Optimal. Leaf size=618 \[ -\frac {6 f^3 \sqrt {a^2-b^2} \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}+\frac {6 f^3 \sqrt {a^2-b^2} \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^4}+\frac {6 i f^2 \sqrt {a^2-b^2} (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}-\frac {6 i f^2 \sqrt {a^2-b^2} (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {3 f \sqrt {a^2-b^2} (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 f \sqrt {a^2-b^2} (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d}+\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^3 \sin (c+d x)}{b d^4}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {(e+f x)^3 \cos (c+d x)}{b d} \]
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Rubi [A] time = 1.07, antiderivative size = 618, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {4525, 32, 3296, 2637, 3323, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {6 i f^2 \sqrt {a^2-b^2} (e+f x) \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}-\frac {6 i f^2 \sqrt {a^2-b^2} (e+f x) \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d^3}+\frac {3 f \sqrt {a^2-b^2} (e+f x)^2 \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 f \sqrt {a^2-b^2} (e+f x)^2 \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d^2}-\frac {6 f^3 \sqrt {a^2-b^2} \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}+\frac {6 f^3 \sqrt {a^2-b^2} \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d^4}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d}+\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {6 f^3 \sin (c+d x)}{b d^4}+\frac {(e+f x)^3 \cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 2637
Rule 3296
Rule 3323
Rule 4525
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {a \int (e+f x)^3 \, dx}{b^2}-\frac {\int (e+f x)^3 \sin (c+d x) \, dx}{b}-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x)^3}{a+b \sin (c+d x)} \, dx}{b^2}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {(e+f x)^3 \cos (c+d x)}{b d}-\frac {\left (2 \left (a^2-b^2\right )\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b^2}-\frac {(3 f) \int (e+f x)^2 \cos (c+d x) \, dx}{b d}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {(e+f x)^3 \cos (c+d x)}{b d}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}+\frac {\left (2 i \sqrt {a^2-b^2}\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b}-\frac {\left (2 i \sqrt {a^2-b^2}\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b}+\frac {\left (6 f^2\right ) \int (e+f x) \sin (c+d x) \, dx}{b d^2}\\ &=\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac {(e+f x)^3 \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac {\left (3 i \sqrt {a^2-b^2} f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d}+\frac {\left (3 i \sqrt {a^2-b^2} f\right ) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d}+\frac {\left (6 f^3\right ) \int \cos (c+d x) \, dx}{b d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac {(e+f x)^3 \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {6 f^3 \sin (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac {\left (6 \sqrt {a^2-b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^2}+\frac {\left (6 \sqrt {a^2-b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^2}\\ &=\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac {(e+f x)^3 \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {6 i \sqrt {a^2-b^2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}-\frac {6 i \sqrt {a^2-b^2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 f^3 \sin (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac {\left (6 i \sqrt {a^2-b^2} f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^3}+\frac {\left (6 i \sqrt {a^2-b^2} f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac {(e+f x)^3 \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {6 i \sqrt {a^2-b^2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}-\frac {6 i \sqrt {a^2-b^2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}+\frac {6 f^3 \sin (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}-\frac {\left (6 \sqrt {a^2-b^2} f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b^2 d^4}+\frac {\left (6 \sqrt {a^2-b^2} f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b^2 d^4}\\ &=\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^2 (e+f x) \cos (c+d x)}{b d^3}+\frac {(e+f x)^3 \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {6 i \sqrt {a^2-b^2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^3}-\frac {6 i \sqrt {a^2-b^2} f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^3}-\frac {6 \sqrt {a^2-b^2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^4}+\frac {6 \sqrt {a^2-b^2} f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^4}+\frac {6 f^3 \sin (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sin (c+d x)}{b d^2}\\ \end {align*}
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Mathematica [A] time = 3.51, size = 1025, normalized size = 1.66 \[ \frac {a x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right ) d^4+4 b (e+f x) \left (d^2 (e+f x)^2-6 f^2\right ) \cos (c+d x) d+\frac {4 \left (b^2-a^2\right ) \left (2 \sqrt {b^2-a^2} e^3 \tan ^{-1}\left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right ) d^3+\sqrt {a^2-b^2} f^3 x^3 \log \left (1-\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) d^3+3 \sqrt {a^2-b^2} e f^2 x^2 \log \left (1-\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) d^3+3 \sqrt {a^2-b^2} e^2 f x \log \left (1-\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) d^3-\sqrt {a^2-b^2} f^3 x^3 \log \left (\frac {e^{i (c+d x)} b}{i a+\sqrt {b^2-a^2}}+1\right ) d^3-3 \sqrt {a^2-b^2} e f^2 x^2 \log \left (\frac {e^{i (c+d x)} b}{i a+\sqrt {b^2-a^2}}+1\right ) d^3-3 \sqrt {a^2-b^2} e^2 f x \log \left (\frac {e^{i (c+d x)} b}{i a+\sqrt {b^2-a^2}}+1\right ) d^3-3 i \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) d^2+3 i \sqrt {a^2-b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{i (c+d x)}}{i a+\sqrt {b^2-a^2}}\right ) d^2+6 \sqrt {a^2-b^2} e f^2 \text {Li}_3\left (\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) d+6 \sqrt {a^2-b^2} f^3 x \text {Li}_3\left (\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right ) d-6 \sqrt {a^2-b^2} e f^2 \text {Li}_3\left (-\frac {b e^{i (c+d x)}}{i a+\sqrt {b^2-a^2}}\right ) d-6 \sqrt {a^2-b^2} f^3 x \text {Li}_3\left (-\frac {b e^{i (c+d x)}}{i a+\sqrt {b^2-a^2}}\right ) d+6 i \sqrt {a^2-b^2} f^3 \text {Li}_4\left (\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right )-6 i \sqrt {a^2-b^2} f^3 \text {Li}_4\left (-\frac {b e^{i (c+d x)}}{i a+\sqrt {b^2-a^2}}\right )\right )}{\sqrt {-\left (a^2-b^2\right )^2}}-12 b f \left (d^2 (e+f x)^2-2 f^2\right ) \sin (c+d x)}{4 b^2 d^4} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.74, size = 2351, normalized size = 3.80 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{3} \cos \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.03, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx \]
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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