Integrand size = 36, antiderivative size = 1014 \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Output:
2*(a^2-b^2)*f^2*sin(d*x+c)/a/b^2/d^3-1/4*(a^2-b^2)*f^2*sin(d*x+c)^2/a^2/b/ d^3+1/2*(a^2-b^2)*(f*x+e)^2*sin(d*x+c)^2/a^2/b/d-1/3*I*(a^2-b^2)^2*(f*x+e) ^3/a^2/b^3/f+2*(a^2-b^2)^2*f^2*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^( 1/2)))/a^2/b^3/d^3+2*(a^2-b^2)^2*f^2*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2- b^2)^(1/2)))/a^2/b^3/d^3+(a^2-b^2)^2*(f*x+e)^2*ln(1-I*b*exp(I*(d*x+c))/(a+ (a^2-b^2)^(1/2)))/a^2/b^3/d+(a^2-b^2)^2*(f*x+e)^2*ln(1-I*b*exp(I*(d*x+c))/ (a-(a^2-b^2)^(1/2)))/a^2/b^3/d-2*f*(f*x+e)*cos(d*x+c)/a/d^2+1/3*I*b*(f*x+e )^3/a^2/f+I*b*f*(f*x+e)*polylog(2,exp(2*I*(d*x+c)))/a^2/d^2-1/4*b*(f*x+e)^ 2/a^2/d+2*f^2*sin(d*x+c)/a/d^3-2*(a^2-b^2)*f*(f*x+e)*cos(d*x+c)/a/b^2/d^2+ 1/2*b*f*(f*x+e)*cos(d*x+c)*sin(d*x+c)/a^2/d^2+1/2*(a^2-b^2)*f*(f*x+e)*cos( d*x+c)*sin(d*x+c)/a^2/b/d^2-2*I*(a^2-b^2)^2*f*(f*x+e)*polylog(2,I*b*exp(I* (d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d^2-2*I*(a^2-b^2)^2*f*(f*x+e)*polylo g(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^3/d^2+2*I*f^2*polylog(2, -exp(I*(d*x+c)))/a/d^3-b*(f*x+e)^2*ln(1-exp(2*I*(d*x+c)))/a^2/d-(f*x+e)^2* csc(d*x+c)/a/d-(a^2-b^2)*(f*x+e)^2*sin(d*x+c)/a/b^2/d-1/4*(a^2-b^2)*(f*x+e )^2/a^2/b/d-1/4*b*f^2*sin(d*x+c)^2/a^2/d^3+1/2*b*(f*x+e)^2*sin(d*x+c)^2/a^ 2/d-4*f*(f*x+e)*arctanh(exp(I*(d*x+c)))/a/d^2-1/2*b*f^2*polylog(3,exp(2*I* (d*x+c)))/a^2/d^3-2*I*f^2*polylog(2,exp(I*(d*x+c)))/a/d^3-(f*x+e)^2*sin(d* x+c)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5075\) vs. \(2(1014)=2028\).
Time = 12.22 (sec) , antiderivative size = 5075, normalized size of antiderivative = 5.00 \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]) ,x]
Output:
Result too large to show
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.
| \(\displaystyle \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle \frac {\int (e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\int (e+f x)^2 \cos ^3(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {2 f^2 \int \cos ^3(c+d x)dx}{9 d^2}-\frac {2}{3} \int (e+f x)^2 \cos (c+d x)dx+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{9 d^2}-\frac {2}{3} \int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {-\frac {2 f^2 \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{9 d^3}-\frac {2}{3} \int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {2}{3} \int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {2}{3} \left (\frac {2 f \int -((e+f x) \sin (c+d x))dx}{d}+\frac {(e+f x)^2 \sin (c+d x)}{d}\right )+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}\right )+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}\right )+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \int \cos (c+d x)dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )+\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\int (e+f x)^2 \cos (c+d x) \cot ^2(c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {-\int (e+f x)^2 \cos (c+d x)dx+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {2 f \int -((e+f x) \sin (c+d x))dx}{d}+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {f \int \cos (c+d x)dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f \left (\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\int (e+f x)^2 \cot (c+d x) \csc (c+d x)dx-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4910 |
| \(\displaystyle \frac {\frac {2 f \int (e+f x) \csc (c+d x)dx}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2 f \int (e+f x) \csc (c+d x)dx}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {2 f \left (-\frac {f \int \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {2 f \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {i f \int e^{-i (c+d x)} \log \left (1+e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^2 \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {2 f \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \cos ^4(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {2 f \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^2 \cos ^2(c+d x) \cot (c+d x)dx-\int (e+f x)^2 \cos ^3(c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {2 f \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 4905 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {f \int (e+f x) \cos ^4(c+d x)dx}{2 d}+\int (e+f x)^2 \cos ^2(c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {2 f \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {-\frac {f \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{2 d}+\int (e+f x)^2 \cos ^2(c+d x) \cot (c+d x)dx+\frac {(e+f x)^2 \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x)^2 \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {2 f \left (-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d^2}\right )}{d}-\frac {2 f^2 \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{9 d^3}-\frac {2 f (e+f x) \cos ^3(c+d x)}{9 d^2}+\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}-\frac {2}{3} \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )-\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {(e+f x)^2 \csc (c+d x)}{d}-\frac {(e+f x)^2 \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}\) |
Input:
Int[((e + f*x)^2*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
Output:
$Aborted
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_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
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[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d _.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x ] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[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]
\[\int \frac {\left (f x +e \right )^{2} \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{2}}{a +b \sin \left (d x +c \right )}d x\]
Input:
int((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
Output:
int((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3131 vs. \(2 (925) = 1850\).
Time = 0.41 (sec) , antiderivative size = 3131, normalized size of antiderivative = 3.09 \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorith m="fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**2*cos(d*x+c)**3*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)
Output:
Integral((e + f*x)**2*cos(c + d*x)**3*cot(c + d*x)**2/(a + b*sin(c + d*x)) , x)
Exception generated. \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorith m="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorith m="giac")
Output:
integrate((f*x + e)^2*cos(d*x + c)^3*cot(d*x + c)^2/(b*sin(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \] Input:
int((cos(c + d*x)^3*cot(c + d*x)^2*(e + f*x)^2)/(a + b*sin(c + d*x)),x)
Output:
\text{Hanged}
\[ \int \frac {(e+f x)^2 \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{2} \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{2}}{\sin \left (d x +c \right ) b +a}d x \] Input:
int((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
Output:
int((f*x+e)^2*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)