Optimal. Leaf size=278 \[ \frac {4 i f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {f^2 \sin (c+d x) \cos (c+d x)}{4 a d^3}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f} \]
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Rubi [A] time = 0.49, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {4515, 3311, 32, 2635, 8, 3296, 2638, 3318, 4184, 3717, 2190, 2279, 2391} \[ \frac {4 i f^2 \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {f^2 \sin (c+d x) \cos (c+d x)}{4 a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2190
Rule 2279
Rule 2391
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3318
Rule 3717
Rule 4184
Rule 4515
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sin ^2(c+d x) \, dx}{a}-\int \frac {(e+f x)^2 \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\\ &=-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}+\frac {\int (e+f x)^2 \, dx}{2 a}-\frac {\int (e+f x)^2 \sin (c+d x) \, dx}{a}-\frac {f^2 \int \sin ^2(c+d x) \, dx}{2 a d^2}+\int \frac {(e+f x)^2 \sin (c+d x)}{a+a \sin (c+d x)} \, dx\\ &=\frac {(e+f x)^3}{6 a f}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}+\frac {\int (e+f x)^2 \, dx}{a}-\frac {(2 f) \int (e+f x) \cos (c+d x) \, dx}{a d}-\frac {f^2 \int 1 \, dx}{4 a d^2}-\int \frac {(e+f x)^2}{a+a \sin (c+d x)} \, dx\\ &=-\frac {f^2 x}{4 a d^2}+\frac {(e+f x)^3}{2 a f}+\frac {(e+f x)^2 \cos (c+d x)}{a d}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\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 {\left (2 f^2\right ) \int \sin (c+d x) \, dx}{a d^2}\\ &=-\frac {f^2 x}{4 a d^2}+\frac {(e+f x)^3}{2 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\frac {(2 f) \int (e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}\\ &=-\frac {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\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 {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\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 {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 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 {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\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 {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}-\frac {\left (4 i f^2\right ) \operatorname {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 {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\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 \text {Li}_2\left (i e^{i (c+d x)}\right )}{a d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2}\\ \end {align*}
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Mathematica [B] time = 3.11, size = 830, normalized size = 2.99 \[ -\frac {-8 f^2 x^3 \sin \left (\frac {1}{2} (c+d x)\right ) d^3-24 e f x^2 \sin \left (\frac {1}{2} (c+d x)\right ) d^3-24 e^2 x \sin \left (\frac {1}{2} (c+d x)\right ) d^3-6 e^2 \cos \left (\frac {3}{2} (c+d x)\right ) d^2-6 f^2 x^2 \cos \left (\frac {3}{2} (c+d x)\right ) d^2-12 e f x \cos \left (\frac {3}{2} (c+d x)\right ) d^2-2 e^2 \cos \left (\frac {5}{2} (c+d x)\right ) d^2-2 f^2 x^2 \cos \left (\frac {5}{2} (c+d x)\right ) d^2-4 e f x \cos \left (\frac {5}{2} (c+d x)\right ) d^2+(24+16 i) e^2 \sin \left (\frac {1}{2} (c+d x)\right ) d^2+(24+16 i) f^2 x^2 \sin \left (\frac {1}{2} (c+d x)\right ) d^2+(48+32 i) e f x \sin \left (\frac {1}{2} (c+d x)\right ) d^2-6 e^2 \sin \left (\frac {3}{2} (c+d x)\right ) d^2-6 f^2 x^2 \sin \left (\frac {3}{2} (c+d x)\right ) d^2-12 e f x \sin \left (\frac {3}{2} (c+d x)\right ) d^2+2 e^2 \sin \left (\frac {5}{2} (c+d x)\right ) d^2+2 f^2 x^2 \sin \left (\frac {5}{2} (c+d x)\right ) d^2+4 e f x \sin \left (\frac {5}{2} (c+d x)\right ) d^2-14 e f \cos \left (\frac {3}{2} (c+d x)\right ) d-14 f^2 x \cos \left (\frac {3}{2} (c+d x)\right ) d+2 e f \cos \left (\frac {5}{2} (c+d x)\right ) d+2 f^2 x \cos \left (\frac {5}{2} (c+d x)\right ) d+16 e f \sin \left (\frac {1}{2} (c+d x)\right ) d+16 f^2 x \sin \left (\frac {1}{2} (c+d x)\right ) d+64 e f \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left (\frac {1}{2} (c+d x)\right ) d+64 f^2 x \log (i \cos (c+d x)+\sin (c+d x)+1) \sin \left (\frac {1}{2} (c+d x)\right ) d+14 e f \sin \left (\frac {3}{2} (c+d x)\right ) d+14 f^2 x \sin \left (\frac {3}{2} (c+d x)\right ) d+2 e f \sin \left (\frac {5}{2} (c+d x)\right ) d+2 f^2 x \sin \left (\frac {5}{2} (c+d x)\right ) d+15 f^2 \cos \left (\frac {3}{2} (c+d x)\right )+f^2 \cos \left (\frac {5}{2} (c+d x)\right )-8 \cos \left (\frac {1}{2} (c+d x)\right ) \left (x \left (3 e^2+3 f x e+f^2 x^2\right ) d^3+(3-2 i) (e+f x)^2 d^2-2 f (e+f x) d-8 f (e+f x) \log (i \cos (c+d x)+\sin (c+d x)+1) d-2 f^2\right )-16 f^2 \sin \left (\frac {1}{2} (c+d x)\right )+64 i f^2 \text {Li}_2(-i \cos (c+d x)-\sin (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+15 f^2 \sin \left (\frac {3}{2} (c+d x)\right )-f^2 \sin \left (\frac {5}{2} (c+d x)\right )}{16 a d^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 844, normalized size = 3.04 \[ \frac {2 \, d^{3} f^{2} x^{3} + 4 \, d^{2} e^{2} + {\left (2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} - 2 \, d e f - f^{2} + 2 \, {\left (2 \, d^{2} e f - d f^{2}\right )} x\right )} \cos \left (d x + c\right )^{3} - 7 \, d e f + 2 \, {\left (3 \, d^{3} e f + 2 \, d^{2} f^{2}\right )} x^{2} + 2 \, {\left (2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} + 3 \, d e f - 4 \, f^{2} + {\left (4 \, d^{2} e f + 3 \, d f^{2}\right )} x\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, d^{3} e^{2} + 8 \, d^{2} e f - 7 \, d f^{2}\right )} x + {\left (2 \, d^{3} f^{2} x^{3} + 6 \, d^{2} e^{2} + d e f + 6 \, {\left (d^{3} e f + d^{2} f^{2}\right )} x^{2} - 7 \, f^{2} + {\left (6 \, d^{3} e^{2} + 12 \, d^{2} e f + d f^{2}\right )} x\right )} \cos \left (d x + c\right ) + {\left (8 i \, f^{2} \cos \left (d x + c\right ) + 8 i \, f^{2} \sin \left (d x + c\right ) + 8 i \, f^{2}\right )} {\rm Li}_2\left (i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) + {\left (-8 i \, f^{2} \cos \left (d x + c\right ) - 8 i \, f^{2} \sin \left (d x + c\right ) - 8 i \, f^{2}\right )} {\rm Li}_2\left (-i \, \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right ) - 8 \, {\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 ) - 8 \, {\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 ) - 8 \, {\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 ) - 8 \, {\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 (2 \, d^{3} f^{2} x^{3} - 4 \, d^{2} e^{2} - 7 \, d e f + 2 \, {\left (3 \, d^{3} e f - 2 \, d^{2} f^{2}\right )} x^{2} - {\left (2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} + 2 \, d e f - f^{2} + 2 \, {\left (2 \, d^{2} e f + d f^{2}\right )} x\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, d^{3} e^{2} - 8 \, d^{2} e f - 7 \, d f^{2}\right )} x + {\left (2 \, d^{2} f^{2} x^{2} + 2 \, d^{2} e^{2} - 8 \, d e f - 7 \, f^{2} + 4 \, {\left (d^{2} e f - 2 \, d f^{2}\right )} x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left (a d^{3} \cos \left (d x + c\right ) + a d^{3} \sin \left (d x + c\right ) + a d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (f x + e\right )}^{2} \sin \left (d x + c\right )^{3}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 481, normalized size = 1.73 \[ \frac {f^{2} x^{3}}{2 a}+\frac {3 f e \,x^{2}}{2 a}+\frac {3 e^{2} x}{2 a}+\frac {\left (f^{2} x^{2} d^{2}+2 d^{2} e f x +2 i d \,f^{2} x +d^{2} e^{2}+2 i d e f -2 f^{2}\right ) {\mathrm e}^{i \left (d x +c \right )}}{2 d^{3} a}+\frac {\left (f^{2} x^{2} d^{2}+2 d^{2} e f x -2 i d \,f^{2} x +d^{2} e^{2}-2 i d e f -2 f^{2}\right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 d^{3} a}+\frac {2 f^{2} x^{2}+4 f e x +2 e^{2}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {4 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) e}{a \,d^{2}}+\frac {4 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right ) e}{a \,d^{2}}+\frac {4 i f^{2} c x}{a \,d^{2}}+\frac {4 i f^{2} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 i f^{2} c^{2}}{a \,d^{3}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}+\frac {2 i f^{2} x^{2}}{a d}+\frac {4 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{3}}-\frac {4 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {f \left (f x +e \right ) \cos \left (2 d x +2 c \right )}{4 d^{2} a}-\frac {\left (2 f^{2} x^{2} d^{2}+4 d^{2} e f x +2 d^{2} e^{2}-f^{2}\right ) \sin \left (2 d x +2 c \right )}{8 d^{3} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\sin \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2}{a+a\,\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {e^{2} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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