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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.913641, antiderivative size = 297, 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} \[ -\frac{a^6 b}{d \left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac{\cos ^6(c+d x) \left (\left (a^2-b^2\right ) \tan (c+d x)+2 a b\right )}{6 d \left (a^2+b^2\right )^2}+\frac{\cos ^4(c+d x) \left (\left (-18 a^2 b^2+13 a^4-7 b^4\right ) \tan (c+d x)+12 a b \left (3 a^2+b^2\right )\right )}{24 d \left (a^2+b^2\right )^3}-\frac{\cos ^2(c+d x) \left (\left (-43 a^4 b^2-7 a^2 b^4+11 a^6-b^6\right ) \tan (c+d x)+48 a^5 b\right )}{16 d \left (a^2+b^2\right )^4}+\frac{2 a^5 b \left (a^2-3 b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^5}+\frac{x \left (-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+5 a^8+b^8\right )}{16 \left (a^2+b^2\right )^5} \]

Antiderivative was successfully verified.

[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 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))^2} \, dx &=\frac{b \operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{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 ) \operatorname{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 ) \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 )^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 [A]  time = 6.47982, size = 526, normalized size = 1.77 \[ \frac{b \left (\frac{12 \left (a^2+b^2\right ) \left (6 a^4 b^2+4 a^2 b^4-3 a^6+b^6\right ) \sin (2 (c+d x))}{b}-144 a^5 \left (a^2+b^2\right ) \cos ^2(c+d x)-16 a \left (a^2+b^2\right )^3 \cos ^6(c+d x)+24 a \left (a^2+b^2\right )^2 \left (3 a^2+b^2\right ) \cos ^4(c+d x)+\frac{24 \left (a^2+b^2\right ) \left (6 a^4 b^2+4 a^2 b^4-3 a^6+b^6\right ) \tan ^{-1}(\tan (c+d x))}{b}-\frac{48 a^6 \left (a^2+b^2\right )}{a+b \tan (c+d x)}-24 a^5 \left (\frac{7 a b^2-a^3}{\sqrt{-b^2}}+2 a^2-6 b^2\right ) \log \left (\sqrt{-b^2}-b \tan (c+d x)\right )+96 a^5 \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))-24 a^5 \left (\frac{a^3-7 a b^2}{\sqrt{-b^2}}+2 a^2-6 b^2\right ) \log \left (\sqrt{-b^2}+b \tan (c+d x)\right )-\frac{9 \left (a^2+b^2\right )^2 \left (3 a^2 b^2-3 a^4+2 b^4\right ) \left (\sin (2 (c+d x))+2 \tan ^{-1}(\tan (c+d x))\right )}{b}+\frac{5 \left (b^2-a^2\right ) \left (a^2+b^2\right )^3 \left (8 \sin (2 (c+d x))+\sin (4 (c+d x))+12 \tan ^{-1}(\tan (c+d x))\right )}{4 b}+\frac{8 \left (b^2-a^2\right ) \left (a^2+b^2\right )^3 \sin (c+d x) \cos ^5(c+d x)}{b}-\frac{12 \left (a^2+b^2\right )^2 \left (3 a^2 b^2-3 a^4+2 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{b}\right )}{48 d \left (a^2+b^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*((24*(a^2 + b^2)*(-3*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*ArcTan[Tan[c + d*x]])/b - 144*a^5*(a^2 + b^2)*Cos[c
+ d*x]^2 + 24*a*(a^2 + b^2)^2*(3*a^2 + b^2)*Cos[c + d*x]^4 - 16*a*(a^2 + b^2)^3*Cos[c + d*x]^6 - 24*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]] + 96*a^5*(a^2 - 3*b^2)*Log[a + b*Tan
[c + d*x]] - 24*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]] - (12*(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])/b + (8*(-a^2 + b^2)*(a^2 + b^2)^3*Cos[c + d*
x]^5*Sin[c + d*x])/b + (12*(a^2 + b^2)*(-3*a^6 + 6*a^4*b^2 + 4*a^2*b^4 + b^6)*Sin[2*(c + d*x)])/b - (9*(a^2 +
b^2)^2*(-3*a^4 + 3*a^2*b^2 + 2*b^4)*(2*ArcTan[Tan[c + d*x]] + Sin[2*(c + d*x)]))/b + (5*(-a^2 + b^2)*(a^2 + b^
2)^3*(12*ArcTan[Tan[c + d*x]] + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(4*b) - (48*a^6*(a^2 + b^2))/(a + b*Ta
n[c + d*x])))/(48*(a^2 + b^2)^5*d)

________________________________________________________________________________________

Maple [B]  time = 0.103, size = 1211, normalized size = 4.1 \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))^2,x)

[Out]

-1/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a^7*b+3/d/(a^2+b^2)^5*ln(1+tan(d*x+c)^2)*a^5*b^3-5/d/(a^2+b^2)^5*arctan(ta
n(d*x+c))*a^6*b^2+25/8/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^4*b^4+1/2/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^2*b^6+2
/d*b*a^7/(a^2+b^2)^5*ln(a+b*tan(d*x+c))-6/d*b^3*a^5/(a^2+b^2)^5*ln(a+b*tan(d*x+c))-1/2/d/(a^2+b^2)^5/(1+tan(d*
x+c)^2)^3*b^3*a^5-11/16/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^8+2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*t
an(d*x+c)*a^6*b^2+15/8/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)*a^4*b^4-1/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^
3*tan(d*x+c)*a^2*b^6-3/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^4*a^7*b-3/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*
tan(d*x+c)^4*a^5*b^3+2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^6*b^2-1/3/d/(a^2+b^2)^5/(1+tan(d*x+c)^2
)^3*tan(d*x+c)^3*a^2*b^6-9/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*a^7*b-5/2/d/(a^2+b^2)^5/(1+tan(d*x+
c)^2)^3*tan(d*x+c)^2*b^3*a^5+5/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^2*a^3*b^5+1/2/d/(a^2+b^2)^5/(1+ta
n(d*x+c)^2)^3*tan(d*x+c)^2*a*b^7+1/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^2*b^6+13/3/d/(a^2+b^2)^5/
(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a^6*b^2+5/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a^4*b^4+25/8/d/(a^2+b^
2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*a^4*b^4+5/16/d/(a^2+b^2)^5*arctan(tan(d*x+c))*a^8+1/16/d/(a^2+b^2)^5*arct
an(tan(d*x+c))*b^8-11/6/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*a^7*b-a^6*b/(a^2+b^2)^4/d/(a+b*tan(d*x+c))+1/16/d/(a^
2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^5*b^8-5/6/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*a^8-1/6/d/(a^2+
b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)^3*b^8-5/16/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)*a^8-1/16/d/(a^2+b^
2)^5/(1+tan(d*x+c)^2)^3*tan(d*x+c)*b^8+3/2/d/(a^2+b^2)^5/(1+tan(d*x+c)^2)^3*a^3*b^5+1/6/d/(a^2+b^2)^5/(1+tan(d
*x+c)^2)^3*a*b^7

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Maxima [B]  time = 1.79554, size = 1079, normalized size = 3.63 \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))^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

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Fricas [B]  time = 2.89634, size = 1388, normalized size = 4.67 \begin{align*} -\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 )}} \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))^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))

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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))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.23314, size = 992, normalized size = 3.34 \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))^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