Integrand size = 34, antiderivative size = 852 \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^4}{4 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac {6 f (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \csc (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {b (e+f x)^3 \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^3}+\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {3 i \left (a^2-b^2\right ) f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {3 i b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {6 f^3 \operatorname {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^4}-\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {6 \left (a^2-b^2\right ) f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^3}-\frac {3 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{2 i (c+d x)}\right )}{2 a^2 d^3}-\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^4}-\frac {6 i \left (a^2-b^2\right ) f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^4}-\frac {3 i b f^3 \operatorname {PolyLog}\left (4,e^{2 i (c+d x)}\right )}{4 a^2 d^4} \] Output:
6*I*f^2*(f*x+e)*polylog(2,-exp(I*(d*x+c)))/a/d^3+3/2*I*b*f*(f*x+e)^2*polyl og(2,exp(2*I*(d*x+c)))/a^2/d^2-6*f*(f*x+e)^2*arctanh(exp(I*(d*x+c)))/a/d^2 -(f*x+e)^3*csc(d*x+c)/a/d-(a^2-b^2)*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a-( a^2-b^2)^(1/2)))/a^2/b/d-(a^2-b^2)*(f*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a ^2-b^2)^(1/2)))/a^2/b/d-b*(f*x+e)^3*ln(1-exp(2*I*(d*x+c)))/a^2/d+3*I*(a^2- b^2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d ^2+1/4*I*(a^2-b^2)*(f*x+e)^4/a^2/b/f-3/4*I*b*f^3*polylog(4,exp(2*I*(d*x+c) ))/a^2/d^4+3*I*(a^2-b^2)*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2- b^2)^(1/2)))/a^2/b/d^2-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-( a^2-b^2)^(1/2)))/a^2/b/d^4-6*f^3*polylog(3,-exp(I*(d*x+c)))/a/d^4+6*f^3*po lylog(3,exp(I*(d*x+c)))/a/d^4-6*(a^2-b^2)*f^2*(f*x+e)*polylog(3,I*b*exp(I* (d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d^3-6*(a^2-b^2)*f^2*(f*x+e)*polylog(3, I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d^3-3/2*b*f^2*(f*x+e)*polylo g(3,exp(2*I*(d*x+c)))/a^2/d^3-6*I*(a^2-b^2)*f^3*polylog(4,I*b*exp(I*(d*x+c ))/(a+(a^2-b^2)^(1/2)))/a^2/b/d^4+1/4*I*b*(f*x+e)^4/a^2/f-6*I*f^2*(f*x+e)* polylog(2,exp(I*(d*x+c)))/a/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3039\) vs. \(2(852)=1704\).
Time = 14.87 (sec) , antiderivative size = 3039, normalized size of antiderivative = 3.57 \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x ]
Output:
-1/2*(((-2*I)*e^2*(b*d*e - 3*a*f)*x)/d - ((2*I)*e^2*(b*d*e + 3*a*f)*x)/d - (I*b*(e + f*x)^4)/((-1 + E^((2*I)*c))*f) + (6*e*f*(b*d*e - 2*a*f)*x*Log[1 - E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e - a*f)*x^2*Log[1 - E^((-I)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 - E^((-I)*(c + d*x))])/d + (6*e*f*(b*d* e + 2*a*f)*x*Log[1 + E^((-I)*(c + d*x))])/d^2 + (6*f^2*(b*d*e + a*f)*x^2*L og[1 + E^((-I)*(c + d*x))])/d^2 + (2*b*f^3*x^3*Log[1 + E^((-I)*(c + d*x))] )/d + (2*e^2*(b*d*e - 3*a*f)*Log[1 - E^(I*(c + d*x))])/d^2 + (2*e^2*(b*d*e + 3*a*f)*Log[1 + E^(I*(c + d*x))])/d^2 + ((6*I)*e*f*(b*d*e + 2*a*f)*PolyL og[2, -E^((-I)*(c + d*x))])/d^3 + ((12*I)*f^2*(b*d*e + a*f)*x*PolyLog[2, - E^((-I)*(c + d*x))])/d^3 + ((6*I)*b*f^3*x^2*PolyLog[2, -E^((-I)*(c + d*x)) ])/d^2 + ((6*I)*e*f*(b*d*e - 2*a*f)*PolyLog[2, E^((-I)*(c + d*x))])/d^3 + ((12*I)*f^2*(b*d*e - a*f)*x*PolyLog[2, E^((-I)*(c + d*x))])/d^3 + ((6*I)*b *f^3*x^2*PolyLog[2, E^((-I)*(c + d*x))])/d^2 + (12*f^2*(b*d*e + a*f)*PolyL og[3, -E^((-I)*(c + d*x))])/d^4 + (12*b*f^3*x*PolyLog[3, -E^((-I)*(c + d*x ))])/d^3 + (12*f^2*(b*d*e - a*f)*PolyLog[3, E^((-I)*(c + d*x))])/d^4 + (12 *b*f^3*x*PolyLog[3, E^((-I)*(c + d*x))])/d^3 - ((12*I)*b*f^3*PolyLog[4, -E ^((-I)*(c + d*x))])/d^4 - ((12*I)*b*f^3*PolyLog[4, E^((-I)*(c + d*x))])/d^ 4)/a^2 + ((a^2 - b^2)*((4*I)*d^4*e^3*E^((2*I)*c)*x + (6*I)*d^4*e^2*E^((2*I )*c)*f*x^2 + (4*I)*d^4*e*E^((2*I)*c)*f^2*x^3 + I*d^4*E^((2*I)*c)*f^3*x^4 + (2*I)*d^3*e^3*ArcTan[(2*a*E^(I*(c + d*x)))/(b*(-1 + E^((2*I)*(c + d*x)...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle \frac {\int (e+f x)^3 \cos (c+d x) \cot ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\int (e+f x)^3 \cos (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\int (e+f x)^3 \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {-\frac {3 f \int -(e+f x)^2 \sin (c+d x)dx}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \sin (c+d x)dx}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \sin (c+d x)dx}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \int (e+f x) \cos (c+d x)dx}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}+\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle \frac {\int (e+f x)^3 \cot (c+d x) \csc (c+d x)dx+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4910 |
| \(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \csc (c+d x)dx}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \int (e+f x)^2 \csc (c+d x)dx}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (-\frac {2 f \int (e+f x) \log \left (1-e^{i (c+d x)}\right )dx}{d}+\frac {2 f \int (e+f x) \log \left (1+e^{i (c+d x)}\right )dx}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )dx}{d}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {b \int \frac {(e+f x)^3 \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \cos ^2(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle -\frac {b \left (\frac {\int (e+f x)^3 \cot (c+d x)dx-\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (\frac {\int -(e+f x)^3 \tan \left (c+d x+\frac {\pi }{2}\right )dx-\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (\frac {-\int (e+f x)^3 \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}+\frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)^3}{1+e^{i (2 c+2 d x+\pi )}}dx-\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {3 i f \int (e+f x)^2 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\int (e+f x)^3 \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )+\frac {3 f \int (e+f x)^2 \sin ^2(c+d x)dx}{2 d}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )+\frac {3 f \int (e+f x)^2 \sin (c+d x)^2dx}{2 d}-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (-\frac {f^2 \int \sin ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^2dx+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}+2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (-\frac {f^2 \int \sin ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (-\frac {f^2 \int \sin (c+d x)^2dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {3 f \left (\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{d}-\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )de^{i (c+d x)}}{d^2}\right )}{d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{i (c+d x)}\right )}{d}\right )}{d}+\frac {3 f \left (\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \sin (c+d x)}{d}-\frac {(e+f x)^3 \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x)^3 \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {3 f \left (-\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}\right )}{2 d}+2 i \left (\frac {3 i f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{2 d}-\frac {i f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )dx}{d}\right )}{2 d}-\frac {i (e+f x)^3 \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x)^3 \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^4}{4 f}}{a}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
Output:
$Aborted
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[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[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_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x _)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) , x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sin[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]
\[\int \frac {\left (f x +e \right )^{3} \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}}{a +b \sin \left (d x +c \right )}d x\]
Input:
int((f*x+e)^3*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
Output:
int((f*x+e)^3*cos(d*x+c)*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. 3903 vs. \(2 (761) = 1522\).
Time = 0.42 (sec) , antiderivative size = 3903, normalized size of antiderivative = 4.58 \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm= "fricas")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{3} \cos {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:
integrate((f*x+e)**3*cos(d*x+c)*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)
Output:
Integral((e + f*x)**3*cos(c + d*x)*cot(c + d*x)**2/(a + b*sin(c + d*x)), x )
Exception generated. \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x+e)^3*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm= "maxima")
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
\[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm= "giac")
Output:
integrate((f*x + e)^3*cos(d*x + c)*cot(d*x + c)^2/(b*sin(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \] Input:
int((cos(c + d*x)*cot(c + d*x)^2*(e + f*x)^3)/(a + b*sin(c + d*x)),x)
Output:
\text{Hanged}
\[ \int \frac {(e+f x)^3 \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (f x +e \right )^{3} \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}}{\sin \left (d x +c \right ) b +a}d x \] Input:
int((f*x+e)^3*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
Output:
int((f*x+e)^3*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)