Optimal. Leaf size=639 \[ -\frac {2 i b^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^3 \sqrt {a^2-b^2}}+\frac {2 i b^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 d^3 \sqrt {a^2-b^2}}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^2 \sqrt {a^2-b^2}}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 d^2 \sqrt {a^2-b^2}}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {2 b f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 i b f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {i f^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i (e+f x)^2}{a d} \]
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Rubi [A] time = 1.21, antiderivative size = 639, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4535, 4184, 3717, 2190, 2279, 2391, 4183, 2531, 2282, 6589, 3323, 2264} \[ -\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^2 \sqrt {a^2-b^2}}+\frac {2 b^2 f (e+f x) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d^2 \sqrt {a^2-b^2}}-\frac {2 i b^2 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^3 \sqrt {a^2-b^2}}+\frac {2 i b^2 f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d^3 \sqrt {a^2-b^2}}-\frac {2 i b f (e+f x) \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 b f^2 \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {i f^2 \text {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{a d^3}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i (e+f x)^2}{a d} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 3323
Rule 3717
Rule 4183
Rule 4184
Rule 4535
Rule 6589
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \csc ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {b \int (e+f x)^2 \csc (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^2}{a+b \sin (c+d x)} \, dx}{a^2}+\frac {(2 f) \int (e+f x) \cot (c+d x) \, dx}{a d}\\ &=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}+\frac {\left (2 b^2\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}{a^2}-\frac {(4 i f) \int \frac {e^{2 i (c+d x)} (e+f x)}{1-e^{2 i (c+d x)}} \, dx}{a d}+\frac {(2 b f) \int (e+f x) \log \left (1-e^{i (c+d x)}\right ) \, dx}{a^2 d}-\frac {(2 b f) \int (e+f x) \log \left (1+e^{i (c+d x)}\right ) \, dx}{a^2 d}\\ &=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {\left (2 i b^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}{a^2 \sqrt {a^2-b^2}}+\frac {\left (2 i b^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}{a^2 \sqrt {a^2-b^2}}-\frac {\left (2 f^2\right ) \int \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (2 i b f^2\right ) \int \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a^2 d^2}-\frac {\left (2 i b f^2\right ) \int \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a^2 d^2}\\ &=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}+\frac {\left (2 i b^2 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}{a^2 \sqrt {a^2-b^2} d}-\frac {\left (2 i b^2 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}{a^2 \sqrt {a^2-b^2} d}+\frac {\left (i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{a d^3}+\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^3}-\frac {\left (2 b f^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^3}\\ &=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {i f^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}+\frac {\left (2 b^2 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}{a^2 \sqrt {a^2-b^2} d^2}-\frac {\left (2 b^2 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}{a^2 \sqrt {a^2-b^2} d^2}\\ &=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {i f^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {\left (2 i b^2 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 )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {\left (2 i b^2 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 )}{a^2 \sqrt {a^2-b^2} d^3}\\ &=-\frac {i (e+f x)^2}{a d}+\frac {2 b (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x)^2 \cot (c+d x)}{a d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {2 f (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d^2}-\frac {2 i b f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {2 i b f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}-\frac {i f^2 \text {Li}_2\left (e^{2 i (c+d x)}\right )}{a d^3}+\frac {2 b f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 b f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a^2 d^3}-\frac {2 i b^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}+\frac {2 i b^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^3}\\ \end {align*}
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Mathematica [A] time = 12.19, size = 911, normalized size = 1.43 \[ \frac {i \left (-2 \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 )+2 \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 )-i \left (\left (2 \sqrt {b^2-a^2} \tan ^{-1}\left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right ) e^2+\sqrt {a^2-b^2} f x (2 e+f x) \left (\log \left (1-\frac {b e^{i (c+d x)}}{\sqrt {b^2-a^2}-i a}\right )-\log \left (\frac {e^{i (c+d x)} b}{i a+\sqrt {b^2-a^2}}+1\right )\right )\right ) d^2+2 \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 )-2 \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 )\right )\right ) b^2}{a^2 \sqrt {-\left (a^2-b^2\right )^2} d^3}+\frac {-b d^2 x^2 \log \left (1-e^{-i (c+d x)}\right ) f^2+b d^2 x^2 \log \left (1+e^{-i (c+d x)}\right ) f^2+2 b \left (i d x \text {Li}_2\left (-e^{-i (c+d x)}\right )+\text {Li}_3\left (-e^{-i (c+d x)}\right )\right ) f^2-2 i b \left (d x \text {Li}_2\left (e^{-i (c+d x)}\right )-i \text {Li}_3\left (e^{-i (c+d x)}\right )\right ) f^2-2 d (b d e-a f) x \log \left (1-e^{-i (c+d x)}\right ) f+2 d (b d e+a f) x \log \left (1+e^{-i (c+d x)}\right ) f+2 i (b d e+a f) \text {Li}_2\left (-e^{-i (c+d x)}\right ) f+2 i (a f-b d e) \text {Li}_2\left (e^{-i (c+d x)}\right ) f-\frac {2 i a d^2 (e+f x)^2}{-1+e^{2 i c}}+i d e (b d e-2 a f) \left (d x+i \log \left (1-e^{i (c+d x)}\right )\right )+d e (b d e+2 a f) \left (\log \left (1+e^{i (c+d x)}\right )-i d x\right )}{a^2 d^3}+\frac {\csc \left (\frac {c}{2}\right ) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\sin \left (\frac {d x}{2}\right ) e^2+2 f x \sin \left (\frac {d x}{2}\right ) e+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\sin \left (\frac {d x}{2}\right ) e^2+2 f x \sin \left (\frac {d x}{2}\right ) e+f^2 x^2 \sin \left (\frac {d x}{2}\right )\right )}{2 a d} \]
Warning: Unable to verify antiderivative.
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fricas [C] time = 0.82, size = 2988, normalized size = 4.68 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 3.59, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{2} \left (\csc ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{2} \csc ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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