Optimal. Leaf size=592 \[ \frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {2 i a^3 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3 \sqrt {a^2-b^2}}-\frac {2 i a^3 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^3 \sqrt {a^2-b^2}}+\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2 \sqrt {a^2-b^2}}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 d^2 \sqrt {a^2-b^2}}+\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d \sqrt {a^2-b^2}}-\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d \sqrt {a^2-b^2}}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {f^2 \sin (c+d x) \cos (c+d x)}{4 b d^3}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {f^2 x}{4 b d^2}+\frac {(e+f x)^3}{6 b f} \]
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Rubi [A] time = 1.18, antiderivative size = 592, normalized size of antiderivative = 1.00, number of steps used = 21, 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, 3323, 2264, 2190, 2531, 2282, 6589} \[ \frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^2 \sqrt {a^2-b^2}}-\frac {2 a^3 f (e+f x) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^2 \sqrt {a^2-b^2}}+\frac {2 i a^3 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d^3 \sqrt {a^2-b^2}}-\frac {2 i a^3 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d^3 \sqrt {a^2-b^2}}+\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 d \sqrt {a^2-b^2}}-\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^3 d \sqrt {a^2-b^2}}+\frac {a^2 (e+f x)^3}{3 b^3 f}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac {f^2 \sin (c+d x) \cos (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {f^2 x}{4 b d^2}+\frac {(e+f x)^3}{6 b f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2635
Rule 2638
Rule 3296
Rule 3311
Rule 3323
Rule 4515
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sin ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^2 \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{b}\\ &=-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac {a \int (e+f x)^2 \sin (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^2 \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {\int (e+f x)^2 \, dx}{2 b}-\frac {f^2 \int \sin ^2(c+d x) \, dx}{2 b d^2}\\ &=\frac {(e+f x)^3}{6 b f}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac {a^2 \int (e+f x)^2 \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^2}{a+b \sin (c+d x)} \, dx}{b^3}-\frac {(2 a f) \int (e+f x) \cos (c+d x) \, dx}{b^2 d}-\frac {f^2 \int 1 \, dx}{4 b d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac {\left (2 a^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^2}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b^3}+\frac {\left (2 a f^2\right ) \int \sin (c+d x) \, dx}{b^2 d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac {\left (2 i a^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^2}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b^2 \sqrt {a^2-b^2}}-\frac {\left (2 i a^3\right ) \int \frac {e^{i (c+d x)} (e+f x)^2}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b^2 \sqrt {a^2-b^2}}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac {\left (2 i a^3 f\right ) \int (e+f x) \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d}+\frac {\left (2 i a^3 f\right ) \int (e+f x) \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}-\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d^2}+\frac {\left (2 a^3 f^2\right ) \int \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^3 \sqrt {a^2-b^2} d^2}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}+\frac {\left (2 i a^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {\left (2 i a^3 f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b^3 \sqrt {a^2-b^2} d^3}\\ &=-\frac {f^2 x}{4 b d^2}+\frac {a^2 (e+f x)^3}{3 b^3 f}+\frac {(e+f x)^3}{6 b f}-\frac {2 a f^2 \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^2 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {2 a^3 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {2 i a^3 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {2 i a^3 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {2 a f (e+f x) \sin (c+d x)}{b^2 d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 b d^2}\\ \end {align*}
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Mathematica [A] time = 4.05, size = 1166, normalized size = 1.97 \[ \frac {-48 \sqrt {b^2-a^2} d^2 e^2 \tan ^{-1}\left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right ) a^3-24 \sqrt {a^2-b^2} d^2 f^2 x^2 \log \left (1-\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) a^3-48 \sqrt {a^2-b^2} d^2 e f x \log \left (1-\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) a^3+24 \sqrt {a^2-b^2} d^2 f^2 x^2 \log \left (\frac {e^{i (c+d x)} b}{i a+\sqrt {b^2-a^2}}+1\right ) a^3+48 \sqrt {a^2-b^2} d^2 e f x \log \left (\frac {e^{i (c+d x)} b}{i a+\sqrt {b^2-a^2}}+1\right ) a^3+48 i \sqrt {a^2-b^2} d f (e+f x) \text {Li}_2\left (\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) a^3-48 i \sqrt {a^2-b^2} d f (e+f x) \text {Li}_2\left (-\frac {b e^{i (c+d x)}}{i a+\sqrt {b^2-a^2}}\right ) a^3-48 \sqrt {a^2-b^2} f^2 \text {Li}_3\left (\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) a^3+48 \sqrt {a^2-b^2} f^2 \text {Li}_3\left (-\frac {b e^{i (c+d x)}}{i a+\sqrt {b^2-a^2}}\right ) a^3+8 \sqrt {-\left (a^2-b^2\right )^2} d^3 f^2 x^3 a^2+24 \sqrt {-\left (a^2-b^2\right )^2} d^3 e f x^2 a^2+24 \sqrt {-\left (a^2-b^2\right )^2} d^3 e^2 x a^2+24 b \sqrt {-\left (a^2-b^2\right )^2} d^2 e^2 \cos (c+d x) a-48 b \sqrt {-\left (a^2-b^2\right )^2} f^2 \cos (c+d x) a+24 b \sqrt {-\left (a^2-b^2\right )^2} d^2 f^2 x^2 \cos (c+d x) a+48 b \sqrt {-\left (a^2-b^2\right )^2} d^2 e f x \cos (c+d x) a-48 b \sqrt {-\left (a^2-b^2\right )^2} d e f \sin (c+d x) a-48 b \sqrt {-\left (a^2-b^2\right )^2} d f^2 x \sin (c+d x) a+4 b^2 \sqrt {-\left (b^2-a^2\right )^2} d^3 f^2 x^3+12 b^2 \sqrt {-\left (b^2-a^2\right )^2} d^3 e f x^2+12 b^2 \sqrt {-\left (b^2-a^2\right )^2} d^3 e^2 x-6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d e f \cos (2 (c+d x))-6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d f^2 x \cos (2 (c+d x))-6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^2 e^2 \sin (2 (c+d x))+3 b^2 \sqrt {-\left (a^2-b^2\right )^2} f^2 \sin (2 (c+d x))-6 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^2 f^2 x^2 \sin (2 (c+d x))-12 b^2 \sqrt {-\left (a^2-b^2\right )^2} d^2 e f x \sin (2 (c+d x))}{24 b^3 \sqrt {-\left (a^2-b^2\right )^2} d^3} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.73, size = 2064, normalized size = 3.49 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} \sin \left (d x + c\right )^{3}}{b \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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