Optimal. Leaf size=517 \[ -\frac {f \left (a^2-b^2\right )^{3/2} \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {f \left (a^2-b^2\right )^{3/2} \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {f \left (a^2-b^2\right ) \sin (c+d x)}{a^2 b d^2}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b^2 d}-\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a^2 b d}+\frac {e x \left (1-\frac {a^2}{b^2}\right )}{a}+\frac {f x^2 \left (1-\frac {a^2}{b^2}\right )}{2 a}-\frac {i b f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {e x}{a}-\frac {f x^2}{2 a} \]
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Rubi [A] time = 1.14, antiderivative size = 517, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 16, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {4543, 4408, 3310, 3720, 3475, 4405, 2633, 3296, 2637, 4183, 2279, 2391, 4525, 3323, 2264, 2190} \[ -\frac {f \left (a^2-b^2\right )^{3/2} \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {f \left (a^2-b^2\right )^{3/2} \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b^2 d^2}-\frac {i b f \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {f \left (a^2-b^2\right ) \sin (c+d x)}{a^2 b d^2}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b^2 d}-\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a^2 b d}+\frac {e x \left (1-\frac {a^2}{b^2}\right )}{a}+\frac {f x^2 \left (1-\frac {a^2}{b^2}\right )}{2 a}+\frac {b f \sin (c+d x)}{a^2 d^2}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {e x}{a}-\frac {f x^2}{2 a} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 2633
Rule 2637
Rule 3296
Rule 3310
Rule 3323
Rule 3475
Rule 3720
Rule 4183
Rule 4405
Rule 4408
Rule 4525
Rule 4543
Rubi steps
\begin {align*} \int \frac {(e+f x) \cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x) \cos ^2(c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {\int (e+f x) \cos ^2(c+d x) \, dx}{a}+\frac {\int (e+f x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos ^3(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac {f \cos ^2(c+d x)}{4 a d^2}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}-\frac {\int (e+f x) \, dx}{2 a}-\frac {\int (e+f x) \, dx}{a}+\frac {\int (e+f x) \cos ^2(c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos (c+d x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx+\frac {f \int \cot (c+d x) \, dx}{a d}\\ &=-\frac {3 e x}{2 a}-\frac {3 f x^2}{4 a}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {f \log (\sin (c+d x))}{a d^2}+\frac {\int (e+f x) \, dx}{2 a}-\frac {b \int (e+f x) \csc (c+d x) \, dx}{a^2}+\frac {b \int (e+f x) \sin (c+d x) \, dx}{a^2}-\frac {\left (a \left (1-\frac {b^2}{a^2}\right )\right ) \int (e+f x) \, dx}{b^2}-\frac {\left (-1+\frac {b^2}{a^2}\right ) \int (e+f x) \sin (c+d x) \, dx}{b}-\frac {\left (\left (a^2-b^2\right ) \left (-1+\frac {b^2}{a^2}\right )\right ) \int \frac {e+f x}{a+b \sin (c+d x)} \, dx}{b^2}\\ &=-\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {f \log (\sin (c+d x))}{a d^2}+\frac {\left (2 \left (a^2-b^2\right )^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a^2 b^2}+\frac {(b f) \int \cos (c+d x) \, dx}{a^2 d}+\frac {(b f) \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a^2 d}-\frac {(b f) \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \cos (c+d x) \, dx}{b d}\\ &=-\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}+\frac {f \log (\sin (c+d x))}{a d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}-\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}+\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a^2 b}-\frac {(i b f) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^2}+\frac {(i b f) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 d^2}\\ &=-\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}+\frac {\left (i \left (a^2-b^2\right )^{3/2} f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d}-\frac {\left (i \left (a^2-b^2\right )^{3/2} f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a^2 b^2 d}\\ &=-\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}+\frac {\left (\left (a^2-b^2\right )^{3/2} f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a-2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^2}-\frac {\left (\left (a^2-b^2\right )^{3/2} f\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a+2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a^2 b^2 d^2}\\ &=-\frac {e x}{a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) e x}{b^2}-\frac {f x^2}{2 a}-\frac {a \left (1-\frac {b^2}{a^2}\right ) f x^2}{2 b^2}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {b (e+f x) \cos (c+d x)}{a^2 d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \cos (c+d x)}{b d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^2 d^2}+\frac {b f \sin (c+d x)}{a^2 d^2}+\frac {\left (1-\frac {b^2}{a^2}\right ) f \sin (c+d x)}{b d^2}\\ \end {align*}
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Mathematica [A] time = 11.89, size = 1019, normalized size = 1.97 \[ \frac {(d e+d f x) \left (\frac {2 (d e-c f) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt {b^2-a^2}}{-i a+b+\sqrt {b^2-a^2}}\right )+\text {Li}_2\left (\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {b^2-a^2}\right )}\right )\right )}{\sqrt {b^2-a^2}}+\frac {i f \left (\log \left (i \tan \left (\frac {1}{2} (c+d x)\right )+1\right ) \log \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt {b^2-a^2}}{i a+b+\sqrt {b^2-a^2}}\right )+\text {Li}_2\left (\frac {a \left (i \tan \left (\frac {1}{2} (c+d x)\right )+1\right )}{a-i \left (b+\sqrt {b^2-a^2}\right )}\right )\right )}{\sqrt {b^2-a^2}}+\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (-\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )-\sqrt {b^2-a^2}}{i a-b+\sqrt {b^2-a^2}}\right )+\text {Li}_2\left (\frac {a \left (\tan \left (\frac {1}{2} (c+d x)\right )+i\right )}{i a-b+\sqrt {b^2-a^2}}\right )\right )}{\sqrt {b^2-a^2}}-\frac {i f \left (\log \left (i \tan \left (\frac {1}{2} (c+d x)\right )+1\right ) \log \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )-\sqrt {b^2-a^2}}{i a+b-\sqrt {b^2-a^2}}\right )+\text {Li}_2\left (\frac {i \tan \left (\frac {1}{2} (c+d x)\right ) a+a}{a+i \left (\sqrt {b^2-a^2}-b\right )}\right )\right )}{\sqrt {b^2-a^2}}\right ) \left (a^2-b^2\right )^2}{a^2 b^2 d^2 \left (d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (i \tan \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}-\frac {a (c+d x) (2 d e-2 c f+f (c+d x))}{2 b^2 d^2}-\frac {(d e-c f+f (c+d x)) \cos (c+d x)}{b d^2}+\frac {\left (-d e \cos \left (\frac {1}{2} (c+d x)\right )+c f \cos \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{2 a d^2}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {b e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d}+\frac {b c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 d^2}-\frac {b f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\text {Li}_2\left (-e^{i (c+d x)}\right )-\text {Li}_2\left (e^{i (c+d x)}\right )\right )\right )}{a^2 d^2}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (d e \sin \left (\frac {1}{2} (c+d x)\right )-c f \sin \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2}+\frac {f \sin (c+d x)}{b d^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.78, size = 1768, normalized size = 3.42 \[ \text {result too large to display} \]
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 [B] time = 1.86, size = 1863, normalized size = 3.60 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (e + f x\right ) \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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