\(\int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1250]

Optimal result

Integrand size = 29, antiderivative size = 210 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (3 a^2+10 b^2\right ) \text {arctanh}(\cos (c+d x))}{256 d}-\frac {2 a b \cot ^7(c+d x)}{7 d}-\frac {2 a b \cot ^9(c+d x)}{9 d}+\frac {\left (3 a^2+10 b^2\right ) \cot (c+d x) \csc (c+d x)}{256 d}+\frac {\left (3 a^2-118 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{384 d}-\frac {\left (93 a^2-170 b^2\right ) \cot (c+d x) \csc ^5(c+d x)}{480 d}+\frac {\left (21 a^2-10 b^2\right ) \cot (c+d x) \csc ^7(c+d x)}{80 d}-\frac {a^2 \cot (c+d x) \csc ^9(c+d x)}{10 d} \] Output:

1/256*(3*a^2+10*b^2)*arctanh(cos(d*x+c))/d-2/7*a*b*cot(d*x+c)^7/d-2/9*a*b* 
cot(d*x+c)^9/d+1/256*(3*a^2+10*b^2)*cot(d*x+c)*csc(d*x+c)/d+1/384*(3*a^2-1 
18*b^2)*cot(d*x+c)*csc(d*x+c)^3/d-1/480*(93*a^2-170*b^2)*cot(d*x+c)*csc(d* 
x+c)^5/d+1/80*(21*a^2-10*b^2)*cot(d*x+c)*csc(d*x+c)^7/d-1/10*a^2*cot(d*x+c 
)*csc(d*x+c)^9/d
 

Mathematica [A] (verified)

Time = 2.88 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.16 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {-80640 \left (3 a^2+10 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+80640 \left (3 a^2+10 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^{10}(c+d x) \left (630 \left (1879 a^2+290 b^2\right ) \cos (c+d x)+1260 \left (519 a^2-62 b^2\right ) \cos (3 (c+d x))+218484 a^2 \cos (5 (c+d x))-24360 b^2 \cos (5 (c+d x))+9135 a^2 \cos (7 (c+d x))-77070 b^2 \cos (7 (c+d x))-945 a^2 \cos (9 (c+d x))-3150 b^2 \cos (9 (c+d x))+537600 a b \sin (2 (c+d x))+522240 a b \sin (4 (c+d x))+207360 a b \sin (6 (c+d x))+25600 a b \sin (8 (c+d x))-2560 a b \sin (10 (c+d x))\right )}{20643840 d} \] Input:

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]
 

Output:

-1/20643840*(-80640*(3*a^2 + 10*b^2)*Log[Cos[(c + d*x)/2]] + 80640*(3*a^2 
+ 10*b^2)*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^10*(630*(1879*a^2 + 290*b^2 
)*Cos[c + d*x] + 1260*(519*a^2 - 62*b^2)*Cos[3*(c + d*x)] + 218484*a^2*Cos 
[5*(c + d*x)] - 24360*b^2*Cos[5*(c + d*x)] + 9135*a^2*Cos[7*(c + d*x)] - 7 
7070*b^2*Cos[7*(c + d*x)] - 945*a^2*Cos[9*(c + d*x)] - 3150*b^2*Cos[9*(c + 
 d*x)] + 537600*a*b*Sin[2*(c + d*x)] + 522240*a*b*Sin[4*(c + d*x)] + 20736 
0*a*b*Sin[6*(c + d*x)] + 25600*a*b*Sin[8*(c + d*x)] - 2560*a*b*Sin[10*(c + 
 d*x)]))/d
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {3042, 3390, 3042, 3087, 244, 2009, 4866, 360, 2345, 1471, 27, 298, 215, 219}

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.

\(\displaystyle \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos (c+d x)^6 (a+b \sin (c+d x))^2}{\sin (c+d x)^{11}}dx\)

\(\Big \downarrow \) 3390

\(\displaystyle \int \cot ^6(c+d x) \csc ^5(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right )dx+2 a b \int \cot ^6(c+d x) \csc ^4(c+d x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^5 \tan (c+d x)^6}dx+2 a b \int \sec \left (c+d x-\frac {\pi }{2}\right )^4 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx\)

\(\Big \downarrow \) 3087

\(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^5 \tan (c+d x)^6}dx+\frac {2 a b \int \cot ^6(c+d x) \left (\cot ^2(c+d x)+1\right )d(-\cot (c+d x))}{d}\)

\(\Big \downarrow \) 244

\(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^5 \tan (c+d x)^6}dx+\frac {2 a b \int \left (\cot ^8(c+d x)+\cot ^6(c+d x)\right )d(-\cot (c+d x))}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^5 \tan (c+d x)^6}dx+\frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}\)

\(\Big \downarrow \) 4866

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\int \frac {\cos ^6(c+d x) \left (a^2+b^2-b^2 \cos ^2(c+d x)\right )}{\left (1-\cos ^2(c+d x)\right )^6}d\cos (c+d x)}{d}\)

\(\Big \downarrow \) 360

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}-\frac {1}{10} \int \frac {-10 b^2 \cos ^6(c+d x)+10 a^2 \cos ^4(c+d x)+10 a^2 \cos ^2(c+d x)+a^2}{\left (1-\cos ^2(c+d x)\right )^5}d\cos (c+d x)}{d}\)

\(\Big \downarrow \) 2345

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {1}{10} \left (\frac {1}{8} \int \frac {-80 b^2 \cos ^4(c+d x)+80 \left (a^2-b^2\right ) \cos ^2(c+d x)+13 a^2-10 b^2}{\left (1-\cos ^2(c+d x)\right )^4}d\cos (c+d x)-\frac {\left (21 a^2-10 b^2\right ) \cos (c+d x)}{8 \left (1-\cos ^2(c+d x)\right )^4}\right )+\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}}{d}\)

\(\Big \downarrow \) 1471

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {\left (93 a^2-170 b^2\right ) \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}-\frac {1}{6} \int \frac {5 \left (3 a^2-22 b^2-96 b^2 \cos ^2(c+d x)\right )}{\left (1-\cos ^2(c+d x)\right )^3}d\cos (c+d x)\right )-\frac {\left (21 a^2-10 b^2\right ) \cos (c+d x)}{8 \left (1-\cos ^2(c+d x)\right )^4}\right )+\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {\left (93 a^2-170 b^2\right ) \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}-\frac {5}{6} \int \frac {3 a^2-22 b^2-96 b^2 \cos ^2(c+d x)}{\left (1-\cos ^2(c+d x)\right )^3}d\cos (c+d x)\right )-\frac {\left (21 a^2-10 b^2\right ) \cos (c+d x)}{8 \left (1-\cos ^2(c+d x)\right )^4}\right )+\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}}{d}\)

\(\Big \downarrow \) 298

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {\left (93 a^2-170 b^2\right ) \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}-\frac {5}{6} \left (\frac {3}{4} \left (3 a^2+10 b^2\right ) \int \frac {1}{\left (1-\cos ^2(c+d x)\right )^2}d\cos (c+d x)+\frac {\left (3 a^2-118 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )\right )-\frac {\left (21 a^2-10 b^2\right ) \cos (c+d x)}{8 \left (1-\cos ^2(c+d x)\right )^4}\right )+\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}}{d}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {\left (93 a^2-170 b^2\right ) \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}-\frac {5}{6} \left (\frac {3}{4} \left (3 a^2+10 b^2\right ) \left (\frac {1}{2} \int \frac {1}{1-\cos ^2(c+d x)}d\cos (c+d x)+\frac {\cos (c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}\right )+\frac {\left (3 a^2-118 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )\right )-\frac {\left (21 a^2-10 b^2\right ) \cos (c+d x)}{8 \left (1-\cos ^2(c+d x)\right )^4}\right )+\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}}{d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 a b \left (-\frac {1}{9} \cot ^9(c+d x)-\frac {1}{7} \cot ^7(c+d x)\right )}{d}-\frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {\left (93 a^2-170 b^2\right ) \cos (c+d x)}{6 \left (1-\cos ^2(c+d x)\right )^3}-\frac {5}{6} \left (\frac {3}{4} \left (3 a^2+10 b^2\right ) \left (\frac {1}{2} \text {arctanh}(\cos (c+d x))+\frac {\cos (c+d x)}{2 \left (1-\cos ^2(c+d x)\right )}\right )+\frac {\left (3 a^2-118 b^2\right ) \cos (c+d x)}{4 \left (1-\cos ^2(c+d x)\right )^2}\right )\right )-\frac {\left (21 a^2-10 b^2\right ) \cos (c+d x)}{8 \left (1-\cos ^2(c+d x)\right )^4}\right )+\frac {a^2 \cos (c+d x)}{10 \left (1-\cos ^2(c+d x)\right )^5}}{d}\)

Input:

Int[Cot[c + d*x]^6*Csc[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]
 

Output:

-(((a^2*Cos[c + d*x])/(10*(1 - Cos[c + d*x]^2)^5) + (-1/8*((21*a^2 - 10*b^ 
2)*Cos[c + d*x])/(1 - Cos[c + d*x]^2)^4 + (((93*a^2 - 170*b^2)*Cos[c + d*x 
])/(6*(1 - Cos[c + d*x]^2)^3) - (5*(((3*a^2 - 118*b^2)*Cos[c + d*x])/(4*(1 
 - Cos[c + d*x]^2)^2) + (3*(3*a^2 + 10*b^2)*(ArcTanh[Cos[c + d*x]]/2 + Cos 
[c + d*x]/(2*(1 - Cos[c + d*x]^2))))/4))/6)/8)/10)/d) + (2*a*b*(-1/7*Cot[c 
 + d*x]^7 - Cot[c + d*x]^9/9))/d
 

Defintions of rubi rules used

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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

rule 219

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

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 298

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( 
b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 
2*p + 3))/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, 
 c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
 

rule 360

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] : 
> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p 
+ 1))), x] + Simp[1/(2*b^(m/2 + 1)*(p + 1))   Int[(a + b*x^2)^(p + 1)*Expan 
dToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 
- 1)*(b*c - a*d))/(a + b*x^2)] - (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; 
FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[m/2, 0] & 
& (IntegerQ[p] || EqQ[m + 2*p + 1, 0])
 

rule 1471

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2 + c*x^4)^p, d + e*x^2 
, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], x 
, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Simp[1/(2*d*(q 
 + 1))   Int[(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), 
x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^ 
2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2009

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

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

rule 3042

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

rule 3087

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_S 
ymbol] :> Simp[1/f   Subst[Int[(b*x)^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + 
f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n - 1) 
/2] && LtQ[0, n, m - 1])
 

rule 3390

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n 
_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[2*a*(b/d) 
Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e + f* 
x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, 
e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0]
 

rule 4866

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_), x_Symbol] :> With[{d = Free 
Factors[Cos[c*(a + b*x)], x]}, Simp[-d/(b*c)   Subst[Int[SubstFor[(1 - d^2* 
x^2)^((n - 1)/2), Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] 
 /; FunctionOfQ[Cos[c*(a + b*x)]/d, u, x]] /; FreeQ[{a, b, c}, x] && Intege 
rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(311\)
default	\(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \cos \left (d x +c \right )^{7}}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \cos \left (d x +c \right )^{7}}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \cos \left (d x +c \right )^{5}}{1280}-\frac {\cos \left (d x +c \right )^{3}}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )}{d}\)	\(311\)
risch	\(-\frac {215040 i a b \,{\mathrm e}^{14 i \left (d x +c \right )}+322560 i a b \,{\mathrm e}^{16 i \left (d x +c \right )}-645120 i a b \,{\mathrm e}^{10 i \left (d x +c \right )}+1075200 i a b \,{\mathrm e}^{12 i \left (d x +c \right )}-1183770 a^{2} {\mathrm e}^{11 i \left (d x +c \right )}-182700 b^{2} {\mathrm e}^{11 i \left (d x +c \right )}+78120 b^{2} {\mathrm e}^{13 i \left (d x +c \right )}+3150 b^{2} {\mathrm e}^{i \left (d x +c \right )}-1183770 a^{2} {\mathrm e}^{9 i \left (d x +c \right )}-182700 b^{2} {\mathrm e}^{9 i \left (d x +c \right )}-218484 a^{2} {\mathrm e}^{15 i \left (d x +c \right )}+24360 b^{2} {\mathrm e}^{15 i \left (d x +c \right )}-653940 a^{2} {\mathrm e}^{13 i \left (d x +c \right )}-653940 a^{2} {\mathrm e}^{7 i \left (d x +c \right )}+78120 b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-218484 a^{2} {\mathrm e}^{5 i \left (d x +c \right )}+24360 b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-9135 a^{2} {\mathrm e}^{3 i \left (d x +c \right )}+5120 i a b +3150 b^{2} {\mathrm e}^{19 i \left (d x +c \right )}-9135 a^{2} {\mathrm e}^{17 i \left (d x +c \right )}+77070 b^{2} {\mathrm e}^{17 i \left (d x +c \right )}+945 a^{2} {\mathrm e}^{19 i \left (d x +c \right )}+77070 b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+945 a^{2} {\mathrm e}^{i \left (d x +c \right )}-829440 i a b \,{\mathrm e}^{6 i \left (d x +c \right )}-92160 i a b \,{\mathrm e}^{4 i \left (d x +c \right )}-51200 i a b \,{\mathrm e}^{2 i \left (d x +c \right )}}{40320 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{10}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{256 d}+\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{128 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{256 d}-\frac {5 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{128 d}\)	\(484\)

Input:

int(cot(d*x+c)^6*csc(d*x+c)^5*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(a^2*(-1/10/sin(d*x+c)^10*cos(d*x+c)^7-3/80/sin(d*x+c)^8*cos(d*x+c)^7- 
1/160/sin(d*x+c)^6*cos(d*x+c)^7+1/640/sin(d*x+c)^4*cos(d*x+c)^7-3/1280/sin 
(d*x+c)^2*cos(d*x+c)^7-3/1280*cos(d*x+c)^5-1/256*cos(d*x+c)^3-3/256*cos(d* 
x+c)-3/256*ln(csc(d*x+c)-cot(d*x+c)))+2*a*b*(-1/9/sin(d*x+c)^9*cos(d*x+c)^ 
7-2/63/sin(d*x+c)^7*cos(d*x+c)^7)+b^2*(-1/8/sin(d*x+c)^8*cos(d*x+c)^7-1/48 
/sin(d*x+c)^6*cos(d*x+c)^7+1/192/sin(d*x+c)^4*cos(d*x+c)^7-1/128/sin(d*x+c 
)^2*cos(d*x+c)^7-1/128*cos(d*x+c)^5-5/384*cos(d*x+c)^3-5/128*cos(d*x+c)-5/ 
128*ln(csc(d*x+c)-cot(d*x+c))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (194) = 388\).

Time = 0.11 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.17 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {630 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 420 \, {\left (21 \, a^{2} - 58 \, b^{2}\right )} \cos \left (d x + c\right )^{7} - 5376 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{5} + 2940 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right ) - 315 \, {\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 315 \, {\left ({\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{8} + 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 10 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 5 \, {\left (3 \, a^{2} + 10 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - 10 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 5120 \, {\left (2 \, a b \cos \left (d x + c\right )^{9} - 9 \, a b \cos \left (d x + c\right )^{7}\right )} \sin \left (d x + c\right )}{161280 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )}} \] Input:

integrate(cot(d*x+c)^6*csc(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="frica 
s")
 

Output:

-1/161280*(630*(3*a^2 + 10*b^2)*cos(d*x + c)^9 - 420*(21*a^2 - 58*b^2)*cos 
(d*x + c)^7 - 5376*(3*a^2 + 10*b^2)*cos(d*x + c)^5 + 2940*(3*a^2 + 10*b^2) 
*cos(d*x + c)^3 - 630*(3*a^2 + 10*b^2)*cos(d*x + c) - 315*((3*a^2 + 10*b^2 
)*cos(d*x + c)^10 - 5*(3*a^2 + 10*b^2)*cos(d*x + c)^8 + 10*(3*a^2 + 10*b^2 
)*cos(d*x + c)^6 - 10*(3*a^2 + 10*b^2)*cos(d*x + c)^4 + 5*(3*a^2 + 10*b^2) 
*cos(d*x + c)^2 - 3*a^2 - 10*b^2)*log(1/2*cos(d*x + c) + 1/2) + 315*((3*a^ 
2 + 10*b^2)*cos(d*x + c)^10 - 5*(3*a^2 + 10*b^2)*cos(d*x + c)^8 + 10*(3*a^ 
2 + 10*b^2)*cos(d*x + c)^6 - 10*(3*a^2 + 10*b^2)*cos(d*x + c)^4 + 5*(3*a^2 
 + 10*b^2)*cos(d*x + c)^2 - 3*a^2 - 10*b^2)*log(-1/2*cos(d*x + c) + 1/2) + 
 5120*(2*a*b*cos(d*x + c)^9 - 9*a*b*cos(d*x + c)^7)*sin(d*x + c))/(d*cos(d 
*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d*x + c)^6 - 10*d*cos(d*x + c)^ 
4 + 5*d*cos(d*x + c)^2 - d)
 

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:

integrate(cot(d*x+c)**6*csc(d*x+c)**5*(a+b*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.30 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {63 \, a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 210 \, b^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {5120 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a b}{\tan \left (d x + c\right )^{9}}}{161280 \, d} \] Input:

integrate(cot(d*x+c)^6*csc(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="maxim 
a")
 

Output:

-1/161280*(63*a^2*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x 
+ c)^5 + 70*cos(d*x + c)^3 - 15*cos(d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x 
 + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1) - 
15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 210*b^2*(2*(15*cos( 
d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c)^3 + 15*cos(d*x + c))/(cos 
(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c)^4 - 4*cos(d*x + c)^2 + 1) 
- 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 5120*(9*tan(d*x + 
 c)^2 + 7)*a*b/tan(d*x + c)^9)/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (194) = 388\).

Time = 0.30 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.23 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx =\text {Too large to display} \] Input:

integrate(cot(d*x+c)^6*csc(d*x+c)^5*(a+b*sin(d*x+c))^2,x, algorithm="giac" 
)
 

Output:

1/1290240*(126*a^2*tan(1/2*d*x + 1/2*c)^10 + 560*a*b*tan(1/2*d*x + 1/2*c)^ 
9 - 315*a^2*tan(1/2*d*x + 1/2*c)^8 + 630*b^2*tan(1/2*d*x + 1/2*c)^8 - 2160 
*a*b*tan(1/2*d*x + 1/2*c)^7 - 630*a^2*tan(1/2*d*x + 1/2*c)^6 - 3360*b^2*ta 
n(1/2*d*x + 1/2*c)^6 + 2520*a^2*tan(1/2*d*x + 1/2*c)^4 + 5040*b^2*tan(1/2* 
d*x + 1/2*c)^4 + 13440*a*b*tan(1/2*d*x + 1/2*c)^3 + 1260*a^2*tan(1/2*d*x + 
 1/2*c)^2 + 10080*b^2*tan(1/2*d*x + 1/2*c)^2 - 30240*a*b*tan(1/2*d*x + 1/2 
*c) - 5040*(3*a^2 + 10*b^2)*log(abs(tan(1/2*d*x + 1/2*c))) + (44286*a^2*ta 
n(1/2*d*x + 1/2*c)^10 + 147620*b^2*tan(1/2*d*x + 1/2*c)^10 + 30240*a*b*tan 
(1/2*d*x + 1/2*c)^9 - 1260*a^2*tan(1/2*d*x + 1/2*c)^8 - 10080*b^2*tan(1/2* 
d*x + 1/2*c)^8 - 13440*a*b*tan(1/2*d*x + 1/2*c)^7 - 2520*a^2*tan(1/2*d*x + 
 1/2*c)^6 - 5040*b^2*tan(1/2*d*x + 1/2*c)^6 + 630*a^2*tan(1/2*d*x + 1/2*c) 
^4 + 3360*b^2*tan(1/2*d*x + 1/2*c)^4 + 2160*a*b*tan(1/2*d*x + 1/2*c)^3 + 3 
15*a^2*tan(1/2*d*x + 1/2*c)^2 - 630*b^2*tan(1/2*d*x + 1/2*c)^2 - 560*a*b*t 
an(1/2*d*x + 1/2*c) - 126*a^2)/tan(1/2*d*x + 1/2*c)^10)/d
 

Mupad [B] (verification not implemented)

Time = 23.51 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.88 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{10240\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^2}{256}+\frac {5\,b^2}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (2\,a^2+4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{4}-\frac {b^2}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{2}+\frac {8\,b^2}{3}\right )+\frac {a^2}{10}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (a^2+8\,b^2\right )-\frac {12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{7}+\frac {32\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-24\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{9}\right )}{1024\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{512}+\frac {b^2}{256}\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{1024}+\frac {b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2}{2048}+\frac {b^2}{384}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^2}{4096}-\frac {b^2}{2048}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {3\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{1792\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{2304\,d}-\frac {3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d} \] Input:

int((cot(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x)^5,x)
 

Output:

(a^2*tan(c/2 + (d*x)/2)^10)/(10240*d) - (log(tan(c/2 + (d*x)/2))*((3*a^2)/ 
256 + (5*b^2)/128))/d - (cot(c/2 + (d*x)/2)^10*(tan(c/2 + (d*x)/2)^6*(2*a^ 
2 + 4*b^2) - tan(c/2 + (d*x)/2)^2*(a^2/4 - b^2/2) - tan(c/2 + (d*x)/2)^4*( 
a^2/2 + (8*b^2)/3) + a^2/10 + tan(c/2 + (d*x)/2)^8*(a^2 + 8*b^2) - (12*a*b 
*tan(c/2 + (d*x)/2)^3)/7 + (32*a*b*tan(c/2 + (d*x)/2)^7)/3 - 24*a*b*tan(c/ 
2 + (d*x)/2)^9 + (4*a*b*tan(c/2 + (d*x)/2))/9))/(1024*d) + (tan(c/2 + (d*x 
)/2)^4*(a^2/512 + b^2/256))/d + (tan(c/2 + (d*x)/2)^2*(a^2/1024 + b^2/128) 
)/d - (tan(c/2 + (d*x)/2)^6*(a^2/2048 + b^2/384))/d - (tan(c/2 + (d*x)/2)^ 
8*(a^2/4096 - b^2/2048))/d + (a*b*tan(c/2 + (d*x)/2)^3)/(96*d) - (3*a*b*ta 
n(c/2 + (d*x)/2)^7)/(1792*d) + (a*b*tan(c/2 + (d*x)/2)^9)/(2304*d) - (3*a* 
b*tan(c/2 + (d*x)/2))/(128*d)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.48 \[ \int \cot ^6(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{9} a b +945 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a^{2}+3150 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b^{2}+2560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a b +630 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}-24780 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-38400 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -15624 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+28560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+48640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +21168 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-10080 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-17920 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -8064 \cos \left (d x +c \right ) a^{2}-945 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{10} a^{2}-3150 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{10} b^{2}}{80640 \sin \left (d x +c \right )^{10} d} \] Input:

int(cot(d*x+c)^6*csc(d*x+c)^5*(a+b*sin(d*x+c))^2,x)
 

Output:

(5120*cos(c + d*x)*sin(c + d*x)**9*a*b + 945*cos(c + d*x)*sin(c + d*x)**8* 
a**2 + 3150*cos(c + d*x)*sin(c + d*x)**8*b**2 + 2560*cos(c + d*x)*sin(c + 
d*x)**7*a*b + 630*cos(c + d*x)*sin(c + d*x)**6*a**2 - 24780*cos(c + d*x)*s 
in(c + d*x)**6*b**2 - 38400*cos(c + d*x)*sin(c + d*x)**5*a*b - 15624*cos(c 
 + d*x)*sin(c + d*x)**4*a**2 + 28560*cos(c + d*x)*sin(c + d*x)**4*b**2 + 4 
8640*cos(c + d*x)*sin(c + d*x)**3*a*b + 21168*cos(c + d*x)*sin(c + d*x)**2 
*a**2 - 10080*cos(c + d*x)*sin(c + d*x)**2*b**2 - 17920*cos(c + d*x)*sin(c 
 + d*x)*a*b - 8064*cos(c + d*x)*a**2 - 945*log(tan((c + d*x)/2))*sin(c + d 
*x)**10*a**2 - 3150*log(tan((c + d*x)/2))*sin(c + d*x)**10*b**2)/(80640*si 
n(c + d*x)**10*d)