Integrand size = 20, antiderivative size = 148 \[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=-\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{a f^4} \]
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Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3399, 4269, 3798, 2221, 2611, 2320, 6724} \[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a f^3}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {f x}{2}+\frac {\pi }{4}\right )}{a f}-\frac {i (c+d x)^3}{a f}+\frac {12 d^3 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{a f^4} \]
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Rule 2221
Rule 2320
Rule 2611
Rule 3399
Rule 3798
Rule 4269
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\int (c+d x)^3 \csc ^2\left (\frac {1}{2} \left (e+\frac {\pi }{2}\right )+\frac {f x}{2}\right ) \, dx}{2 a} \\ & = -\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {(3 d) \int (c+d x)^2 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right ) \, dx}{a f} \\ & = -\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {(6 d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a f} \\ & = -\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {\left (12 d^2\right ) \int (c+d x) \log \left (1-i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^2} \\ & = -\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a f^3}+\frac {\left (12 i d^3\right ) \int \operatorname {PolyLog}\left (2,i e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a f^3} \\ & = -\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a f^3}+\frac {\left (12 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a f^4} \\ & = -\frac {i (c+d x)^3}{a f}-\frac {(c+d x)^3 \cot \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f x}{2}\right )}{a f}+\frac {6 d (c+d x)^2 \log \left (1-i e^{i (e+f x)}\right )}{a f^2}-\frac {12 i d^2 (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{a f^3}+\frac {12 d^3 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{a f^4} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.85 \[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=\frac {-12 i d^2 f (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )+12 d^3 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )+f^2 (c+d x)^2 \left (-i f (c+d x)+6 d \log \left (1-i e^{i (e+f x)}\right )+f (c+d x) \tan \left (\frac {1}{4} (2 e-\pi +2 f x)\right )\right )}{a f^4} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (131 ) = 262\).
Time = 0.25 (sec) , antiderivative size = 484, normalized size of antiderivative = 3.27
method | result | size |
risch | \(-\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {12 i c \,d^{2} e x}{a \,f^{2}}+\frac {4 i d^{3} e^{3}}{a \,f^{4}}-\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right ) c^{2} d}{a \,f^{2}}+\frac {12 c \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{2}}-\frac {12 i c \,d^{2} \operatorname {Li}_{2}\left (i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {6 i c \,d^{2} x^{2}}{a f}+\frac {12 c \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{a \,f^{3}}+\frac {6 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2} d}{a \,f^{2}}+\frac {6 e^{2} d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a \,f^{4}}-\frac {6 e^{2} d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {6 i c \,d^{2} e^{2}}{a \,f^{3}}-\frac {12 e c \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{a \,f^{3}}+\frac {6 d^{3} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{a \,f^{2}}+\frac {12 e c \,d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{3}}-\frac {12 i d^{3} \operatorname {Li}_{2}\left (i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{a \,f^{3}}-\frac {6 e^{2} d^{3} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}+\frac {6 i d^{3} e^{2} x}{a \,f^{3}}+\frac {12 d^{3} \operatorname {Li}_{3}\left (i {\mathrm e}^{i \left (f x +e \right )}\right )}{a \,f^{4}}-\frac {2 i d^{3} x^{3}}{a f}\) | \(484\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 915 vs. \(2 (125) = 250\).
Time = 0.32 (sec) , antiderivative size = 915, normalized size of antiderivative = 6.18 \[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=-\frac {d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3} + {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3}\right )} \cos \left (f x + e\right ) + 6 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right ) + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 6 \, {\left (-i \, d^{3} f x - i \, c d^{2} f + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right ) + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \sin \left (f x + e\right )\right )} {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \cos \left (f x + e\right ) + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \sin \left (f x + e\right )\right )} \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \cos \left (f x + e\right ) + {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x - d^{3} e^{2} + 2 \, c d^{2} e f\right )} \sin \left (f x + e\right )\right )} \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2} + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sin \left (f x + e\right )\right )} \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right ) + d^{3} \sin \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 6 \, {\left (d^{3} \cos \left (f x + e\right ) + d^{3} \sin \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - {\left (d^{3} f^{3} x^{3} + 3 \, c d^{2} f^{3} x^{2} + 3 \, c^{2} d f^{3} x + c^{3} f^{3}\right )} \sin \left (f x + e\right )}{a f^{4} \cos \left (f x + e\right ) + a f^{4} \sin \left (f x + e\right ) + a f^{4}} \]
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\[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=\frac {\int \frac {c^{3}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 979 vs. \(2 (125) = 250\).
Time = 0.30 (sec) , antiderivative size = 979, normalized size of antiderivative = 6.61 \[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{a \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^3}{a+a \sin (e+f x)} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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