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

Optimal result

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

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
 

Mathematica [B] (verified)

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

Time = 6.58 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.24 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {b \left (-\frac {5 \left (a^2-b^2\right ) \arctan (\tan (c+d x))}{16 b \left (a^2+b^2\right )^2}+\frac {3 \left (3 a^4-3 a^2 b^2-2 b^4\right ) \arctan (\tan (c+d x))}{8 b \left (a^2+b^2\right )^3}-\frac {\left (3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right ) \arctan (\tan (c+d x))}{2 b \left (a^2+b^2\right )^4}-\frac {3 a^5 \cos ^2(c+d x)}{\left (a^2+b^2\right )^4}+\frac {a \left (3 a^2+b^2\right ) \cos ^4(c+d x)}{2 \left (a^2+b^2\right )^3}-\frac {a \cos ^6(c+d x)}{3 \left (a^2+b^2\right )^2}-\frac {a^5 \left (2 a^2-6 b^2-\frac {a^3-7 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}+\frac {2 a^5 \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}-\frac {a^5 \left (2 a^2-6 b^2+\frac {a^3-7 a b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^5}-\frac {5 (a-b) (a+b) \cos (c+d x) \sin (c+d x)}{16 b \left (a^2+b^2\right )^2}+\frac {3 \left (3 a^4-3 a^2 b^2-2 b^4\right ) \cos (c+d x) \sin (c+d x)}{8 b \left (a^2+b^2\right )^3}-\frac {\left (3 a^6-6 a^4 b^2-4 a^2 b^4-b^6\right ) \cos (c+d x) \sin (c+d x)}{2 b \left (a^2+b^2\right )^4}-\frac {5 \left (a^2-b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{24 b \left (a^2+b^2\right )^2}+\frac {\left (3 a^4-3 a^2 b^2-2 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{4 b \left (a^2+b^2\right )^3}-\frac {\left (a^2-b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{6 b \left (a^2+b^2\right )^2}-\frac {a^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}\right )}{d} \] Input:

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

Output:

(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)*Co 
s[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*Tan[c + d*x] 
))))/d
 

Rubi [A] (verified)

Time = 1.51 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3042, 3999, 601, 25, 2178, 27, 2178, 2160, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3999

\(\displaystyle \frac {b \int \frac {b^6 \tan ^6(c+d x)}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^4}d(b \tan (c+d x))}{d}\)

\(\Big \downarrow \) 601

\(\displaystyle \frac {b \left (-\frac {\int -\frac {-\frac {2 a \left (5 a^2+b^2\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^2}+6 \tan ^4(c+d x) b^6-\frac {\left (6 a^4+17 b^2 a^2+b^4\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^2}+\frac {a^2 \left (a^2-b^2\right ) b^6}{\left (a^2+b^2\right )^2}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \left (\frac {\int \frac {-\frac {2 a \left (5 a^2+b^2\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^2}+6 \tan ^4(c+d x) b^6-\frac {\left (6 a^4+17 b^2 a^2+b^4\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^2}+\frac {a^2 \left (a^2-b^2\right ) b^6}{\left (a^2+b^2\right )^2}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^3}d(b \tan (c+d x))}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {\int \frac {3 \left (-\frac {2 a \left (13 a^4+6 b^2 a^2+b^4\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^3}-\frac {\left (8 a^6+37 b^2 a^4+6 b^4 a^2+b^6\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (3 a^4-6 b^2 a^2-b^4\right ) b^6}{\left (a^2+b^2\right )^3}\right )}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \int \frac {-\frac {2 a \left (13 a^4+6 b^2 a^2+b^4\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^3}-\frac {\left (8 a^6+37 b^2 a^4+6 b^4 a^2+b^6\right ) \tan ^2(c+d x) b^6}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (3 a^4-6 b^2 a^2-b^4\right ) b^6}{\left (a^2+b^2\right )^3}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )^2}d(b \tan (c+d x))}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

\(\Big \downarrow \) 2178

\(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \left (\frac {b^4 \left (48 a^5 b^2+b \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {\int \frac {-\frac {\left (11 a^6-43 b^2 a^4-7 b^4 a^2-b^6\right ) \tan ^2(c+d x) b^8}{\left (a^2+b^2\right )^4}-\frac {2 a \left (11 a^4-6 b^2 a^2-b^4\right ) \tan (c+d x) b^7}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (5 a^6-37 b^2 a^4+7 b^4 a^2+b^6\right ) b^6}{\left (a^2+b^2\right )^4}}{(a+b \tan (c+d x))^2 \left (\tan ^2(c+d x) b^2+b^2\right )}d(b \tan (c+d x))}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

\(\Big \downarrow \) 2160

\(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \left (\frac {b^4 \left (48 a^5 b^2+b \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {\int \left (\frac {32 a^5 \left (a^2-3 b^2\right ) b^6}{\left (a^2+b^2\right )^5 (a+b \tan (c+d x))}+\frac {\left (5 a^8-80 b^2 a^6-32 b \left (a^2-3 b^2\right ) \tan (c+d x) a^5+50 b^4 a^4+8 b^6 a^2+b^8\right ) b^6}{\left (a^2+b^2\right )^5 \left (\tan ^2(c+d x) b^2+b^2\right )}+\frac {16 a^6 b^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))^2}\right )d(b \tan (c+d x))}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (\frac {\frac {b^4 \left (12 a b^2 \left (3 a^2+b^2\right )+b \left (13 a^4-18 a^2 b^2-7 b^4\right ) \tan (c+d x)\right )}{4 \left (a^2+b^2\right )^3 \left (b^2 \tan ^2(c+d x)+b^2\right )^2}-\frac {3 \left (\frac {b^4 \left (48 a^5 b^2+b \left (11 a^6-43 a^4 b^2-7 a^2 b^4-b^6\right ) \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 \left (b^2 \tan ^2(c+d x)+b^2\right )}-\frac {-\frac {16 a^6 b^6}{\left (a^2+b^2\right )^4 (a+b \tan (c+d x))}-\frac {16 a^5 b^6 \left (a^2-3 b^2\right ) \log \left (b^2 \tan ^2(c+d x)+b^2\right )}{\left (a^2+b^2\right )^5}+\frac {32 a^5 b^6 \left (a^2-3 b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^5}+\frac {b^5 \left (5 a^8-80 a^6 b^2+50 a^4 b^4+8 a^2 b^6+b^8\right ) \arctan (\tan (c+d x))}{\left (a^2+b^2\right )^5}}{2 b^2}\right )}{4 b^2}}{6 b^2}-\frac {b^4 \left (b \left (a^2-b^2\right ) \tan (c+d x)+2 a b^2\right )}{6 \left (a^2+b^2\right )^2 \left (b^2 \tan ^2(c+d x)+b^2\right )^3}\right )}{d}\)

Input:

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

Output:

(b*(-1/6*(b^4*(2*a*b^2 + b*(a^2 - b^2)*Tan[c + d*x]))/((a^2 + b^2)^2*(b^2 
+ b^2*Tan[c + d*x]^2)^3) + ((b^4*(12*a*b^2*(3*a^2 + b^2) + b*(13*a^4 - 18* 
a^2*b^2 - 7*b^4)*Tan[c + d*x]))/(4*(a^2 + b^2)^3*(b^2 + b^2*Tan[c + d*x]^2 
)^2) - (3*((b^4*(48*a^5*b^2 + b*(11*a^6 - 43*a^4*b^2 - 7*a^2*b^4 - b^6)*Ta 
n[c + d*x]))/(2*(a^2 + b^2)^4*(b^2 + b^2*Tan[c + d*x]^2)) - ((b^5*(5*a^8 - 
 80*a^6*b^2 + 50*a^4*b^4 + 8*a^2*b^6 + b^8)*ArcTan[Tan[c + d*x]])/(a^2 + b 
^2)^5 + (32*a^5*b^6*(a^2 - 3*b^2)*Log[a + b*Tan[c + d*x]])/(a^2 + b^2)^5 - 
 (16*a^5*b^6*(a^2 - 3*b^2)*Log[b^2 + b^2*Tan[c + d*x]^2])/(a^2 + b^2)^5 - 
(16*a^6*b^6)/((a^2 + b^2)^4*(a + b*Tan[c + d*x])))/(2*b^2)))/(4*b^2))/(6*b 
^2)))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2160

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

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3999

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ 
), x_Symbol] :> Simp[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]
 

Maple [A] (verified)

Time = 62.46 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.29

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

Input:

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

Output:

1/d*(1/(a^2+b^2)^5*(((-11/16*a^8+2*a^6*b^2+25/8*a^4*b^4+1/2*a^2*b^6+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*a^2*b^6-1/6*b^8)*tan(d*x+c)^3+(-9/2*a^7*b-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*a^2*b^ 
6-1/16*b^8)*tan(d*x+c)-11/6*a^7*b-1/2*b^3*a^5+3/2*a^3*b^5+1/6*a*b^7)/(1+ta 
n(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^8-8 
0*a^6*b^2+50*a^4*b^4+8*a^2*b^6+b^8)*arctan(tan(d*x+c)))-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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (289) = 578\).

Time = 0.15 (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 )}} \] Input:

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

Output:

-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))*l 
og(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 + 5 
0*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. \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\text {Timed out} \] Input:

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

Output:

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 = 0.13 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.69 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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^1 
0 + 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 + 10*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 [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.94 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx=\frac {{\left (5 \, a^{8} - 80 \, a^{6} b^{2} + 50 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\right )} {\left (d x + c\right )}}{16 \, {\left (a^{10} d + 5 \, a^{8} b^{2} d + 10 \, a^{6} b^{4} d + 10 \, a^{4} b^{6} d + 5 \, a^{2} b^{8} d + b^{10} d\right )}} - \frac {{\left (a^{7} b - 3 \, a^{5} b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{10} d + 5 \, a^{8} b^{2} d + 10 \, a^{6} b^{4} d + 10 \, a^{4} b^{6} d + 5 \, a^{2} b^{8} d + b^{10} d} + \frac {2 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b d + 5 \, a^{8} b^{3} d + 10 \, a^{6} b^{5} d + 10 \, a^{4} b^{7} d + 5 \, a^{2} b^{9} d + b^{11} d} - \frac {136 \, a^{8} b + 72 \, a^{6} b^{3} - 72 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + 3 \, {\left (27 \, a^{8} b - 16 \, a^{6} b^{3} - 50 \, a^{4} b^{5} - 8 \, a^{2} b^{7} - b^{9}\right )} \tan \left (d x + c\right )^{6} + 3 \, {\left (11 \, a^{9} + 16 \, a^{7} b^{2} - 2 \, a^{5} b^{4} - 8 \, a^{3} b^{6} - a b^{8}\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (41 \, a^{8} b + 10 \, a^{6} b^{3} - 30 \, a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{4} + 8 \, {\left (5 \, a^{9} + a^{7} b^{2} - 15 \, a^{5} b^{4} - 13 \, a^{3} b^{6} - 2 \, a b^{8}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (125 \, a^{8} b + 56 \, a^{6} b^{3} - 70 \, a^{4} b^{5} + b^{9}\right )} \tan \left (d x + c\right )^{2} + {\left (15 \, a^{9} - 8 \, a^{7} b^{2} - 66 \, a^{5} b^{4} - 48 \, a^{3} b^{6} - 5 \, a b^{8}\right )} \tan \left (d x + c\right )}{48 \, {\left (a^{2} + b^{2}\right )}^{5} {\left (b \tan \left (d x + c\right ) + a\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3} d} \] Input:

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

Output:

1/16*(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*d 
 + 5*a^8*b^2*d + 10*a^6*b^4*d + 10*a^4*b^6*d + 5*a^2*b^8*d + b^10*d) - (a^ 
7*b - 3*a^5*b^3)*log(tan(d*x + c)^2 + 1)/(a^10*d + 5*a^8*b^2*d + 10*a^6*b^ 
4*d + 10*a^4*b^6*d + 5*a^2*b^8*d + b^10*d) + 2*(a^7*b^2 - 3*a^5*b^4)*log(a 
bs(b*tan(d*x + c) + a))/(a^10*b*d + 5*a^8*b^3*d + 10*a^6*b^5*d + 10*a^4*b^ 
7*d + 5*a^2*b^9*d + b^11*d) - 1/48*(136*a^8*b + 72*a^6*b^3 - 72*a^4*b^5 - 
8*a^2*b^7 + 3*(27*a^8*b - 16*a^6*b^3 - 50*a^4*b^5 - 8*a^2*b^7 - b^9)*tan(d 
*x + c)^6 + 3*(11*a^9 + 16*a^7*b^2 - 2*a^5*b^4 - 8*a^3*b^6 - a*b^8)*tan(d* 
x + c)^5 + 8*(41*a^8*b + 10*a^6*b^3 - 30*a^4*b^5 + 2*a^2*b^7 + b^9)*tan(d* 
x + c)^4 + 8*(5*a^9 + a^7*b^2 - 15*a^5*b^4 - 13*a^3*b^6 - 2*a*b^8)*tan(d*x 
 + c)^3 + 3*(125*a^8*b + 56*a^6*b^3 - 70*a^4*b^5 + b^9)*tan(d*x + c)^2 + ( 
15*a^9 - 8*a^7*b^2 - 66*a^5*b^4 - 48*a^3*b^6 - 5*a*b^8)*tan(d*x + c))/((a^ 
2 + b^2)^5*(b*tan(d*x + c) + a)*(tan(d*x + c)^2 + 1)^3*d)
 

Mupad [B] (verification not implemented)

Time = 2.71 (sec) , antiderivative size = 757, normalized size of antiderivative = 2.55 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(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*tan(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))
 

Reduce [B] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 1222, normalized size of antiderivative = 4.11 \[ \int \frac {\sin ^6(c+d x)}{(a+b \tan (c+d x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 96*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**8*b**2 + 288*cos(c + d 
*x)*log(tan((c + d*x)/2)**2 + 1)*a**6*b**4 + 96*cos(c + d*x)*log(tan((c + 
d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**8*b**2 - 288*cos(c + d*x)*log( 
tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**6*b**4 + 8*cos(c + d* 
x)*sin(c + d*x)**6*a**8*b**2 + 32*cos(c + d*x)*sin(c + d*x)**6*a**6*b**4 + 
 48*cos(c + d*x)*sin(c + d*x)**6*a**4*b**6 + 32*cos(c + d*x)*sin(c + d*x)* 
*6*a**2*b**8 + 8*cos(c + d*x)*sin(c + d*x)**6*b**10 + 14*cos(c + d*x)*sin( 
c + d*x)**4*a**8*b**2 + 40*cos(c + d*x)*sin(c + d*x)**4*a**6*b**4 + 36*cos 
(c + d*x)*sin(c + d*x)**4*a**4*b**6 + 8*cos(c + d*x)*sin(c + d*x)**4*a**2* 
b**8 - 2*cos(c + d*x)*sin(c + d*x)**4*b**10 + 33*cos(c + d*x)*sin(c + d*x) 
**2*a**8*b**2 + 48*cos(c + d*x)*sin(c + d*x)**2*a**6*b**4 - 6*cos(c + d*x) 
*sin(c + d*x)**2*a**4*b**6 - 24*cos(c + d*x)*sin(c + d*x)**2*a**2*b**8 - 3 
*cos(c + d*x)*sin(c + d*x)**2*b**10 + 15*cos(c + d*x)*a**10 + 15*cos(c + d 
*x)*a**9*b*d*x - 144*cos(c + d*x)*a**8*b**2 - 240*cos(c + d*x)*a**7*b**3*d 
*x - 138*cos(c + d*x)*a**6*b**4 + 150*cos(c + d*x)*a**5*b**5*d*x + 24*cos( 
c + d*x)*a**4*b**6 + 24*cos(c + d*x)*a**3*b**7*d*x + 3*cos(c + d*x)*a**2*b 
**8 + 3*cos(c + d*x)*a*b**9*d*x - 96*log(tan((c + d*x)/2)**2 + 1)*sin(c + 
d*x)*a**7*b**3 + 288*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**5*b**5 + 
 96*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*a** 
7*b**3 - 288*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(...