Integrand size = 29, antiderivative size = 215 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {A \log (\sin (c+d x))}{a^3 d}-\frac {b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right )^3 d}+\frac {b (A b-a B)}{2 a \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^2 A b+A b^3-2 a^3 B\right )}{a^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \] Output:
-(3*A*a^2*b-A*b^3-B*a^3+3*B*a*b^2)*x/(a^2+b^2)^3+A*ln(sin(d*x+c))/a^3/d-b* (6*A*a^4*b+3*A*a^2*b^3+A*b^5-3*B*a^5+B*a^3*b^2)*ln(a*cos(d*x+c)+b*sin(d*x+ c))/a^3/(a^2+b^2)^3/d+1/2*b*(A*b-B*a)/a/(a^2+b^2)/d/(a+b*tan(d*x+c))^2+b*( 3*A*a^2*b+A*b^3-2*B*a^3)/a^2/(a^2+b^2)^2/d/(a+b*tan(d*x+c))
Result contains complex when optimal does not.
Time = 2.55 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.18 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {-\frac {a (a-i b) (A+i B) \log (i-\tan (c+d x))}{(a+i b)^2}+\frac {2 A \left (a^2+b^2\right ) \log (\tan (c+d x))}{a^2}-\frac {a (a+i b) (A-i B) \log (i+\tan (c+d x))}{(a-i b)^2}-\frac {2 b \left (6 a^4 A b+3 a^2 A b^3+A b^5-3 a^5 B+a^3 b^2 B\right ) \log (a+b \tan (c+d x))}{a^2 \left (a^2+b^2\right )^2}+\frac {b (A b-a B)}{(a+b \tan (c+d x))^2}+\frac {4 a b (A b-a B)}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {2 A b^2}{a^2+a b \tan (c+d x)}}{2 a \left (a^2+b^2\right ) d} \] Input:
Integrate[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
Output:
(-((a*(a - I*b)*(A + I*B)*Log[I - Tan[c + d*x]])/(a + I*b)^2) + (2*A*(a^2 + b^2)*Log[Tan[c + d*x]])/a^2 - (a*(a + I*b)*(A - I*B)*Log[I + Tan[c + d*x ]])/(a - I*b)^2 - (2*b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a^3*b^ 2*B)*Log[a + b*Tan[c + d*x]])/(a^2*(a^2 + b^2)^2) + (b*(A*b - a*B))/(a + b *Tan[c + d*x])^2 + (4*a*b*(A*b - a*B))/((a^2 + b^2)*(a + b*Tan[c + d*x])) + (2*A*b^2)/(a^2 + a*b*Tan[c + d*x]))/(2*a*(a^2 + b^2)*d)
Time = 1.34 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 4092, 27, 3042, 4132, 3042, 4134, 3042, 25, 3956, 4013}
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.
| \(\displaystyle \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \tan (c+d x)}{\tan (c+d x) (a+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4092 |
| \(\displaystyle \frac {\int \frac {2 \cot (c+d x) \left (b (A b-a B) \tan ^2(c+d x)-a (A b-a B) \tan (c+d x)+A \left (a^2+b^2\right )\right )}{(a+b \tan (c+d x))^2}dx}{2 a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cot (c+d x) \left (b (A b-a B) \tan ^2(c+d x)-a (A b-a B) \tan (c+d x)+A \left (a^2+b^2\right )\right )}{(a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {b (A b-a B) \tan (c+d x)^2-a (A b-a B) \tan (c+d x)+A \left (a^2+b^2\right )}{\tan (c+d x) (a+b \tan (c+d x))^2}dx}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\int \frac {\cot (c+d x) \left (-\left (\left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2\right )+A \left (a^2+b^2\right )^2+b \left (-2 B a^3+3 A b a^2+A b^3\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-\left (\left (-B a^2+2 A b a+b^2 B\right ) \tan (c+d x) a^2\right )+A \left (a^2+b^2\right )^2+b \left (-2 B a^3+3 A b a^2+A b^3\right ) \tan (c+d x)^2}{\tan (c+d x) (a+b \tan (c+d x))}dx}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4134 |
| \(\displaystyle \frac {\frac {\frac {A \left (a^2+b^2\right )^2 \int \cot (c+d x)dx}{a}-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^2 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {A \left (a^2+b^2\right )^2 \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^2 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {A \left (a^2+b^2\right )^2 \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{a}-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}-\frac {a^2 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)}dx}{a \left (a^2+b^2\right )}+\frac {A \left (a^2+b^2\right )^2 \log (-\sin (c+d x))}{a d}-\frac {a^2 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}}{a \left (a^2+b^2\right )}+\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}}{a \left (a^2+b^2\right )}+\frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}\) |
| \(\Big \downarrow \) 4013 |
| \(\displaystyle \frac {b (A b-a B)}{2 a d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\frac {b \left (-2 a^3 B+3 a^2 A b+A b^3\right )}{a d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac {\frac {A \left (a^2+b^2\right )^2 \log (-\sin (c+d x))}{a d}-\frac {a^2 x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{a^2+b^2}-\frac {b \left (-3 a^5 B+6 a^4 A b+a^3 b^2 B+3 a^2 A b^3+A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a d \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}}{a \left (a^2+b^2\right )}\) |
Input:
Int[(Cot[c + d*x]*(A + B*Tan[c + d*x]))/(a + b*Tan[c + d*x])^3,x]
Output:
(b*(A*b - a*B))/(2*a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^2) + ((-((a^2*(3*a ^2*A*b - A*b^3 - a^3*B + 3*a*b^2*B)*x)/(a^2 + b^2)) + (A*(a^2 + b^2)^2*Log [-Sin[c + d*x]])/(a*d) - (b*(6*a^4*A*b + 3*a^2*A*b^3 + A*b^5 - 3*a^5*B + a ^3*b^2*B)*Log[a*Cos[c + d*x] + b*Sin[c + d*x]])/(a*(a^2 + b^2)*d))/(a*(a^2 + b^2)) + (b*(3*a^2*A*b + A*b^3 - 2*a^3*B))/(a*(a^2 + b^2)*d*(a + b*Tan[c + d*x])))/(a*(a^2 + b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)* (x_)]), x_Symbol] :> Simp[(c/(b*f))*Log[RemoveContent[a*Cos[e + f*x] + b*Si n[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[b*(A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1) /(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^ 2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b* B*(b*c*(m + 1) + a*d*(n + 1)) + A*(a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2 )) - (A*b - a*B)*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^ 2)/(((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)])), x_Symbol] :> Simp[(a*(A*c - c*C + B*d) + b*(B*c - A*d + C*d))*(x/ ((a^2 + b^2)*(c^2 + d^2))), x] + (Simp[(A*b^2 - a*b*B + a^2*C)/((b*c - a*d) *(a^2 + b^2)) Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x] - Sim p[(c^2*C - B*c*d + A*d^2)/((b*c - a*d)*(c^2 + d^2)) Int[(d - c*Tan[e + f* x])/(c + d*Tan[e + f*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Time = 0.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (A b -B a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(243\) |
| default | \(\frac {\frac {\frac {\left (-A \,a^{3}+3 A a \,b^{2}-3 B \,a^{2} b +B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-3 A \,a^{2} b +A \,b^{3}+B \,a^{3}-3 B a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3}}+\frac {b \left (3 A \,a^{2} b +A \,b^{3}-2 B \,a^{3}\right )}{\left (a^{2}+b^{2}\right )^{2} a^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3} a^{3}}+\frac {\left (A b -B a \right ) b}{2 \left (a^{2}+b^{2}\right ) a \left (a +b \tan \left (d x +c \right )\right )^{2}}}{d}\) | \(243\) |
| parallelrisch | \(\frac {-12 \left (A \,a^{4} b +\frac {1}{2} A \,a^{2} b^{3}+\frac {1}{6} A \,b^{5}-\frac {1}{2} B \,a^{5}+\frac {1}{6} B \,a^{3} b^{2}\right ) b \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (a +b \tan \left (d x +c \right )\right )-a^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )^{2}\right )+2 A \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2} \ln \left (\tan \left (d x +c \right )\right )-6 b^{2} \left (-\frac {B \,a^{6} d x}{3}+b \left (A d x -\frac {B}{3}\right ) a^{5}+\frac {b^{2} \left (2 B d x +A \right ) a^{4}}{2}-\frac {b^{3} \left (A d x +B \right ) a^{3}}{3}+\frac {2 A \,a^{2} b^{4}}{3}+\frac {A \,b^{6}}{6}\right ) \tan \left (d x +c \right )^{2}-12 a^{4} b x d \left (A \,a^{2} b -\frac {1}{3} A \,b^{3}-\frac {1}{3} B \,a^{3}+B a \,b^{2}\right ) \tan \left (d x +c \right )-6 a^{2} \left (-\frac {B \,a^{6} d x}{3}+b \left (A d x +\frac {B}{2}\right ) a^{5}-\frac {2 b^{2} \left (-\frac {3 B d x}{2}+A \right ) a^{4}}{3}-\frac {b^{3} \left (A d x -2 B \right ) a^{3}}{3}-A \,a^{2} b^{4}+\frac {B a \,b^{5}}{6}-\frac {A \,b^{6}}{3}\right )}{2 d \,a^{3} \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )^{2}}\) | \(375\) |
| norman | \(\frac {-\frac {\left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a^{2} x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b \left (4 A \,a^{2} b^{2}+2 A \,b^{4}-3 B \,a^{3} b -B a \,b^{3}\right ) \tan \left (d x +c \right )}{d \,a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{2} \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) x \tan \left (d x +c \right )^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}-\frac {b^{2} \left (7 A \,a^{2} b^{2}+3 A \,b^{4}-5 B \,a^{3} b -B a \,b^{3}\right ) \tan \left (d x +c \right )^{2}}{2 d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 b \left (3 A \,a^{2} b -A \,b^{3}-B \,a^{3}+3 B a \,b^{2}\right ) a x \tan \left (d x +c \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}}{\left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {\left (A \,a^{3}-3 A a \,b^{2}+3 B \,a^{2} b -B \,b^{3}\right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {b \left (6 A \,a^{4} b +3 A \,a^{2} b^{3}+A \,b^{5}-3 B \,a^{5}+B \,a^{3} b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right ) a^{3} d}\) | \(497\) |
| risch | \(-\frac {x B}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}-\frac {6 i a^{2} b B x}{a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}}+\frac {6 i b^{4} A c}{a d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {2 i A c}{a^{3} d}-\frac {2 i A x}{a^{3}}+\frac {2 i b^{6} A c}{a^{3} d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {2 i B \,b^{3} x}{a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}}+\frac {2 i B \,b^{3} c}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {2 i b^{6} A x}{a^{3} \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {6 i a^{2} b B c}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {12 i a \,b^{2} A c}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {2 i b^{2} \left (3 i A \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+i A \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i B \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-4 A \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 A a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 B \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+B \,a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-4 i A \,a^{2} b^{2}-i A \,b^{4}+3 i B \,a^{3} b -4 A \,a^{3} b -A a \,b^{3}+3 B \,a^{4}\right )}{a^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right )^{2} \left (i b +a \right )^{2} d \left (-i b +a \right )^{3}}-\frac {i x A}{3 i a^{2} b -i b^{3}-a^{3}+3 a \,b^{2}}+\frac {6 i b^{4} A x}{a \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {12 i a \,b^{2} A x}{a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}}-\frac {6 a \,b^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {3 b^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{a d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {b^{6} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) A}{a^{3} d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B \,b^{3}}{d \left (a^{6}+3 b^{2} a^{4}+3 b^{4} a^{2}+b^{6}\right )}+\frac {A \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{3} d}\) | \(1010\) |
Input:
int(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x,method=_RETURNVERBOSE )
Output:
1/d*(1/(a^2+b^2)^3*(1/2*(-A*a^3+3*A*a*b^2-3*B*a^2*b+B*b^3)*ln(1+tan(d*x+c) ^2)+(-3*A*a^2*b+A*b^3+B*a^3-3*B*a*b^2)*arctan(tan(d*x+c)))+1/a^3*A*ln(tan( d*x+c))+b*(3*A*a^2*b+A*b^3-2*B*a^3)/(a^2+b^2)^2/a^2/(a+b*tan(d*x+c))-b*(6* A*a^4*b+3*A*a^2*b^3+A*b^5-3*B*a^5+B*a^3*b^2)/(a^2+b^2)^3/a^3*ln(a+b*tan(d* x+c))+1/2*(A*b-B*a)*b/(a^2+b^2)/a/(a+b*tan(d*x+c))^2)
Leaf count of result is larger than twice the leaf count of optimal. 683 vs. \(2 (213) = 426\).
Time = 0.14 (sec) , antiderivative size = 683, normalized size of antiderivative = 3.18 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=-\frac {7 \, B a^{5} b^{3} - 9 \, A a^{4} b^{4} + B a^{3} b^{5} - 3 \, A a^{2} b^{6} - 2 \, {\left (B a^{8} - 3 \, A a^{7} b - 3 \, B a^{6} b^{2} + A a^{5} b^{3}\right )} d x - {\left (5 \, B a^{5} b^{3} - 7 \, A a^{4} b^{4} - B a^{3} b^{5} - A a^{2} b^{6} + 2 \, {\left (B a^{6} b^{2} - 3 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + A a^{3} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left (A a^{8} + 3 \, A a^{6} b^{2} + 3 \, A a^{4} b^{4} + A a^{2} b^{6} + {\left (A a^{6} b^{2} + 3 \, A a^{4} b^{4} + 3 \, A a^{2} b^{6} + A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (A a^{7} b + 3 \, A a^{5} b^{3} + 3 \, A a^{3} b^{5} + A a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (3 \, B a^{7} b - 6 \, A a^{6} b^{2} - B a^{5} b^{3} - 3 \, A a^{4} b^{4} - A a^{2} b^{6} + {\left (3 \, B a^{5} b^{3} - 6 \, A a^{4} b^{4} - B a^{3} b^{5} - 3 \, A a^{2} b^{6} - A b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{6} b^{2} - 6 \, A a^{5} b^{3} - B a^{4} b^{4} - 3 \, A a^{3} b^{5} - A a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, B a^{6} b^{2} - 4 \, A a^{5} b^{3} - 3 \, B a^{4} b^{4} + 3 \, A a^{3} b^{5} + A a b^{7} + 2 \, {\left (B a^{7} b - 3 \, A a^{6} b^{2} - 3 \, B a^{5} b^{3} + A a^{4} b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{9} b^{2} + 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}\right )} d \tan \left (d x + c\right ) + {\left (a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}\right )} d\right )}} \] Input:
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="fri cas")
Output:
-1/2*(7*B*a^5*b^3 - 9*A*a^4*b^4 + B*a^3*b^5 - 3*A*a^2*b^6 - 2*(B*a^8 - 3*A *a^7*b - 3*B*a^6*b^2 + A*a^5*b^3)*d*x - (5*B*a^5*b^3 - 7*A*a^4*b^4 - B*a^3 *b^5 - A*a^2*b^6 + 2*(B*a^6*b^2 - 3*A*a^5*b^3 - 3*B*a^4*b^4 + A*a^3*b^5)*d *x)*tan(d*x + c)^2 - (A*a^8 + 3*A*a^6*b^2 + 3*A*a^4*b^4 + A*a^2*b^6 + (A*a ^6*b^2 + 3*A*a^4*b^4 + 3*A*a^2*b^6 + A*b^8)*tan(d*x + c)^2 + 2*(A*a^7*b + 3*A*a^5*b^3 + 3*A*a^3*b^5 + A*a*b^7)*tan(d*x + c))*log(tan(d*x + c)^2/(tan (d*x + c)^2 + 1)) - (3*B*a^7*b - 6*A*a^6*b^2 - B*a^5*b^3 - 3*A*a^4*b^4 - A *a^2*b^6 + (3*B*a^5*b^3 - 6*A*a^4*b^4 - B*a^3*b^5 - 3*A*a^2*b^6 - A*b^8)*t an(d*x + c)^2 + 2*(3*B*a^6*b^2 - 6*A*a^5*b^3 - B*a^4*b^4 - 3*A*a^3*b^5 - A *a*b^7)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/ (tan(d*x + c)^2 + 1)) - 2*(3*B*a^6*b^2 - 4*A*a^5*b^3 - 3*B*a^4*b^4 + 3*A*a ^3*b^5 + A*a*b^7 + 2*(B*a^7*b - 3*A*a^6*b^2 - 3*B*a^5*b^3 + A*a^4*b^4)*d*x )*tan(d*x + c))/((a^9*b^2 + 3*a^7*b^4 + 3*a^5*b^6 + a^3*b^8)*d*tan(d*x + c )^2 + 2*(a^10*b + 3*a^8*b^3 + 3*a^6*b^5 + a^4*b^7)*d*tan(d*x + c) + (a^11 + 3*a^9*b^2 + 3*a^7*b^4 + a^5*b^6)*d)
Exception generated. \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**3,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Time = 0.14 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.73 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, B a^{5} b - 6 \, A a^{4} b^{2} - B a^{3} b^{3} - 3 \, A a^{2} b^{4} - A b^{6}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{9} + 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} + a^{3} b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, B a^{4} b - 7 \, A a^{3} b^{2} + B a^{2} b^{3} - 3 \, A a b^{4} + 2 \, {\left (2 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - A b^{5}\right )} \tan \left (d x + c\right )}{a^{8} + 2 \, a^{6} b^{2} + a^{4} b^{4} + {\left (a^{6} b^{2} + 2 \, a^{4} b^{4} + a^{2} b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}\right )} \tan \left (d x + c\right )} + \frac {2 \, A \log \left (\tan \left (d x + c\right )\right )}{a^{3}}}{2 \, d} \] Input:
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="max ima")
Output:
1/2*(2*(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(3*B*a^5*b - 6*A*a^4*b^2 - B*a^3*b^3 - 3*A*a^2*b^4 - A*b^6)*log(b*tan(d*x + c) + a)/(a^9 + 3*a^7*b^2 + 3*a^5*b^4 + a^3*b^6) - (A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*log(tan(d*x + c)^2 + 1)/(a^6 + 3* a^4*b^2 + 3*a^2*b^4 + b^6) - (5*B*a^4*b - 7*A*a^3*b^2 + B*a^2*b^3 - 3*A*a* b^4 + 2*(2*B*a^3*b^2 - 3*A*a^2*b^3 - A*b^5)*tan(d*x + c))/(a^8 + 2*a^6*b^2 + a^4*b^4 + (a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*tan(d*x + c)^2 + 2*(a^7*b + 2 *a^5*b^3 + a^3*b^5)*tan(d*x + c)) + 2*A*log(tan(d*x + c))/a^3)/d
Time = 0.33 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.73 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, {\left (a^{6} d + 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d + b^{6} d\right )}} + \frac {{\left (3 \, B a^{5} b^{2} - 6 \, A a^{4} b^{3} - B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{9} b d + 3 \, a^{7} b^{3} d + 3 \, a^{5} b^{5} d + a^{3} b^{7} d} + \frac {A \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3} d} - \frac {5 \, B a^{7} b - 7 \, A a^{6} b^{2} + 6 \, B a^{5} b^{3} - 10 \, A a^{4} b^{4} + B a^{3} b^{5} - 3 \, A a^{2} b^{6} + 2 \, {\left (2 \, B a^{6} b^{2} - 3 \, A a^{5} b^{3} + 2 \, B a^{4} b^{4} - 4 \, A a^{3} b^{5} - A a b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} + b^{2}\right )}^{3} {\left (b \tan \left (d x + c\right ) + a\right )}^{2} a^{3} d} \] Input:
integrate(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x, algorithm="gia c")
Output:
(B*a^3 - 3*A*a^2*b - 3*B*a*b^2 + A*b^3)*(d*x + c)/(a^6*d + 3*a^4*b^2*d + 3 *a^2*b^4*d + b^6*d) - 1/2*(A*a^3 + 3*B*a^2*b - 3*A*a*b^2 - B*b^3)*log(tan( d*x + c)^2 + 1)/(a^6*d + 3*a^4*b^2*d + 3*a^2*b^4*d + b^6*d) + (3*B*a^5*b^2 - 6*A*a^4*b^3 - B*a^3*b^4 - 3*A*a^2*b^5 - A*b^7)*log(abs(b*tan(d*x + c) + a))/(a^9*b*d + 3*a^7*b^3*d + 3*a^5*b^5*d + a^3*b^7*d) + A*log(abs(tan(d*x + c)))/(a^3*d) - 1/2*(5*B*a^7*b - 7*A*a^6*b^2 + 6*B*a^5*b^3 - 10*A*a^4*b^ 4 + B*a^3*b^5 - 3*A*a^2*b^6 + 2*(2*B*a^6*b^2 - 3*A*a^5*b^3 + 2*B*a^4*b^4 - 4*A*a^3*b^5 - A*a*b^7)*tan(d*x + c))/((a^2 + b^2)^3*(b*tan(d*x + c) + a)^ 2*a^3*d)
Time = 5.67 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.47 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx=\frac {\frac {-5\,B\,a^3\,b+7\,A\,a^2\,b^2-B\,a\,b^3+3\,A\,b^4}{2\,a\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (-2\,B\,a^3\,b^2+3\,A\,a^2\,b^3+A\,b^5\right )}{a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {A\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )}-\frac {b\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,B\,a^5+6\,A\,a^4\,b+B\,a^3\,b^2+3\,A\,a^2\,b^3+A\,b^5\right )}{a^3\,d\,{\left (a^2+b^2\right )}^3} \] Input:
int((cot(c + d*x)*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^3,x)
Output:
((3*A*b^4 + 7*A*a^2*b^2 - B*a*b^3 - 5*B*a^3*b)/(2*a*(a^4 + b^4 + 2*a^2*b^2 )) + (tan(c + d*x)*(A*b^5 + 3*A*a^2*b^3 - 2*B*a^3*b^2))/(a^2*(a^4 + b^4 + 2*a^2*b^2)))/(d*(a^2 + b^2*tan(c + d*x)^2 + 2*a*b*tan(c + d*x))) + (A*log( tan(c + d*x)))/(a^3*d) + (log(tan(c + d*x) - 1i)*(A*1i - B))/(2*d*(a*b^2*3 i + 3*a^2*b - a^3*1i - b^3)) + (log(tan(c + d*x) + 1i)*(A - B*1i))/(2*d*(3 *a*b^2 + a^2*b*3i - a^3 - b^3*1i)) - (b*log(a + b*tan(c + d*x))*(A*b^5 - 3 *B*a^5 + 3*A*a^2*b^3 + B*a^3*b^2 + 6*A*a^4*b))/(a^3*d*(a^2 + b^2)^3)
Time = 0.22 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.52 \[ \int \frac {\cot (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^3,x)
Output:
( - cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**5 + cos(c + d*x)*log(tan( (c + d*x)/2)**2 + 1)*a**3*b**2 - 3*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*a**3*b**2 - cos(c + d*x)*log(tan((c + d*x)/2)* *2*a - 2*tan((c + d*x)/2)*b - a)*a*b**4 + cos(c + d*x)*log(tan((c + d*x)/2 ))*a**5 + 2*cos(c + d*x)*log(tan((c + d*x)/2))*a**3*b**2 + cos(c + d*x)*lo g(tan((c + d*x)/2))*a*b**4 - 2*cos(c + d*x)*a**4*b*d*x + cos(c + d*x)*a**3 *b**2 + cos(c + d*x)*a*b**4 - log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a* *4*b + log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*a**2*b**3 - 3*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*a**2*b**3 - log(ta n((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*sin(c + d*x)*b**5 + log(ta n((c + d*x)/2))*sin(c + d*x)*a**4*b + 2*log(tan((c + d*x)/2))*sin(c + d*x) *a**2*b**3 + log(tan((c + d*x)/2))*sin(c + d*x)*b**5 - 2*sin(c + d*x)*a**3 *b**2*d*x)/(a**2*d*(cos(c + d*x)*a**5 + 2*cos(c + d*x)*a**3*b**2 + cos(c + d*x)*a*b**4 + sin(c + d*x)*a**4*b + 2*sin(c + d*x)*a**2*b**3 + sin(c + d* x)*b**5))