c)^5*a*b^8-9/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^4*a^8*b+3/d/(a^2+b^2)^6/(1+tan
(d*x+c)^2)^3*tan(d*x+c)^4*a^6*b^3+15/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^4*a^4*b^5+12/d/(a^2+b^2)^6/
(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a^7*b^2+2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a^5*b^4-34/3/d/(a^2+b^
2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a^3*b^6-1/2/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a*b^8-1/2*a^6*b
/(a^2+b^2)^4/d/(a+b*tan(d*x+c))^2-27/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*a^8*b+19/2/d/(a^2+b^2)^6/
(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*a^6*b^3+15/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*a^4*b^5-3/2/d/(a^2+b^
2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*a^2*b^7+21/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)*a^7*b^2-3/8/d/(a
^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)*a^5*b^4-23/4/d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)*a^3*b^6+3/16/
d/(a^2+b^2)^6/(1+tan(d*x+c)^2)^3*tan(d*x+c)*a*b^8

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Maxima [B]  time = 1.93332, size = 1469, normalized size = 3.85 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*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

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Fricas [B]  time = 3.58986, size = 2137, normalized size = 5.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.28551, size = 1246, normalized size = 3.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^6/(a+b*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