Integrand size = 29, antiderivative size = 193 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 b^2 \left (3 a^2-4 b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}-\frac {b \left (a^2-4 b^2\right ) \text {arctanh}(\cos (c+d x))}{a^5 d}+\frac {\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac {4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))} \] Output:
-2*b^2*(3*a^2-4*b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/a^5/ (a^2-b^2)^(1/2)/d-b*(a^2-4*b^2)*arctanh(cos(d*x+c))/a^5/d+1/3*(a^2-12*b^2) *cot(d*x+c)/a^4/d+2*b*cot(d*x+c)*csc(d*x+c)/a^3/d-4/3*cot(d*x+c)*csc(d*x+c )^2/a^2/d+cot(d*x+c)*csc(d*x+c)^2/a/d/(a+b*sin(d*x+c))
Time = 8.22 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.99 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {2 b^2 \left (3 a^2-4 b^2\right ) \arctan \left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \sqrt {a^2-b^2} d}+\frac {\left (a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 a^4 d}+\frac {b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {\left (-a^2 b+4 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}+\frac {\left (a^2 b-4 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^5 d}-\frac {b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{4 a^3 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-a^2 \sin \left (\frac {1}{2} (c+d x)\right )+9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}-\frac {b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 a^2 d} \] Input:
Integrate[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]
Output:
(-2*b^2*(3*a^2 - 4*b^2)*ArcTan[(Sec[(c + d*x)/2]*(b*Cos[(c + d*x)/2] + a*S in[(c + d*x)/2]))/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]*d) + ((a^2*Cos[(c + d*x)/2] - 9*b^2*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(6*a^4*d) + (b*Csc[ (c + d*x)/2]^2)/(4*a^3*d) - (Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(24*a^2* d) + ((-(a^2*b) + 4*b^3)*Log[Cos[(c + d*x)/2]])/(a^5*d) + ((a^2*b - 4*b^3) *Log[Sin[(c + d*x)/2]])/(a^5*d) - (b*Sec[(c + d*x)/2]^2)/(4*a^3*d) + (Sec[ (c + d*x)/2]*(-(a^2*Sin[(c + d*x)/2]) + 9*b^2*Sin[(c + d*x)/2]))/(6*a^4*d) - (b^3*Cos[c + d*x])/(a^4*d*(a + b*Sin[c + d*x])) + (Sec[(c + d*x)/2]^2*T an[(c + d*x)/2])/(24*a^2*d)
Time = 2.04 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.42, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {3042, 3368, 3042, 3535, 3042, 3535, 25, 3042, 3534, 27, 3042, 3534, 27, 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.
| \(\displaystyle \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^2}{\sin (c+d x)^4 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3368 |
| \(\displaystyle \int \frac {\left (1-\sin ^2(c+d x)\right ) \csc ^4(c+d x)}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1-\sin (c+d x)^2}{\sin (c+d x)^4 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\int \frac {\csc ^4(c+d x) \left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^4 (a+b \sin (c+d x))}dx}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle \frac {\frac {\int -\frac {\csc ^3(c+d x) \left (-8 b \left (a^2-b^2\right ) \sin ^2(c+d x)+a \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {\csc ^3(c+d x) \left (-8 b \left (a^2-b^2\right ) \sin ^2(c+d x)+a \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^2-b^2\right )\right )}{a+b \sin (c+d x)}dx}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {-8 b \left (a^2-b^2\right ) \sin (c+d x)^2+a \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^2-b^2\right )}{\sin (c+d x)^3 (a+b \sin (c+d x))}dx}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {2 \csc ^2(c+d x) \left (a^4-13 b^2 a^2-2 b \left (a^2-b^2\right ) \sin (c+d x) a+12 b^4+6 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{2 a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {\csc ^2(c+d x) \left (a^4-13 b^2 a^2-2 b \left (a^2-b^2\right ) \sin (c+d x) a+12 b^4+6 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {a^4-13 b^2 a^2-2 b \left (a^2-b^2\right ) \sin (c+d x) a+12 b^4+6 b^2 \left (a^2-b^2\right ) \sin (c+d x)^2}{\sin (c+d x)^2 (a+b \sin (c+d x))}dx}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3534 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\int -\frac {3 \csc (c+d x) \left (b \left (a^4-5 b^2 a^2+4 b^4\right )-2 a b^2 \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \int \frac {\csc (c+d x) \left (b \left (a^4-5 b^2 a^2+4 b^4\right )-2 a b^2 \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}dx}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \int \frac {b \left (a^4-5 b^2 a^2+4 b^4\right )-2 a b^2 \left (a^2-b^2\right ) \sin (c+d x)}{\sin (c+d x) (a+b \sin (c+d x))}dx}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \left (\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {b^2 \left (3 a^4-7 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \left (\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {b^2 \left (3 a^4-7 a^2 b^2+4 b^4\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a}\right )}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \left (\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {2 b^2 \left (3 a^4-7 a^2 b^2+4 b^4\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{a d}\right )}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \left (\frac {4 b^2 \left (3 a^4-7 a^2 b^2+4 b^4\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \csc (c+d x)dx}{a}\right )}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {-\frac {\frac {-\frac {3 \left (\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \int \csc (c+d x)dx}{a}-\frac {2 b^2 \left (3 a^4-7 a^2 b^2+4 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}\right )}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {-\frac {4 \left (a^2-b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{3 a d}-\frac {\frac {-\frac {3 \left (-\frac {2 b^2 \left (3 a^4-7 a^2 b^2+4 b^4\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {b \left (a^4-5 a^2 b^2+4 b^4\right ) \text {arctanh}(\cos (c+d x))}{a d}\right )}{a}-\frac {\left (a^4-13 a^2 b^2+12 b^4\right ) \cot (c+d x)}{a d}}{a}-\frac {6 b \left (a^2-b^2\right ) \cot (c+d x) \csc (c+d x)}{a d}}{3 a}}{a \left (a^2-b^2\right )}+\frac {\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\) |
Input:
Int[(Cot[c + d*x]^2*Csc[c + d*x]^2)/(a + b*Sin[c + d*x])^2,x]
Output:
((-4*(a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x]^2)/(3*a*d) - (((-3*((-2*b^2*(3* a^4 - 7*a^2*b^2 + 4*b^4)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2])])/(a*Sqrt[a^2 - b^2]*d) - (b*(a^4 - 5*a^2*b^2 + 4*b^4)*ArcTanh[Cos[ c + d*x]])/(a*d)))/a - ((a^4 - 13*a^2*b^2 + 12*b^4)*Cot[c + d*x])/(a*d))/a - (6*b*(a^2 - b^2)*Cot[c + d*x]*Csc[c + d*x])/(a*d))/(3*a))/(a*(a^2 - b^2 )) + (Cot[c + d*x]*Csc[c + d*x]^2)/(a*d*(a + b*Sin[c + d*x]))
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 = 1.11 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.37
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}}{3}-2 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}}-\frac {2 b^{2} \left (\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a b}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (3 a^{2}-4 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{a^{5}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(264\) |
| default | \(\frac {\frac {\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} a^{2}}{3}-2 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}+12 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{4}}-\frac {2 b^{2} \left (\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a b}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a}+\frac {\left (3 a^{2}-4 b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{a^{5}}-\frac {1}{24 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {-a^{2}+12 b^{2}}{8 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{4 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b \left (a^{2}-4 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5}}}{d}\) | \(264\) |
| risch | \(-\frac {2 \left (6 a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+36 i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+12 i b^{3} {\mathrm e}^{6 i \left (d x +c \right )}-30 a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-36 i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+i a^{2} b +6 a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+42 a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+3 i a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}+2 a^{3} {\mathrm e}^{i \left (d x +c \right )}-7 i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+3 i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-18 \,{\mathrm e}^{i \left (d x +c \right )} a \,b^{2}-12 i b^{3}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) d \,a^{4}}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{\sqrt {-a^{2}+b^{2}}\, d \,a^{3}}+\frac {4 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a +a^{2}-b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,a^{5}}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right ) b^{2}}{\sqrt {-a^{2}+b^{2}}\, d \,a^{3}}-\frac {4 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \sqrt {-a^{2}+b^{2}}\, a -a^{2}+b^{2}}{b \sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, d \,a^{5}}-\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{3}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d \,a^{5}}+\frac {b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{3}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d \,a^{5}}\) | \(623\) |
Input:
int(cot(d*x+c)^2*csc(d*x+c)^2/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/8/a^4*(1/3*tan(1/2*d*x+1/2*c)^3*a^2-2*a*b*tan(1/2*d*x+1/2*c)^2-tan( 1/2*d*x+1/2*c)*a^2+12*b^2*tan(1/2*d*x+1/2*c))-2*b^2/a^5*((b^2*tan(1/2*d*x+ 1/2*c)+a*b)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)+(3*a^2-4*b^2 )/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^2)^(1/2)) )-1/24/a^2/tan(1/2*d*x+1/2*c)^3-1/8*(-a^2+12*b^2)/a^4/tan(1/2*d*x+1/2*c)+1 /4*b/a^3/tan(1/2*d*x+1/2*c)^2+1/a^5*b*(a^2-4*b^2)*ln(tan(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 1471, normalized size of antiderivative = 7.62 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2/(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
[-1/6*(2*(a^6 - 7*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c)^3 - 3*(3*a^2*b^3 - 4*b ^5 + (3*a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - 2*(3*a^2*b^3 - 4*b^5)*cos(d*x + c)^2 + (3*a^3*b^2 - 4*a*b^4 - (3*a^3*b^2 - 4*a*b^4)*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)) + 12*( a^4*b^2 - a^2*b^4)*cos(d*x + c) + 3*(a^4*b^2 - 5*a^2*b^4 + 4*b^6 + (a^4*b^ 2 - 5*a^2*b^4 + 4*b^6)*cos(d*x + c)^4 - 2*(a^4*b^2 - 5*a^2*b^4 + 4*b^6)*co s(d*x + c)^2 + (a^5*b - 5*a^3*b^3 + 4*a*b^5 - (a^5*b - 5*a^3*b^3 + 4*a*b^5 )*cos(d*x + c)^2)*sin(d*x + c))*log(1/2*cos(d*x + c) + 1/2) - 3*(a^4*b^2 - 5*a^2*b^4 + 4*b^6 + (a^4*b^2 - 5*a^2*b^4 + 4*b^6)*cos(d*x + c)^4 - 2*(a^4 *b^2 - 5*a^2*b^4 + 4*b^6)*cos(d*x + c)^2 + (a^5*b - 5*a^3*b^3 + 4*a*b^5 - (a^5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))*log(-1/2*cos(d *x + c) + 1/2) + 2*((a^5*b - 13*a^3*b^3 + 12*a*b^5)*cos(d*x + c)^3 - 3*(a^ 5*b - 5*a^3*b^3 + 4*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^7*b - a^5*b^3)* d*cos(d*x + c)^4 - 2*(a^7*b - a^5*b^3)*d*cos(d*x + c)^2 + (a^7*b - a^5*b^3 )*d - ((a^8 - a^6*b^2)*d*cos(d*x + c)^2 - (a^8 - a^6*b^2)*d)*sin(d*x + c)) , -1/6*(2*(a^6 - 7*a^4*b^2 + 6*a^2*b^4)*cos(d*x + c)^3 - 6*(3*a^2*b^3 - 4* b^5 + (3*a^2*b^3 - 4*b^5)*cos(d*x + c)^4 - 2*(3*a^2*b^3 - 4*b^5)*cos(d*x + c)^2 + (3*a^3*b^2 - 4*a*b^4 - (3*a^3*b^2 - 4*a*b^4)*cos(d*x + c)^2)*si...
\[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\cot ^{2}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(cot(d*x+c)**2*csc(d*x+c)**2/(a+b*sin(d*x+c))**2,x)
Output:
Integral(cot(c + d*x)**2*csc(c + d*x)**2/(a + b*sin(c + d*x))**2, x)
Exception generated. \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2/(a+b*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
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
Time = 0.16 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.70 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {24 \, {\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac {48 \, {\left (3 \, a^{2} b^{2} - 4 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}} - \frac {48 \, {\left (b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a b^{3}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )} a^{5}} - \frac {44 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 176 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \] Input:
integrate(cot(d*x+c)^2*csc(d*x+c)^2/(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/24*(24*(a^2*b - 4*b^3)*log(abs(tan(1/2*d*x + 1/2*c)))/a^5 - 48*(3*a^2*b^ 2 - 4*b^4)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*d* x + 1/2*c) + b)/sqrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*a^5) + (a^4*tan(1/2*d*x + 1/2*c)^3 - 6*a^3*b*tan(1/2*d*x + 1/2*c)^2 - 3*a^4*tan(1/2*d*x + 1/2*c) + 36*a^2*b^2*tan(1/2*d*x + 1/2*c))/a^6 - 48*(b^4*tan(1/2*d*x + 1/2*c) + a* b^3)/((a*tan(1/2*d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)*a^5) - (44 *a^2*b*tan(1/2*d*x + 1/2*c)^3 - 176*b^3*tan(1/2*d*x + 1/2*c)^3 - 3*a^3*tan (1/2*d*x + 1/2*c)^2 + 36*a*b^2*tan(1/2*d*x + 1/2*c)^2 - 6*a^2*b*tan(1/2*d* x + 1/2*c) + a^3)/(a^5*tan(1/2*d*x + 1/2*c)^3))/d
Time = 34.23 (sec) , antiderivative size = 1089, normalized size of antiderivative = 5.64 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:
int(cot(c + d*x)^2/(sin(c + d*x)^2*(a + b*sin(c + d*x))^2),x)
Output:
tan(c/2 + (d*x)/2)^3/(24*a^2*d) - (tan(c/2 + (d*x)/2)^2*(8*a*b^2 - (2*a^3) /3) - tan(c/2 + (d*x)/2)^3*(4*a^2*b - 40*b^3) + a^3/3 - (4*a^2*b*tan(c/2 + (d*x)/2))/3 + (tan(c/2 + (d*x)/2)^4*(16*b^4 - a^4 + 12*a^2*b^2))/a)/(d*(8 *a^5*tan(c/2 + (d*x)/2)^3 + 8*a^5*tan(c/2 + (d*x)/2)^5 + 16*a^4*b*tan(c/2 + (d*x)/2)^4)) + (tan(c/2 + (d*x)/2)*(1/(8*a^2) - (16*a^2 + 32*b^2)/(64*a^ 4) + (2*b^2)/a^4))/d - (b*tan(c/2 + (d*x)/2)^2)/(4*a^3*d) + (b*log(tan(c/2 + (d*x)/2))*(a^2 - 4*b^2))/(a^5*d) - (b^2*atan(((b^2*(-(a + b)*(a - b))^( 1/2)*(3*a^2 - 4*b^2)*((2*(8*a^5*b^4 - 4*a^7*b^2))/a^8 + (2*tan(c/2 + (d*x) /2)*(a^7*b + 16*a^3*b^5 - 12*a^5*b^3))/a^7 + (b^2*(-(a + b)*(a - b))^(1/2) *(2*a^2*b - (2*tan(c/2 + (d*x)/2)*(3*a^10 - 4*a^8*b^2))/a^7)*(3*a^2 - 4*b^ 2))/(a^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2) + (b^2*(-(a + b)*(a - b))^(1/2)*( 3*a^2 - 4*b^2)*((2*(8*a^5*b^4 - 4*a^7*b^2))/a^8 + (2*tan(c/2 + (d*x)/2)*(a ^7*b + 16*a^3*b^5 - 12*a^5*b^3))/a^7 - (b^2*(-(a + b)*(a - b))^(1/2)*(2*a^ 2*b - (2*tan(c/2 + (d*x)/2)*(3*a^10 - 4*a^8*b^2))/a^7)*(3*a^2 - 4*b^2))/(a ^7 - a^5*b^2))*1i)/(a^7 - a^5*b^2))/((4*(16*b^7 - 16*a^2*b^5 + 3*a^4*b^3)) /a^8 + (4*tan(c/2 + (d*x)/2)*(16*b^6 - 12*a^2*b^4))/a^7 + (b^2*(-(a + b)*( a - b))^(1/2)*(3*a^2 - 4*b^2)*((2*(8*a^5*b^4 - 4*a^7*b^2))/a^8 + (2*tan(c/ 2 + (d*x)/2)*(a^7*b + 16*a^3*b^5 - 12*a^5*b^3))/a^7 + (b^2*(-(a + b)*(a - b))^(1/2)*(2*a^2*b - (2*tan(c/2 + (d*x)/2)*(3*a^10 - 4*a^8*b^2))/a^7)*(3*a ^2 - 4*b^2))/(a^7 - a^5*b^2)))/(a^7 - a^5*b^2) - (b^2*(-(a + b)*(a - b)...
Time = 0.18 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.07 \[ \int \frac {\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:
int(cot(d*x+c)^2*csc(d*x+c)^2/(a+b*sin(d*x+c))^2,x)
Output:
( - 18*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))* sin(c + d*x)**4*a**2*b**3 + 24*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**4*b**5 - 18*sqrt(a**2 - b**2)*atan(( tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*a**3*b**2 + 24* sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*sin(c + d*x)**3*a*b**4 + cos(c + d*x)*sin(c + d*x)**3*a**5*b - 13*cos(c + d*x)*si n(c + d*x)**3*a**3*b**3 + 12*cos(c + d*x)*sin(c + d*x)**3*a*b**5 + cos(c + d*x)*sin(c + d*x)**2*a**6 - 7*cos(c + d*x)*sin(c + d*x)**2*a**4*b**2 + 6* cos(c + d*x)*sin(c + d*x)**2*a**2*b**4 + 2*cos(c + d*x)*sin(c + d*x)*a**5* b - 2*cos(c + d*x)*sin(c + d*x)*a**3*b**3 - cos(c + d*x)*a**6 + cos(c + d* x)*a**4*b**2 + 3*log(tan((c + d*x)/2))*sin(c + d*x)**4*a**4*b**2 - 15*log( tan((c + d*x)/2))*sin(c + d*x)**4*a**2*b**4 + 12*log(tan((c + d*x)/2))*sin (c + d*x)**4*b**6 + 3*log(tan((c + d*x)/2))*sin(c + d*x)**3*a**5*b - 15*lo g(tan((c + d*x)/2))*sin(c + d*x)**3*a**3*b**3 + 12*log(tan((c + d*x)/2))*s in(c + d*x)**3*a*b**5)/(3*sin(c + d*x)**3*a**5*d*(sin(c + d*x)*a**2*b - si n(c + d*x)*b**3 + a**3 - a*b**2))