Optimal. Leaf size=765 \[ -\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d^3}+\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d^3}-\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d^2}+\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d^2}+\frac{6 f^3 \sqrt{a^2-b^2} \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d^4}-\frac{6 f^3 \sqrt{a^2-b^2} \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac{6 i f^3 \text{PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d}+\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^4}{4 b f} \]
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Rubi [A] time = 1.42568, antiderivative size = 765, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 14, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {4543, 4408, 3296, 2637, 4183, 2531, 6609, 2282, 6589, 4525, 32, 3323, 2264, 2190} \[ -\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d^3}+\frac{6 i f^2 \sqrt{a^2-b^2} (e+f x) \text{PolyLog}\left (3,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d^3}-\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d^2}+\frac{3 f \sqrt{a^2-b^2} (e+f x)^2 \text{PolyLog}\left (2,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d^2}+\frac{6 f^3 \sqrt{a^2-b^2} \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d^4}-\frac{6 f^3 \sqrt{a^2-b^2} \text{PolyLog}\left (4,\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac{6 i f^3 \text{PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}-\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{a-\sqrt{a^2-b^2}}\right )}{a b d}+\frac{i \sqrt{a^2-b^2} (e+f x)^3 \log \left (1-\frac{i b e^{i (c+d x)}}{\sqrt{a^2-b^2}+a}\right )}{a b d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac{(e+f x)^4}{4 b f} \]
Antiderivative was successfully verified.
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Rule 4543
Rule 4408
Rule 3296
Rule 2637
Rule 4183
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 4525
Rule 32
Rule 3323
Rule 2264
Rule 2190
Rubi steps
\begin{align*} \int \frac{(e+f x)^3 \cos (c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{\int (e+f x)^3 \cos (c+d x) \cot (c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac{\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\frac{\int (e+f x)^3 \, dx}{b}+\left (\frac{a}{b}-\frac{b}{a}\right ) \int \frac{(e+f x)^3}{a+b \sin (c+d x)} \, dx\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\left (2 \left (\frac{a}{b}-\frac{b}{a}\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-\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a 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}{a}+\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}{a}-\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac{\left (6 i f^2\right ) \int (e+f x) \text{Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a 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 )}{a 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 )}{a b d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\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}{a b 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}{a b d}+\frac{\left (6 f^3\right ) \int \text{Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac{\left (6 f^3\right ) \int \text{Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a d^3}\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a 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 )}{a 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 )}{a b d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\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}{a 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}{a b d^2}-\frac{\left (6 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac{\left (6 i f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a 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 )}{a 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 )}{a b d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^3}-\frac{6 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a d^4}+\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}{a b 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}{a b d^3}\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a 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 )}{a 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 )}{a b d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^3}-\frac{6 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^4}\\ &=-\frac{(e+f x)^4}{4 b f}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a 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 )}{a 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 )}{a b d}+\frac{3 i f (e+f x)^2 \text{Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac{3 i f (e+f x)^2 \text{Li}_2\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^2}-\frac{6 f^2 (e+f x) \text{Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_3\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^3}-\frac{6 i f^3 \text{Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac{6 i f^3 \text{Li}_4\left (e^{i (c+d x)}\right )}{a 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 )}{a b 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 )}{a b d^4}\\ \end{align*}
Mathematica [A] time = 2.24113, size = 1194, normalized size = 1.56 \[ -\frac{x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right )}{4 b}+\frac{\left (a^2-b^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{PolyLog}\left (2,\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{PolyLog}\left (2,-\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{PolyLog}\left (3,\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{PolyLog}\left (3,\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{PolyLog}\left (3,-\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{PolyLog}\left (3,-\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{PolyLog}\left (4,\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{PolyLog}\left (4,-\frac{b e^{i (c+d x)}}{i a+\sqrt{b^2-a^2}}\right )\right )}{a b \sqrt{-\left (a^2-b^2\right )^2} d^4}+\frac{i \left (2 i \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) (e+f x)^3+\frac{3 f \left (-2 \text{PolyLog}(4,-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text{PolyLog}(3,-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,-\cos (c+d x)-i \sin (c+d x))\right )}{d^3}-\frac{3 f \left (-2 \text{PolyLog}(4,\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text{PolyLog}(3,\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,\cos (c+d x)+i \sin (c+d x))\right )}{d^3}\right )}{a d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 2.293, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\cos \left ( dx+c \right ) \cot \left ( dx+c \right ) }{a+b\sin \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 3.96738, size = 7471, normalized size = 9.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right )^{3} \cos{\left (c + d x \right )} \cot{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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