Integrand size = 27, antiderivative size = 269 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{3/2} d}+\frac {\left (a^2-12 b^2\right ) \text {arctanh}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]
b*(6*a^4-19*a^2*b^2+12*b^4)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2 ))/a^5/(a^2-b^2)^(3/2)/d+1/2*(a^2-12*b^2)*arctanh(cos(d*x+c))/a^5/d+1/2*b* (11*a^2-12*b^2)*cot(d*x+c)/a^4/(a^2-b^2)/d-1/2*(5*a^2-6*b^2)*cot(d*x+c)*cs c(d*x+c)/a^3/(a^2-b^2)/d+1/2*cot(d*x+c)*csc(d*x+c)/a/d/(a+b*sin(d*x+c))^2+ 1/2*(3*a^2-4*b^2)*cot(d*x+c)*csc(d*x+c)/a^2/(a^2-b^2)/d/(a+b*sin(d*x+c))
Time = 7.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.23 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{3/2} d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\left (a^2-12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\left (-a^2+12 b^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {b^2 \cos (c+d x)}{2 a^3 d (a+b \sin (c+d x))^2}+\frac {5 a^2 b^2 \cos (c+d x)-6 b^4 \cos (c+d x)}{2 a^4 (a-b) (a+b) d (a+b \sin (c+d x))}-\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d} \]
(b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x) /2] + a*Sin[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*(a^2 - b^2)^(3/2)*d) + ( 3*b*Cot[(c + d*x)/2])/(2*a^4*d) - Csc[(c + d*x)/2]^2/(8*a^3*d) + ((a^2 - 1 2*b^2)*Log[Cos[(c + d*x)/2]])/(2*a^5*d) + ((-a^2 + 12*b^2)*Log[Sin[(c + d* x)/2]])/(2*a^5*d) + Sec[(c + d*x)/2]^2/(8*a^3*d) + (b^2*Cos[c + d*x])/(2*a ^3*d*(a + b*Sin[c + d*x])^2) + (5*a^2*b^2*Cos[c + d*x] - 6*b^4*Cos[c + d*x ])/(2*a^4*(a - b)*(a + b)*d*(a + b*Sin[c + d*x])) - (3*b*Tan[(c + d*x)/2]) /(2*a^4*d)
Time = 2.07 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.17, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3368, 3042, 3535, 3042, 3535, 3042, 3534, 27, 3042, 3534, 3042, 3480, 3042, 3139, 1083, 217, 4257}
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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(c+d x)\right ) \csc ^3(c+d x)}{(a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-\sin (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int \frac {\csc ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^3 (a+b \sin (c+d x))^2}dx}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\frac {\int \frac {\csc ^3(c+d x) \left (-2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (5 a^4-11 b^2 a^2+6 b^4\right )\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {-2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin (c+d x)^2-a b \left (a^2-b^2\right ) \sin (c+d x)+2 \left (5 a^4-11 b^2 a^2+6 b^4\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {2 \csc ^2(c+d x) \left (-b \left (5 a^4-11 b^2 a^2+6 b^4\right ) \sin ^2(c+d x)+a \left (a^4-3 b^2 a^2+2 b^4\right ) \sin (c+d x)+b \left (11 a^4-23 b^2 a^2+12 b^4\right )\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {\csc ^2(c+d x) \left (-b \left (5 a^4-11 b^2 a^2+6 b^4\right ) \sin ^2(c+d x)+a \left (a^4-3 b^2 a^2+2 b^4\right ) \sin (c+d x)+b \left (11 a^4-23 b^2 a^2+12 b^4\right )\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-b \left (5 a^4-11 b^2 a^2+6 b^4\right ) \sin (c+d x)^2+a \left (a^4-3 b^2 a^2+2 b^4\right ) \sin (c+d x)+b \left (11 a^4-23 b^2 a^2+12 b^4\right )}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {\csc (c+d x) \left (\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2-a b \left (5 a^4-11 b^2 a^2+6 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2-a b \left (5 a^4-11 b^2 a^2+6 b^4\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}-\frac {b \left (6 a^6-25 a^4 b^2+31 a^2 b^4-12 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}-\frac {b \left (6 a^6-25 a^4 b^2+31 a^2 b^4-12 b^6\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}-\frac {2 b \left (6 a^6-25 a^4 b^2+31 a^2 b^4-12 b^6\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}+\frac {4 b \left (6 a^6-25 a^4 b^2+31 a^2 b^4-12 b^6\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \int \csc (c+d x)dx}{a}-\frac {2 b \left (6 a^6-25 a^4 b^2+31 a^2 b^4-12 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}}{a \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{a d (a+b \sin (c+d x))}+\frac {-\frac {\left (5 a^4-11 a^2 b^2+6 b^4\right ) \cot (c+d x) \csc (c+d x)}{a d}-\frac {\frac {-\frac {\left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2 \text {arctanh}(\cos (c+d x))}{a d}-\frac {2 b \left (6 a^6-25 a^4 b^2+31 a^2 b^4-12 b^6\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}}{a}-\frac {b \left (11 a^4-23 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}}{a \left (a^2-b^2\right )}}{2 a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}\) |
(Cot[c + d*x]*Csc[c + d*x])/(2*a*d*(a + b*Sin[c + d*x])^2) + ((-((((-2*b*( 6*a^6 - 25*a^4*b^2 + 31*a^2*b^4 - 12*b^6)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/ 2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - ((a^2 - 12*b^2)*(a^2 - b ^2)^2*ArcTanh[Cos[c + d*x]])/(a*d))/a - (b*(11*a^4 - 23*a^2*b^2 + 12*b^4)* Cot[c + d*x])/(a*d))/a) - ((5*a^4 - 11*a^2*b^2 + 6*b^4)*Cot[c + d*x]*Csc[c + d*x])/(a*d))/(a*(a^2 - b^2)) + ((3*a^2 - 4*b^2)*Cot[c + d*x]*Csc[c + d* x])/(a*d*(a + b*Sin[c + d*x])))/(2*a*(a^2 - b^2))
3.11.90.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, m, n }, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[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*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A *b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[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] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ [n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) | | EqQ[a, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[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*Sin[e + f*x])^(m + 1)*(c + d*Sin [e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.80 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {b \left (6 a^{4}+5 a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {a \,b^{2} \left (17 a^{2}-20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (6 a^{2}-7 b^{2}\right )}{2 a^{2}-2 b^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{4}-19 a^{2} b^{2}+12 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(345\) |
| default | \(\frac {\frac {\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{2}-6 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{4}}+\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {b \left (6 a^{4}+5 a^{2} b^{2}-14 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a^{2}-2 b^{2}}+\frac {a \,b^{2} \left (17 a^{2}-20 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a^{2}-2 b^{2}}+\frac {a^{2} b \left (6 a^{2}-7 b^{2}\right )}{2 a^{2}-2 b^{2}}}{{\left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{2}}+\frac {\left (6 a^{4}-19 a^{2} b^{2}+12 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}-\frac {1}{8 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-2 a^{2}+24 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{5}}+\frac {3 b}{2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\) | \(345\) |
| risch | \(\frac {-30 i b \,{\mathrm e}^{2 i \left (d x +c \right )} a^{4}-36 i b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+44 i a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-5 a^{3} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+6 a \,b^{4} {\mathrm e}^{7 i \left (d x +c \right )}+5 i a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-15 i a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-14 i a^{4} b \,{\mathrm e}^{6 i \left (d x +c \right )}+4 a^{5} {\mathrm e}^{5 i \left (d x +c \right )}+45 a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-54 a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}+11 i a^{2} b^{3}+12 i b^{5} {\mathrm e}^{6 i \left (d x +c \right )}-i a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-87 a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+90 a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}-12 i b^{5}+36 i b^{5} {\mathrm e}^{2 i \left (d x +c \right )}+39 a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}-42 a \,b^{4} {\mathrm e}^{i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) a^{4} d}-\frac {3 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}+\frac {19 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}-\frac {6 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {3 i b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d a}-\frac {19 i b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{3}}+\frac {6 i b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,a^{5}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d \,a^{3}}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b^{2}}{a^{5} d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d \,a^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) b^{2}}{a^{5} d}\) | \(976\) |
1/d*(1/4/a^4*(1/2*tan(1/2*d*x+1/2*c)^2*a-6*b*tan(1/2*d*x+1/2*c))+2*b/a^5*( (1/2*a*b^2*(7*a^2-8*b^2)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^3+1/2*b*(6*a^4+5*a^2 *b^2-14*b^4)/(a^2-b^2)*tan(1/2*d*x+1/2*c)^2+1/2*a*b^2*(17*a^2-20*b^2)/(a^2 -b^2)*tan(1/2*d*x+1/2*c)+1/2*a^2*b*(6*a^2-7*b^2)/(a^2-b^2))/(tan(1/2*d*x+1 /2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+1/2*(6*a^4-19*a^2*b^2+12*b^4)/(a^2-b ^2)^(3/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)))-1/8/a^ 3/tan(1/2*d*x+1/2*c)^2+1/4/a^5*(-2*a^2+24*b^2)*ln(tan(1/2*d*x+1/2*c))+3/2* b/a^4/tan(1/2*d*x+1/2*c))
Leaf count of result is larger than twice the leaf count of optimal. 919 vs. \(2 (254) = 508\).
Time = 0.79 (sec) , antiderivative size = 1922, normalized size of antiderivative = 7.14 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
[-1/4*(2*(17*a^6*b^2 - 35*a^4*b^4 + 18*a^2*b^6)*cos(d*x + c)^3 - (6*a^6*b - 13*a^4*b^3 - 7*a^2*b^5 + 12*b^7 + (6*a^4*b^3 - 19*a^2*b^5 + 12*b^7)*cos( d*x + c)^4 - (6*a^6*b - 7*a^4*b^3 - 26*a^2*b^5 + 24*b^7)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6 - (6*a^5*b^2 - 19*a^3*b^4 + 12*a*b^6) *cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*log(-((2*a^2 - b^2)*cos(d* x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 - 2*(a*cos(d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 2*(a^8 - 19*a^6*b^2 + 36*a^4*b^4 - 18*a^2*b^6)*cos(d*x + c) - (a^8 - 13*a^6*b^2 + 11*a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a ^4*b^4 + 38*a^2*b^6 - 24*b^8)*cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25* a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + (a^8 - 13*a^6*b^2 + 11 *a^4*b^4 + 13*a^2*b^6 - 12*b^8 + (a^6*b^2 - 14*a^4*b^4 + 25*a^2*b^6 - 12*b ^8)*cos(d*x + c)^4 - (a^8 - 12*a^6*b^2 - 3*a^4*b^4 + 38*a^2*b^6 - 24*b^8)* cos(d*x + c)^2 + 2*(a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7 - (a^7*b - 14*a^5*b^3 + 25*a^3*b^5 - 12*a*b^7)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2 *cos(d*x + c) + 1/2) + 2*((11*a^5*b^3 - 23*a^3*b^5 + 12*a*b^7)*cos(d*x + c )^3 - (4*a^7*b + 3*a^5*b^3 - 19*a^3*b^5 + 12*a*b^7)*cos(d*x + c))*sin(d*x + c))/((a^9*b^2 - 2*a^7*b^4 + a^5*b^6)*d*cos(d*x + c)^4 - (a^11 - 3*a^7...
\[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (254) = 508\).
Time = 0.52 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.96 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {8 \, {\left (6 \, a^{4} b - 19 \, a^{2} b^{3} + 12 \, b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{7} - a^{5} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 26 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 20 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 32 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 53 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 64 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 28 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 112 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 68 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 76 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{6} + a^{4} b^{2}}{{\left (a^{7} - a^{5} b^{2}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2}} - \frac {4 \, {\left (a^{2} - 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \]
1/8*(8*(6*a^4*b - 19*a^2*b^3 + 12*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*s gn(a) + arctan((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^7 - a^5* b^2)*sqrt(a^2 - b^2)) + (2*a^6*tan(1/2*d*x + 1/2*c)^6 - 26*a^4*b^2*tan(1/2 *d*x + 1/2*c)^6 + 24*a^2*b^4*tan(1/2*d*x + 1/2*c)^6 + 20*a^5*b*tan(1/2*d*x + 1/2*c)^5 - 60*a^3*b^3*tan(1/2*d*x + 1/2*c)^5 + 32*a*b^5*tan(1/2*d*x + 1 /2*c)^5 + 3*a^6*tan(1/2*d*x + 1/2*c)^4 + 53*a^4*b^2*tan(1/2*d*x + 1/2*c)^4 - 64*a^2*b^4*tan(1/2*d*x + 1/2*c)^4 - 16*b^6*tan(1/2*d*x + 1/2*c)^4 + 28* a^5*b*tan(1/2*d*x + 1/2*c)^3 + 60*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 - 112*a*b ^5*tan(1/2*d*x + 1/2*c)^3 + 68*a^4*b^2*tan(1/2*d*x + 1/2*c)^2 - 76*a^2*b^4 *tan(1/2*d*x + 1/2*c)^2 + 8*a^5*b*tan(1/2*d*x + 1/2*c) - 8*a^3*b^3*tan(1/2 *d*x + 1/2*c) - a^6 + a^4*b^2)/((a^7 - a^5*b^2)*(a*tan(1/2*d*x + 1/2*c)^3 + 2*b*tan(1/2*d*x + 1/2*c)^2 + a*tan(1/2*d*x + 1/2*c))^2) - 4*(a^2 - 12*b^ 2)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 + (a^3*tan(1/2*d*x + 1/2*c)^2 - 12*a ^2*b*tan(1/2*d*x + 1/2*c))/a^6)/d
Time = 12.38 (sec) , antiderivative size = 1906, normalized size of antiderivative = 7.09 \[ \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \]
tan(c/2 + (d*x)/2)^2/(8*a^3*d) - (a^3/2 - (2*tan(c/2 + (d*x)/2)^5*(3*a^4*b - 16*b^5 + 11*a^2*b^3))/(a^2 - b^2) - (2*tan(c/2 + (d*x)/2)^3*(5*a^4*b - 52*b^5 + 41*a^2*b^3))/(a^2 - b^2) - 4*a^2*b*tan(c/2 + (d*x)/2) + (tan(c/2 + (d*x)/2)^2*(50*a*b^4 + a^5 - 47*a^3*b^2))/(a^2 - b^2) + (tan(c/2 + (d*x) /2)^4*(a^6 + 112*b^6 + 8*a^2*b^4 - 97*a^4*b^2))/(2*a*(a^2 - b^2)))/(d*(4*a ^6*tan(c/2 + (d*x)/2)^2 + 4*a^6*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^ 4*(8*a^6 + 16*a^4*b^2) + 16*a^5*b*tan(c/2 + (d*x)/2)^3 + 16*a^5*b*tan(c/2 + (d*x)/2)^5)) - (3*b*tan(c/2 + (d*x)/2))/(2*a^4*d) - (log(tan(c/2 + (d*x) /2))*(a^2 - 12*b^2))/(2*a^5*d) - (b*atan(((b*(-(a + b)^3*(a - b)^3)^(1/2)* (6*a^4 + 12*b^4 - 19*a^2*b^2)*((7*a^9*b + 24*a^5*b^5 - 32*a^7*b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(a^11 + 48*a^3*b^8 - 124*a^5*b^6 + 103*a^7 *b^4 - 28*a^9*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2) + (b*((2*a^12*b - 2*a^10* b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(6*a^14 - 8*a^8*b^6 + 22*a^10* b^4 - 20*a^12*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2))*(-(a + b)^3*(a - b)^3)^( 1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2))/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9 *b^2)))*1i)/(2*(a^11 - a^5*b^6 + 3*a^7*b^4 - 3*a^9*b^2)) - (b*(-(a + b)^3* (a - b)^3)^(1/2)*(6*a^4 + 12*b^4 - 19*a^2*b^2)*((tan(c/2 + (d*x)/2)*(a^11 + 48*a^3*b^8 - 124*a^5*b^6 + 103*a^7*b^4 - 28*a^9*b^2))/(a^11 + a^7*b^4 - 2*a^9*b^2) - (7*a^9*b + 24*a^5*b^5 - 32*a^7*b^3)/(a^10 - a^8*b^2) + (b*((2 *a^12*b - 2*a^10*b^3)/(a^10 - a^8*b^2) - (tan(c/2 + (d*x)/2)*(6*a^14 - ...