Integrand size = 34, antiderivative size = 517 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {e x}{a}+\frac {\left (1-\frac {a^2}{b^2}\right ) e x}{a}-\frac {f x^2}{2 a}+\frac {\left (1-\frac {a^2}{b^2}\right ) f x^2}{2 a}+\frac {2 b (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a^2 b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (a^2-b^2\right ) f \sin (c+d x)}{a^2 b d^2} \]
-e*x/a+(1-a^2/b^2)*e*x/a-1/2*f*x^2/a+1/2*(1-a^2/b^2)*f*x^2/a+2*b*(f*x+e)*a rctanh(exp(I*(d*x+c)))/a^2/d-b*(f*x+e)*cos(d*x+c)/a^2/d-(a^2-b^2)*(f*x+e)* cos(d*x+c)/a^2/b/d-(f*x+e)*cot(d*x+c)/a/d+f*ln(sin(d*x+c))/a/d^2-I*b*f*pol ylog(2,-exp(I*(d*x+c)))/a^2/d^2+I*b*f*polylog(2,exp(I*(d*x+c)))/a^2/d^2-I* (a^2-b^2)^(3/2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b ^2/d+I*(a^2-b^2)^(3/2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)) )/a^2/b^2/d-(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1 /2)))/a^2/b^2/d^2+(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b ^2)^(1/2)))/a^2/b^2/d^2+b*f*sin(d*x+c)/a^2/d^2+(a^2-b^2)*f*sin(d*x+c)/a^2/ b/d^2
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1091\) vs. \(2(517)=1034\).
Time = 12.25 (sec) , antiderivative size = 1091, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a (c+d x) (2 d e-2 c f+f (c+d x))}{2 b^2 d^2}-\frac {(d e-c f+f (c+d x)) \cos (c+d x)}{b d^2}+\frac {\left (-d e \cos \left (\frac {1}{2} (c+d x)\right )+c f \cos \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{2 a d^2}-\frac {b e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d}+\frac {b c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d^2}+\frac {f (\log (\cos (c+d x))+\log (\tan (c+d x)))}{a d^2}-\frac {b f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )\right )\right )}{a^2 d^2}+\frac {\left (a^2-b^2\right )^2 (d e+d f x) \left (\frac {2 (d e-c f) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (-\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}\right )}{a^2 b^2 d^2 \left (d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (d e \sin \left (\frac {1}{2} (c+d x)\right )-c f \sin \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}+\frac {f \sin (c+d x)}{b d^2} \]
-1/2*(a*(c + d*x)*(2*d*e - 2*c*f + f*(c + d*x)))/(b^2*d^2) - ((d*e - c*f + f*(c + d*x))*Cos[c + d*x])/(b*d^2) + ((-(d*e*Cos[(c + d*x)/2]) + c*f*Cos[ (c + d*x)/2] - f*(c + d*x)*Cos[(c + d*x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) - (b*e*Log[Tan[(c + d*x)/2]])/(a^2*d) + (b*c*f*Log[Tan[(c + d*x)/2]])/(a^2* d^2) + (f*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/(a*d^2) - (b*f*((c + d* x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, - E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])))/(a^2*d^2) + ((a^2 - b^2) ^2*(d*e + d*f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[ -a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] + (I*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + b^2] + a *Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2] - (I*f *Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2 ])/((-I)*a + b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] + (I*f*Log[1 + I*Tan [(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] - (I*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2] + (I*f*PolyLog [2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^ 2 + b^2] + (I*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] - (I*f*PolyLog[2, (a + I*a*Tan[(c + d*x)/2]...
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.
| \(\displaystyle \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle \frac {\int (e+f x) \cos ^2(c+d x) \cot ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {\int (e+f x) \cot ^2(c+d x)dx-\int (e+f x) \cos ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {\int (e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {1}{2} \int (e+f x)dx-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\int (e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {(e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \frac {-\frac {f \int -\cot (c+d x)dx}{d}-\int (e+f x)dx-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {(e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f \int -\cot (c+d x)dx}{d}-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {f \int \cot (c+d x)dx}{d}-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {f \int -\tan \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {f \int \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx}{d}-\frac {f \cos ^2(c+d x)}{4 d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^3(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx-\int (e+f x) \cos ^2(c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4905 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx-\frac {f \int \cos ^3(c+d x)dx}{3 d}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx-\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{3 d}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\frac {f \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{3 d^2}+\int (e+f x) \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {-\int (e+f x) \sin (c+d x)dx+\int (e+f x) \csc (c+d x)dx+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {-\int (e+f x) \sin (c+d x)dx+\int (e+f x) \csc (c+d x)dx+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \csc (c+d x)dx-\frac {f \int \cos (c+d x)dx}{d}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \csc (c+d x)dx-\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (\frac {\int (e+f x) \csc (c+d x)dx+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\frac {f \int \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {i f \int e^{-i (c+d x)} \log \left (1-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 5036 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x) \cos ^2(c+d x)dx}{b^2}-\frac {\int (e+f x) \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{b^2}-\frac {\int (e+f x) \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {1}{2} \int (e+f x)dx+\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}\right )}{b^2}-\frac {\int (e+f x) \cos ^2(c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\int (e+f x) \cos ^2(c+d x) \sin (c+d x)dx}{b}+\frac {a \left (\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{b^2}\right )}{a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4905 |
| \(\displaystyle \frac {-\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {f \log (-\sin (c+d x))}{d^2}-\frac {(e+f x) \cot (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}-\frac {3 (e+f x)^2}{4 f}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {f \int \cos ^3(c+d x)dx}{3 d}-\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{b}+\frac {a \left (\frac {f \cos ^2(c+d x)}{4 d^2}+\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{b^2}\right )}{a}+\frac {-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}-\frac {f \sin (c+d x)}{d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}+\frac {(e+f x) \cos (c+d x)}{d}}{a}\right )}{a}\) |
3.4.43.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[a + b*x]^(n + 1)/(b*(n + 1 ))), x] + Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d _.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^ (p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; Fr eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.) *Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[a/b^2 Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] + (-Simp[1/b Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)* Sin[c + d*x], x], x] - Simp[(a^2 - b^2)/b^2 Int[(e + f*x)^m*(Cos[c + d*x] ^(n - 2)/(a + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]^(p_.)*Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + ( f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [1/a Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Simp[b/a I nt[(e + f*x)^m*Cos[c + d*x]^(p + 1)*(Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d* x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5309 vs. \(2 (475 ) = 950\).
Time = 2.82 (sec) , antiderivative size = 5310, normalized size of antiderivative = 10.27
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1751 vs. \(2 (462) = 924\).
Time = 0.55 (sec) , antiderivative size = 1751, normalized size of antiderivative = 3.39 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(2*a^2*b*f*cos(d*x + c)^2 - I*b^3*f*dilog(cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + I*b^3*f*dilog(cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - I*b^3*f*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + I*b^3*f* dilog(-cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - I*(a^2*b - b^3)*f*sqr t(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + I*(a^2*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*dilog((I*a*cos(d*x + c) - a*sin( d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b) /b + 1)*sin(d*x + c) + I*(a^2*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*dilog((-I* a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt (-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) - I*(a^2*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) - 2*a^2 *b*f - ((a^2*b - b^3)*d*e - (a^2*b - b^3)*c*f)*sqrt(-(a^2 - b^2)/b^2)*log( 2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a )*sin(d*x + c) - ((a^2*b - b^3)*d*e - (a^2*b - b^3)*c*f)*sqrt(-(a^2 - b^2) /b^2)*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^ 2) - 2*I*a)*sin(d*x + c) + ((a^2*b - b^3)*d*e - (a^2*b - b^3)*c*f)*sqrt(-( a^2 - b^2)/b^2)*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^ 2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) + ((a^2*b - b^3)*d*e - (a^2*b - b^3...
\[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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
Timed out. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]