Integrand size = 26, antiderivative size = 129 \[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3} \]
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Time = 0.16 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4611, 32, 3399, 4269, 3798, 2221, 2317, 2438} \[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f} \]
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Rule 32
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3798
Rule 4269
Rule 4611
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^2 \, dx}{a}-\int \frac {(e+f x)^2}{a+a \sin (c+d x)} \, dx \\ & = \frac {(e+f x)^3}{3 a f}-\frac {\int (e+f x)^2 \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a} \\ & = \frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(2 f) \int (e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d} \\ & = \frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {(4 f) \int \frac {e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )} (e+f x)}{1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}} \, dx}{a d} \\ & = \frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {\left (4 f^2\right ) \int \log \left (1-i e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right ) \, dx}{a d^2} \\ & = \frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}-\frac {\left (4 i f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{2 i \left (\frac {c}{2}+\frac {d x}{2}\right )}\right )}{a d^3} \\ & = \frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{3 a f}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3} \\ \end{align*}
Time = 1.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.65 \[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {x \left (3 e^2+3 e f x+f^2 x^2\right )+\frac {12 f (\cos (c)+i \sin (c)) \left (\frac {(e+f x)^2 (\cos (c)-i \sin (c))}{2 f}-\frac {(e+f x) \log (1+i \cos (c+d x)+\sin (c+d x)) (1+i \cos (c)+\sin (c))}{d}+\frac {f \operatorname {PolyLog}(2,-i \cos (c+d x)-\sin (c+d x)) (\cos (c)-i (1+\sin (c)))}{d^2}\right )}{d (\cos (c)+i (1+\sin (c)))}-\frac {6 (e+f x)^2 \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}}{3 a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (113 ) = 226\).
Time = 0.23 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.60
method | result | size |
risch | \(\frac {f^{2} x^{3}}{3 a}+\frac {f e \,x^{2}}{a}+\frac {e^{2} x}{a}+\frac {e^{3}}{3 a f}+\frac {2 x^{2} f^{2}+4 f e x +2 e^{2}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {2 f e \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {4 i f^{2} c x}{a \,d^{2}}+\frac {4 f e \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {4 i f e \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {4 i f^{2} \operatorname {Li}_{2}\left (i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 i f^{2} c^{2}}{a \,d^{3}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}-\frac {4 i f^{2} c \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 f^{2} c \ln \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 i f^{2} x^{2}}{a d}-\frac {4 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}\) | \(335\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (108) = 216\).
Time = 0.30 (sec) , antiderivative size = 583, normalized size of antiderivative = 4.52 \[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {d^{3} f^{2} x^{3} + 3 \, d^{2} e^{2} + 3 \, {\left (d^{3} e f + d^{2} f^{2}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} + 2 \, d^{2} e f\right )} x + {\left (d^{3} f^{2} x^{3} + 3 \, d^{2} e^{2} + 3 \, {\left (d^{3} e f + d^{2} f^{2}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} + 2 \, d^{2} e f\right )} x\right )} \cos \left (d x + c\right ) - 6 \, {\left (-i \, f^{2} \cos \left (d x + c\right ) - i \, f^{2} \sin \left (d x + c\right ) - i \, f^{2}\right )} {\rm Li}_2\left (i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 6 \, {\left (i \, f^{2} \cos \left (d x + c\right ) + i \, f^{2} \sin \left (d x + c\right ) + i \, f^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 6 \, {\left (d e f - c f^{2} + {\left (d e f - c f^{2}\right )} \cos \left (d x + c\right ) + {\left (d e f - c f^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) - 6 \, {\left (d f^{2} x + c f^{2} + {\left (d f^{2} x + c f^{2}\right )} \cos \left (d x + c\right ) + {\left (d f^{2} x + c f^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - 6 \, {\left (d f^{2} x + c f^{2} + {\left (d f^{2} x + c f^{2}\right )} \cos \left (d x + c\right ) + {\left (d f^{2} x + c f^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-i \, \cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right ) - 6 \, {\left (d e f - c f^{2} + {\left (d e f - c f^{2}\right )} \cos \left (d x + c\right ) + {\left (d e f - c f^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + i\right ) + {\left (d^{3} f^{2} x^{3} - 3 \, d^{2} e^{2} + 3 \, {\left (d^{3} e f - d^{2} f^{2}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} - 2 \, d^{2} e f\right )} x\right )} \sin \left (d x + c\right )}{3 \, {\left (a d^{3} \cos \left (d x + c\right ) + a d^{3} \sin \left (d x + c\right ) + a d^{3}\right )}} \]
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\[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{2} \sin {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \sin {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \sin {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 402 vs. \(2 (108) = 216\).
Time = 0.39 (sec) , antiderivative size = 402, normalized size of antiderivative = 3.12 \[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2} + 3 \, d^{3} e^{2} x - 6 i \, d^{2} e^{2} - 12 \, {\left (d e f \cos \left (d x + c\right ) + i \, d e f \sin \left (d x + c\right ) + i \, d e f\right )} \arctan \left (\sin \left (d x + c\right ) + 1, \cos \left (d x + c\right )\right ) + 12 \, {\left (d f^{2} x \cos \left (d x + c\right ) + i \, d f^{2} x \sin \left (d x + c\right ) + i \, d f^{2} x\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) - {\left (i \, d^{3} f^{2} x^{3} - 3 \, {\left (-i \, d^{3} e f + 2 \, d^{2} f^{2}\right )} x^{2} - 3 \, {\left (-i \, d^{3} e^{2} + 4 \, d^{2} e f\right )} x\right )} \cos \left (d x + c\right ) + 12 \, {\left (f^{2} \cos \left (d x + c\right ) + i \, f^{2} \sin \left (d x + c\right ) + i \, f^{2}\right )} {\rm Li}_2\left (i \, e^{\left (i \, d x + i \, c\right )}\right ) - 6 \, {\left (d f^{2} x + d e f - {\left (i \, d f^{2} x + i \, d e f\right )} \cos \left (d x + c\right ) + {\left (d f^{2} x + d e f\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + {\left (d^{3} f^{2} x^{3} + 3 \, {\left (d^{3} e f + 2 i \, d^{2} f^{2}\right )} x^{2} + 3 \, {\left (d^{3} e^{2} + 4 i \, d^{2} e f\right )} x\right )} \sin \left (d x + c\right )}{-3 i \, a d^{3} \cos \left (d x + c\right ) + 3 \, a d^{3} \sin \left (d x + c\right ) + 3 \, a d^{3}} \]
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\[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \sin \left (d x + c\right )}{a \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {\sin \left (c+d\,x\right )\,{\left (e+f\,x\right )}^2}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
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