Integrand size = 28, antiderivative size = 369 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {3 f (e+f x)^2}{8 a d^2}+\frac {i (e+f x)^3}{a d}+\frac {3 (e+f x)^4}{8 a f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{a d^3}+\frac {(e+f x)^3 \cos (c+d x)}{a d}+\frac {(e+f x)^3 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {6 f (e+f x)^2 \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right )}{a d^4}+\frac {6 f^3 \sin (c+d x)}{a d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{a d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {3 f^3 \sin ^2(c+d x)}{8 a d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 a d^2} \] Output:
-3/8*f*(f*x+e)^2/a/d^2+I*(f*x+e)^3/a/d+3/8*(f*x+e)^4/a/f-6*f^2*(f*x+e)*cos (d*x+c)/a/d^3+(f*x+e)^3*cos(d*x+c)/a/d+(f*x+e)^3*cot(1/2*c+1/4*Pi+1/2*d*x) /a/d-6*f*(f*x+e)^2*ln(1-I*exp(I*(d*x+c)))/a/d^2+12*I*f^2*(f*x+e)*polylog(2 ,I*exp(I*(d*x+c)))/a/d^3-12*f^3*polylog(3,I*exp(I*(d*x+c)))/a/d^4+6*f^3*si n(d*x+c)/a/d^4-3*f*(f*x+e)^2*sin(d*x+c)/a/d^2+3/4*f^2*(f*x+e)*cos(d*x+c)*s in(d*x+c)/a/d^3-1/2*(f*x+e)^3*cos(d*x+c)*sin(d*x+c)/a/d-3/8*f^3*sin(d*x+c) ^2/a/d^4+3/4*f*(f*x+e)^2*sin(d*x+c)^2/a/d^2
Time = 4.14 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.46 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {48 e^3 x+72 e^2 f x^2+48 e f^2 x^3+12 f^3 x^4+\frac {192 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^3 (\cos (c)-i \sin (c))}{3 f}-\frac {(e+f x)^2 \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {2 f (d (e+f x) \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x))-i f \operatorname {PolyLog}(3,-i \cos (c+d x)-\sin (c+d x))) (\cos (c)-i (1+\sin (c)))}{d^3}\right )}{d (\cos (c)+i (1+\sin (c)))}-\frac {64 (e+f x)^3 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {16 \left (6 i f^3-6 d f^2 (e+f x)-3 i d^2 f (e+f x)^2+d^3 (e+f x)^3\right ) (\cos (c+d x)-i \sin (c+d x))}{d^4}+\frac {16 \left (-6 i f^3-6 d f^2 (e+f x)+3 i d^2 f (e+f x)^2+d^3 (e+f x)^3\right ) (\cos (c+d x)+i \sin (c+d x))}{d^4}+\frac {\left (3 f^3+6 i d f^2 (e+f x)-6 d^2 f (e+f x)^2-4 i d^3 (e+f x)^3\right ) (\cos (2 (c+d x))-i \sin (2 (c+d x)))}{d^4}+\frac {\left (3 f^3-6 i d f^2 (e+f x)-6 d^2 f (e+f x)^2+4 i d^3 (e+f x)^3\right ) (\cos (2 (c+d x))+i \sin (2 (c+d x)))}{d^4}}{32 a} \] Input:
Integrate[((e + f*x)^3*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
(48*e^3*x + 72*e^2*f*x^2 + 48*e*f^2*x^3 + 12*f^3*x^4 + (192*f*(Cos[c] + I* Sin[c])*(((e + f*x)^3*(Cos[c] - I*Sin[c]))/(3*f) - ((e + f*x)^2*Log[1 + I* Cos[c + d*x] + Sin[c + d*x]]*(1 + I*Cos[c] + Sin[c]))/d + (2*f*(d*(e + f*x )*PolyLog[2, (-I)*Cos[c + d*x] - Sin[c + d*x]] - I*f*PolyLog[3, (-I)*Cos[c + d*x] - Sin[c + d*x]])*(Cos[c] - I*(1 + Sin[c])))/d^3))/(d*(Cos[c] + I*( 1 + Sin[c]))) - (64*(e + f*x)^3*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Co s[(c + d*x)/2] + Sin[(c + d*x)/2])) + (16*((6*I)*f^3 - 6*d*f^2*(e + f*x) - (3*I)*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(Cos[c + d*x] - I*Sin[c + d*x] ))/d^4 + (16*((-6*I)*f^3 - 6*d*f^2*(e + f*x) + (3*I)*d^2*f*(e + f*x)^2 + d ^3*(e + f*x)^3)*(Cos[c + d*x] + I*Sin[c + d*x]))/d^4 + ((3*f^3 + (6*I)*d*f ^2*(e + f*x) - 6*d^2*f*(e + f*x)^2 - (4*I)*d^3*(e + f*x)^3)*(Cos[2*(c + d* x)] - I*Sin[2*(c + d*x)]))/d^4 + ((3*f^3 - (6*I)*d*f^2*(e + f*x) - 6*d^2*f *(e + f*x)^2 + (4*I)*d^3*(e + f*x)^3)*(Cos[2*(c + d*x)] + I*Sin[2*(c + d*x )]))/d^4)/(32*a)
Time = 3.13 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.11, number of steps used = 31, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {5026, 3042, 3792, 17, 3042, 3791, 17, 5026, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117, 5026, 17, 3042, 3799, 3042, 4672, 3042, 25, 4202, 2620, 3011, 2720, 7143}
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.
| \(\displaystyle \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5026 |
| \(\displaystyle \frac {\int (e+f x)^3 \sin ^2(c+d x)dx}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^3 \sin (c+d x)^2dx}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \sin ^2(c+d x)dx}{2 d^2}+\frac {1}{2} \int (e+f x)^3dx+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \sin ^2(c+d x)dx}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \sin (c+d x)^2dx}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3791 |
| \(\displaystyle \frac {-\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\int \frac {(e+f x)^3 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 5026 |
| \(\displaystyle -\frac {\int (e+f x)^3 \sin (c+d x)dx}{a}+\int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (e+f x)^3 \sin (c+d x)dx}{a}+\int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \int (e+f x)^2 \cos (c+d x)dx}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \left (\frac {2 f \int -((e+f x) \sin (c+d x))dx}{d}+\frac {(e+f x)^2 \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \int \cos (c+d x)dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \int \frac {(e+f x)^3 \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 5026 |
| \(\displaystyle -\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)^3dx}{a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {(e+f x)^3}{\sin (c+d x) a+a}dx+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -\frac {\int (e+f x)^3 \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int (e+f x)^3 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {\frac {6 f \int (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {6 f \int -(e+f x)^2 \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {-\frac {6 f \int (e+f x)^2 \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)^2}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \int (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {i f \int \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}+\frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{a}-\frac {-\frac {2 (e+f x)^3 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {6 f \left (\frac {i (e+f x)^3}{3 f}-2 i \left (\frac {2 i f \left (\frac {i (e+f x) \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}-\frac {f \operatorname {PolyLog}\left (3,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}-\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^4}{4 a f}\) |
Input:
Int[((e + f*x)^3*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
Output:
(e + f*x)^4/(4*a*f) - ((-2*(e + f*x)^3*Cot[c/2 + Pi/4 + (d*x)/2])/d - (6*f *(((I/3)*(e + f*x)^3)/f - (2*I)*(((-I)*(e + f*x)^2*Log[1 + E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d + ((2*I)*f*((I*(e + f*x)*PolyLog[2, -E^((I/2)*(2*c + 3 *Pi + 2*d*x))])/d - (f*PolyLog[3, -E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d^2))/ d)))/d)/(2*a) + ((e + f*x)^4/(8*f) - ((e + f*x)^3*Cos[c + d*x]*Sin[c + d*x ])/(2*d) + (3*f*(e + f*x)^2*Sin[c + d*x]^2)/(4*d^2) - (3*f^2*((e + f*x)^2/ (4*f) - ((e + f*x)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (f*Sin[c + d*x]^2)/( 4*d^2)))/(2*d^2))/a - (-(((e + f*x)^3*Cos[c + d*x])/d) + (3*f*(((e + f*x)^ 2*Sin[c + d*x])/d - (2*f*(-(((e + f*x)*Cos[c + d*x])/d) + (f*Sin[c + d*x]) /d^2))/d))/d)/a
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[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[((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_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
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_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_. )*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sin[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*(Sin[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]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (340 ) = 680\).
Time = 2.20 (sec) , antiderivative size = 1054, normalized size of antiderivative = 2.86
| method | result | size |
| risch | \(-\frac {3 f \,e^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a \,d^{2}}-\frac {12 f^{3} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}+\frac {3 f^{2} e \,x^{3}}{2 a}+\frac {6 f \,e^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {6 f^{3} c^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}+\frac {6 f^{3} c^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {6 f^{3} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x^{2}}{a \,d^{2}}+\frac {2 i f^{3} x^{3}}{a d}-\frac {4 i f^{3} c^{3}}{a \,d^{4}}+\frac {12 i f^{2} e c x}{a \,d^{2}}-\frac {12 i f^{2} c e \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {12 i f^{3} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{3}}+\frac {6 i f^{2} e \,x^{2}}{a d}+\frac {6 i f^{2} e \,c^{2}}{a \,d^{3}}+\frac {12 i f^{2} e \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {6 i f^{3} c^{2} \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{4}}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}-\frac {12 f^{2} c e \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {12 f^{2} e \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}+\frac {3 e^{4}}{8 a f}+\frac {6 f^{2} e c \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a \,d^{3}}+\frac {6 i f \,e^{2} \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {\left (d^{3} x^{3} f^{3}+3 e \,f^{2} x^{2} d^{3}+3 i d^{2} f^{3} x^{2}+3 e^{2} f x \,d^{3}+6 i d^{2} e \,f^{2} x +d^{3} e^{3}+3 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}-6 i f^{3}\right ) {\mathrm e}^{i \left (d x +c \right )}}{2 a \,d^{4}}+\frac {\left (d^{3} x^{3} f^{3}+3 e \,f^{2} x^{2} d^{3}-3 i d^{2} f^{3} x^{2}+3 e^{2} f x \,d^{3}-6 i d^{2} e \,f^{2} x +d^{3} e^{3}-3 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}+6 i f^{3}\right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 a \,d^{4}}-\frac {6 i f^{3} c^{2} x}{d^{3} a}+\frac {3 f^{3} x^{4}}{8 a}+\frac {2 f^{3} x^{3}+6 e \,f^{2} x^{2}+6 e^{2} f x +2 e^{3}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {9 f \,e^{2} x^{2}}{4 a}+\frac {3 e^{3} x}{2 a}+\frac {i \left (4 d^{3} x^{3} f^{3}+12 e \,f^{2} x^{2} d^{3}+6 i d^{2} f^{3} x^{2}+12 e^{2} f x \,d^{3}+12 i d^{2} e \,f^{2} x +4 d^{3} e^{3}+6 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}-3 i f^{3}\right ) {\mathrm e}^{2 i \left (d x +c \right )}}{32 a \,d^{4}}-\frac {i \left (4 d^{3} x^{3} f^{3}+12 e \,f^{2} x^{2} d^{3}-6 i d^{2} f^{3} x^{2}+12 e^{2} f x \,d^{3}-12 i d^{2} e \,f^{2} x +4 d^{3} e^{3}-6 i d^{2} e^{2} f -6 d \,f^{3} x -6 d e \,f^{2}+3 i f^{3}\right ) {\mathrm e}^{-2 i \left (d x +c \right )}}{32 a \,d^{4}}-\frac {3 f^{3} c^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a \,d^{4}}\) | \(1054\) |
Input:
int((f*x+e)^3*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/2*(d^3*x^3*f^3+3*I*d^2*f^3*x^2+3*e*f^2*x^2*d^3+6*I*d^2*e*f^2*x+3*e^2*f*x *d^3+3*I*d^2*e^2*f+d^3*e^3-6*d*f^3*x-6*I*f^3-6*d*e*f^2)/a/d^4*exp(I*(d*x+c ))+1/2*(d^3*x^3*f^3-3*I*d^2*f^3*x^2+3*e*f^2*x^2*d^3-6*I*d^2*e*f^2*x+3*e^2* f*x*d^3-3*I*d^2*e^2*f+d^3*e^3-6*d*f^3*x+6*I*f^3-6*d*e*f^2)/a/d^4*exp(-I*(d *x+c))+2*(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e^3)/d/a/(exp(I*(d*x+c))+I)+3/2/a* f^2*e*x^3+3/8/a/f*e^4+6/a/d^4*f^3*c^2*ln(exp(I*(d*x+c)))+6/a/d^2*f*e^2*ln( exp(I*(d*x+c)))-6/a/d^2*f^3*ln(1-I*exp(I*(d*x+c)))*x^2+6/a/d^4*f^3*c^2*ln( 1-I*exp(I*(d*x+c)))+2*I/a/d*f^3*x^3-4*I/a/d^4*f^3*c^3-12*f^3*polylog(3,I*e xp(I*(d*x+c)))/a/d^4-12*I/a/d^3*f^2*e*c*arctan(exp(I*(d*x+c)))+12*I/a/d^2* f^2*e*c*x-12/a/d^3*f^2*e*c*ln(exp(I*(d*x+c)))+6/a/d^3*f^2*e*c*ln(exp(2*I*( d*x+c))+1)-12/a/d^2*f^2*e*ln(1-I*exp(I*(d*x+c)))*x+6*I/a/d^2*f*e^2*arctan( exp(I*(d*x+c)))+12*I/a/d^3*f^2*e*polylog(2,I*exp(I*(d*x+c)))-6*I/a/d^3*f^3 *c^2*x+12*I/a/d^3*f^3*polylog(2,I*exp(I*(d*x+c)))*x+6*I/a/d*f^2*e*x^2+6*I/ a/d^3*f^2*e*c^2+6*I/a/d^4*f^3*c^2*arctan(exp(I*(d*x+c)))-12/a/d^3*f^2*e*ln (1-I*exp(I*(d*x+c)))*c+3/8/a*f^3*x^4+1/32*I*(4*d^3*x^3*f^3+6*I*d^2*f^3*x^2 +12*e*f^2*x^2*d^3+12*I*d^2*e*f^2*x+12*e^2*f*x*d^3+6*I*d^2*e^2*f+4*d^3*e^3- 6*d*f^3*x-3*I*f^3-6*d*e*f^2)/a/d^4*exp(2*I*(d*x+c))-1/32*I*(4*d^3*x^3*f^3- 6*I*d^2*f^3*x^2+12*e*f^2*x^2*d^3-12*I*d^2*e*f^2*x+12*e^2*f*x*d^3-6*I*d^2*e ^2*f+4*d^3*e^3-6*d*f^3*x+3*I*f^3-6*d*e*f^2)/a/d^4*exp(-2*I*(d*x+c))+9/4/a* f*e^2*x^2+3/2/a*e^3*x-3/a/d^2*f*e^2*ln(exp(2*I*(d*x+c))+1)-3/a/d^4*f^3*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1565 vs. \(2 (334) = 668\).
Time = 0.14 (sec) , antiderivative size = 1565, normalized size of antiderivative = 4.24 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/16*(6*d^4*f^3*x^4 + 16*d^3*e^3 - 42*d^2*e^2*f + 8*(3*d^4*e*f^2 + 2*d^3*f ^3)*x^3 + 2*(4*d^3*f^3*x^3 + 4*d^3*e^3 - 6*d^2*e^2*f - 6*d*e*f^2 + 3*f^3 + 6*(2*d^3*e*f^2 - d^2*f^3)*x^2 + 6*(2*d^3*e^2*f - 2*d^2*e*f^2 - d*f^3)*x)* cos(d*x + c)^3 + 93*f^3 + 6*(6*d^4*e^2*f + 8*d^3*e*f^2 - 7*d^2*f^3)*x^2 + 2*(8*d^3*f^3*x^3 + 8*d^3*e^3 + 18*d^2*e^2*f - 48*d*e*f^2 - 45*f^3 + 6*(4*d ^3*e*f^2 + 3*d^2*f^3)*x^2 + 12*(2*d^3*e^2*f + 3*d^2*e*f^2 - 4*d*f^3)*x)*co s(d*x + c)^2 + 12*(2*d^4*e^3 + 4*d^3*e^2*f - 7*d^2*e*f^2)*x + 3*(2*d^4*f^3 *x^4 + 8*d^3*e^3 + 2*d^2*e^2*f - 28*d*e*f^2 + 8*(d^4*e*f^2 + d^3*f^3)*x^3 - f^3 + 2*(6*d^4*e^2*f + 12*d^3*e*f^2 + d^2*f^3)*x^2 + 4*(2*d^4*e^3 + 6*d^ 3*e^2*f + d^2*e*f^2 - 7*d*f^3)*x)*cos(d*x + c) - 96*(-I*d*f^3*x - I*d*e*f^ 2 + (-I*d*f^3*x - I*d*e*f^2)*cos(d*x + c) + (-I*d*f^3*x - I*d*e*f^2)*sin(d *x + c))*dilog(I*cos(d*x + c) - sin(d*x + c)) - 96*(I*d*f^3*x + I*d*e*f^2 + (I*d*f^3*x + I*d*e*f^2)*cos(d*x + c) + (I*d*f^3*x + I*d*e*f^2)*sin(d*x + c))*dilog(-I*cos(d*x + c) - sin(d*x + c)) - 48*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*cos(d*x + c) + (d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3)*sin(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - 48*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 + (d^2*f^3*x^ 2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*cos(d*x + c) + (d^2*f^3*x^2 + 2 *d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3)*sin(d*x + c))*log(I*cos(d*x + c) + s in(d*x + c) + 1) - 48*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*...
\[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{3} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{3} x^{3} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e f^{2} x^{2} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {3 e^{2} f x \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:
integrate((f*x+e)**3*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)
Output:
(Integral(e**3*sin(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**3*x**3 *sin(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(3*e*f**2*x**2*sin(c + d *x)**3/(sin(c + d*x) + 1), x) + Integral(3*e**2*f*x*sin(c + d*x)**3/(sin(c + d*x) + 1), x))/a
Exception generated. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((f*x+e)^3*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \sin \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*sin(d*x + c)^3/(a*sin(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^3}{a+a\,\sin \left (c+d\,x\right )} \,d x \] Input:
int((sin(c + d*x)^3*(e + f*x)^3)/(a + a*sin(c + d*x)),x)
Output:
int((sin(c + d*x)^3*(e + f*x)^3)/(a + a*sin(c + d*x)), x)
\[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {too large to display} \] Input:
int((f*x+e)^3*sin(d*x+c)^3/(a+a*sin(d*x+c)),x)
Output:
( - 192*cos(c + d*x)*int(x**2/(tan((c + d*x)/2)**6 + 2*tan((c + d*x)/2)**5 + 3*tan((c + d*x)/2)**4 + 4*tan((c + d*x)/2)**3 + 3*tan((c + d*x)/2)**2 + 2*tan((c + d*x)/2) + 1),x)*d**3*f**3 - 384*cos(c + d*x)*int(x/(tan((c + d *x)/2)**6 + 2*tan((c + d*x)/2)**5 + 3*tan((c + d*x)/2)**4 + 4*tan((c + d*x )/2)**3 + 3*tan((c + d*x)/2)**2 + 2*tan((c + d*x)/2) + 1),x)*d**3*e*f**2 + 128*cos(c + d*x)*int(x/(tan((c + d*x)/2)**6 + 2*tan((c + d*x)/2)**5 + 3*t an((c + d*x)/2)**4 + 4*tan((c + d*x)/2)**3 + 3*tan((c + d*x)/2)**2 + 2*tan ((c + d*x)/2) + 1),x)*d**2*f**3 + 72*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*d**2*e**2*f - 48*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*d*e*f**2 + 16*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*f**3 - 144*cos(c + d*x)*log( tan((c + d*x)/2) + 1)*d**2*e**2*f + 96*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*d*e*f**2 - 32*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*f**3 + 12*cos(c + d*x)*sin(c + d*x)**2*d**3*e**3 + 36*cos(c + d*x)*sin(c + d*x)**2*d**3*e** 2*f*x + 36*cos(c + d*x)*sin(c + d*x)**2*d**3*e*f**2*x**2 + 12*cos(c + d*x) *sin(c + d*x)**2*d**3*f**3*x**3 + 18*cos(c + d*x)*sin(c + d*x)**2*d**2*e** 2*f + 12*cos(c + d*x)*sin(c + d*x)**2*d**2*e*f**2*x + 6*cos(c + d*x)*sin(c + d*x)**2*d**2*f**3*x**2 - 6*cos(c + d*x)*sin(c + d*x)**2*d*e*f**2 + 2*co s(c + d*x)*sin(c + d*x)**2*d*f**3*x - 7*cos(c + d*x)*sin(c + d*x)**2*f**3 - 12*cos(c + d*x)*sin(c + d*x)*d**3*e**3 - 36*cos(c + d*x)*sin(c + d*x)*d* *3*e**2*f*x - 36*cos(c + d*x)*sin(c + d*x)*d**3*e*f**2*x**2 - 12*cos(c ...