Integrand size = 20, antiderivative size = 671 \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]
I*(d*x+c)^2/(a^2-b^2)/f-2*d*(d*x+c)*ln(1-I*b*exp(I*(f*x+e))/(a-(a^2-b^2)^( 1/2)))/(a^2-b^2)/f^2-I*a*(d*x+c)^2*ln(1-I*b*exp(I*(f*x+e))/(a-(a^2-b^2)^(1 /2)))/(a^2-b^2)^(3/2)/f-2*d*(d*x+c)*ln(1-I*b*exp(I*(f*x+e))/(a+(a^2-b^2)^( 1/2)))/(a^2-b^2)/f^2+I*a*(d*x+c)^2*ln(1-I*b*exp(I*(f*x+e))/(a+(a^2-b^2)^(1 /2)))/(a^2-b^2)^(3/2)/f+2*I*d^2*polylog(2,I*b*exp(I*(f*x+e))/(a-(a^2-b^2)^ (1/2)))/(a^2-b^2)/f^3-2*a*d*(d*x+c)*polylog(2,I*b*exp(I*(f*x+e))/(a-(a^2-b ^2)^(1/2)))/(a^2-b^2)^(3/2)/f^2+2*I*d^2*polylog(2,I*b*exp(I*(f*x+e))/(a+(a ^2-b^2)^(1/2)))/(a^2-b^2)/f^3+2*a*d*(d*x+c)*polylog(2,I*b*exp(I*(f*x+e))/( a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f^2-2*I*a*d^2*polylog(3,I*b*exp(I*(f*x +e))/(a-(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f^3+2*I*a*d^2*polylog(3,I*b*exp( I*(f*x+e))/(a+(a^2-b^2)^(1/2)))/(a^2-b^2)^(3/2)/f^3+b*(d*x+c)^2*cos(f*x+e) /(a^2-b^2)/f/(a+b*sin(f*x+e))
Time = 1.20 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\frac {i f^2 (c+d x)^2-2 d f (c+d x) \log \left (1+\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )-2 d f (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )+2 i d^2 \operatorname {PolyLog}\left (2,-\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )+2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )-\frac {i a \left (f^2 (c+d x)^2 \log \left (1+\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )-f^2 (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )-2 i d f (c+d x) \operatorname {PolyLog}\left (2,-\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )+2 i d f (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )-2 d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2}}+\frac {b f^2 (c+d x)^2 \cos (e+f x)}{a+b \sin (e+f x)}}{\left (a^2-b^2\right ) f^3} \]
(I*f^2*(c + d*x)^2 - 2*d*f*(c + d*x)*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + S qrt[a^2 - b^2])] - 2*d*f*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt [a^2 - b^2])] + (2*I)*d^2*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a ^2 - b^2])] + (2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b ^2])] - (I*a*(f^2*(c + d*x)^2*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - f^2*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - (2*I)*d*f*(c + d*x)*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqr t[a^2 - b^2])] + (2*I)*d*f*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + 2*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])] - 2*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])) /Sqrt[a^2 - b^2] + (b*f^2*(c + d*x)^2*Cos[e + f*x])/(a + b*Sin[e + f*x]))/ ((a^2 - b^2)*f^3)
Time = 2.78 (sec) , antiderivative size = 631, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3805, 3042, 3804, 2694, 27, 2620, 3011, 2720, 5030, 2620, 2715, 2838, 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.
| \(\displaystyle \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2}dx\) |
| \(\Big \downarrow \) 3805 |
| \(\displaystyle \frac {a \int \frac {(c+d x)^2}{a+b \sin (e+f x)}dx}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \frac {(c+d x)^2}{a+b \sin (e+f x)}dx}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3804 |
| \(\displaystyle \frac {2 a \int \frac {e^{i (e+f x)} (c+d x)^2}{2 e^{i (e+f x)} a-i b e^{2 i (e+f x)}+i b}dx}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle \frac {2 a \left (\frac {i b \int \frac {e^{i (e+f x)} (c+d x)^2}{2 \left (a-i b e^{i (e+f x)}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i (e+f x)} (c+d x)^2}{2 \left (a-i b e^{i (e+f x)}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a \left (\frac {i b \int \frac {e^{i (e+f x)} (c+d x)^2}{a-i b e^{i (e+f x)}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i (e+f x)} (c+d x)^2}{a-i b e^{i (e+f x)}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \int (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )dx}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \int (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )dx}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {i d \int \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )dx}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {i d \int \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )dx}{f}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)}dx}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 5030 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \left (\int \frac {e^{i (e+f x)} (c+d x)}{a-i b e^{i (e+f x)}-\sqrt {a^2-b^2}}dx+\int \frac {e^{i (e+f x)} (c+d x)}{a-i b e^{i (e+f x)}+\sqrt {a^2-b^2}}dx-\frac {i (c+d x)^2}{2 b d}\right )}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \left (-\frac {d \int \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )dx}{b f}-\frac {d \int \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )dx}{b f}+\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}+\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {i (c+d x)^2}{2 b d}\right )}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \left (\frac {i d \int e^{-i (e+f x)} \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{b f^2}+\frac {i d \int e^{-i (e+f x)} \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{b f^2}+\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}+\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {i (c+d x)^2}{2 b d}\right )}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \int e^{-i (e+f x)} \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (e+f x)}}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}-\frac {2 b d \left (\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}+\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {i d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f^2}-\frac {i d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{b f^2}-\frac {i (c+d x)^2}{2 b d}\right )}{f \left (a^2-b^2\right )}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {2 b d \left (\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}+\frac {(c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {i d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f^2}-\frac {i d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{b f^2}-\frac {i (c+d x)^2}{2 b d}\right )}{f \left (a^2-b^2\right )}+\frac {2 a \left (\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{b f}-\frac {2 d \left (\frac {i (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f}-\frac {d \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^2}\right )}{b f}\right )}{2 \sqrt {a^2-b^2}}\right )}{a^2-b^2}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}\) |
(-2*b*d*(((-1/2*I)*(c + d*x)^2)/(b*d) + ((c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(b*f) + ((c + d*x)*Log[1 - (I*b*E^(I*(e + f *x)))/(a + Sqrt[a^2 - b^2])])/(b*f) - (I*d*PolyLog[2, (I*b*E^(I*(e + f*x)) )/(a - Sqrt[a^2 - b^2])])/(b*f^2) - (I*d*PolyLog[2, (I*b*E^(I*(e + f*x)))/ (a + Sqrt[a^2 - b^2])])/(b*f^2)))/((a^2 - b^2)*f) + (2*a*(((-1/2*I)*b*(((c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/(b*f) - (2 *d*((I*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/ f - (d*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/f^2))/(b*f )))/Sqrt[a^2 - b^2] + ((I/2)*b*(((c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x))) /(a + Sqrt[a^2 - b^2])])/(b*f) - (2*d*((I*(c + d*x)*PolyLog[2, (I*b*E^(I*( e + f*x)))/(a + Sqrt[a^2 - b^2])])/f - (d*PolyLog[3, (I*b*E^(I*(e + f*x))) /(a + Sqrt[a^2 - b^2])])/f^2))/(b*f)))/Sqrt[a^2 - b^2]))/(a^2 - b^2) + (b* (c + d*x)^2*Cos[e + f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))
3.2.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
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[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[(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[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[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy mbol] :> Simp[2 Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x )) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ [a^2 - b^2, 0] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f *x]))), x] + (Simp[a/(a^2 - b^2) Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] , x] - Simp[b*d*(m/(f*(a^2 - b^2))) Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ (c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 ))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^( I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
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]
\[\int \frac {\left (d x +c \right )^{2}}{\left (a +b \sin \left (f x +e \right )\right )^{2}}d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3091 vs. \(2 (581) = 1162\).
Time = 0.52 (sec) , antiderivative size = 3091, normalized size of antiderivative = 4.61 \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Too large to display} \]
1/2*(2*(a*b^2*d^2*sin(f*x + e) + a^2*b*d^2)*sqrt(-(a^2 - b^2)/b^2)*polylog (3, -(I*a*cos(f*x + e) + a*sin(f*x + e) + (b*cos(f*x + e) - I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) - 2*(a*b^2*d^2*sin(f*x + e) + a^2*b*d^2)*sq rt(-(a^2 - b^2)/b^2)*polylog(3, -(I*a*cos(f*x + e) + a*sin(f*x + e) - (b*c os(f*x + e) - I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) + 2*(a*b^2*d^2* sin(f*x + e) + a^2*b*d^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(-I*a*cos(f*x + e) + a*sin(f*x + e) + (b*cos(f*x + e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) - 2*(a*b^2*d^2*sin(f*x + e) + a^2*b*d^2)*sqrt(-(a^2 - b^2)/b ^2)*polylog(3, -(-I*a*cos(f*x + e) + a*sin(f*x + e) - (b*cos(f*x + e) + I* b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) + 2*((a^2*b - b^3)*d^2*f^2*x^2 + 2*(a^2*b - b^3)*c*d*f^2*x + (a^2*b - b^3)*c^2*f^2)*cos(f*x + e) - 2*(-I* (a^2*b - b^3)*d^2*sin(f*x + e) - I*(a^3 - a*b^2)*d^2 + (-I*a^2*b*d^2*f*x - I*a^2*b*c*d*f + (-I*a*b^2*d^2*f*x - I*a*b^2*c*d*f)*sin(f*x + e))*sqrt(-(a ^2 - b^2)/b^2))*dilog((I*a*cos(f*x + e) - a*sin(f*x + e) + (b*cos(f*x + e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - 2*(-I*(a^2*b - b ^3)*d^2*sin(f*x + e) - I*(a^3 - a*b^2)*d^2 + (I*a^2*b*d^2*f*x + I*a^2*b*c* d*f + (I*a*b^2*d^2*f*x + I*a*b^2*c*d*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^ 2))*dilog((I*a*cos(f*x + e) - a*sin(f*x + e) - (b*cos(f*x + e) + I*b*sin(f *x + e))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - 2*(I*(a^2*b - b^3)*d^2*sin(f *x + e) + I*(a^3 - a*b^2)*d^2 + (I*a^2*b*d^2*f*x + I*a^2*b*c*d*f + (I*a...
Timed out. \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
\[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Hanged} \]