Optimal. Leaf size=288 \[ \frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {20 i d^2 (c+d x) \text {PolyLog}\left (2,-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {20 d^3 \text {PolyLog}\left (3,-e^{i (e+f x)}\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Rubi [A]
time = 0.49, antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3399,
4271, 4269, 3556, 3800, 2221, 2611, 2320, 6724} \begin {gather*} \frac {20 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {20 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3399
Rule 3556
Rule 3800
Rule 4269
Rule 4271
Rule 4276
Rule 6724
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+a \sec (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^3}{a^2}+\frac {(c+d x)^3}{a^2 (1+\cos (e+f x))^2}-\frac {2 (c+d x)^3}{a^2 (1+\cos (e+f x))}\right ) \, dx\\ &=\frac {(c+d x)^4}{4 a^2 d}+\frac {\int \frac {(c+d x)^3}{(1+\cos (e+f x))^2} \, dx}{a^2}-\frac {2 \int \frac {(c+d x)^3}{1+\cos (e+f x)} \, dx}{a^2}\\ &=\frac {(c+d x)^4}{4 a^2 d}+\frac {\int (c+d x)^3 \csc ^4\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{4 a^2}-\frac {\int (c+d x)^3 \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{a^2}\\ &=\frac {(c+d x)^4}{4 a^2 d}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\int (c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f^2}+\frac {(6 d) \int (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac {2 i (c+d x)^3}{a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (2 d^3\right ) \int \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f^3}-\frac {(12 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}-\frac {d \int (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {12 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (24 d^2\right ) \int (c+d x) \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}+\frac {(2 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)^2}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}\\ &=\frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {24 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (24 i d^3\right ) \int \text {Li}_2\left (-e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3}-\frac {\left (4 d^2\right ) \int (c+d x) \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}\\ &=\frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {20 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}-\frac {\left (24 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4}+\frac {\left (4 i d^3\right ) \int \text {Li}_2\left (-e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^3}\\ &=\frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {20 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {24 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}+\frac {\left (4 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^4}\\ &=\frac {5 i (c+d x)^3}{3 a^2 f}+\frac {(c+d x)^4}{4 a^2 d}-\frac {10 d (c+d x)^2 \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{a^2 f^4}+\frac {20 i d^2 (c+d x) \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {20 d^3 \text {Li}_3\left (-e^{i (e+f x)}\right )}{a^2 f^4}-\frac {d (c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{2 a^2 f^2}+\frac {2 d^2 (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f^3}-\frac {5 (c+d x)^3 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(1455\) vs. \(2(288)=576\).
time = 7.19, size = 1455, normalized size = 5.05 \begin {gather*} \frac {20 d^3 e^{-\frac {i e}{2}} \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \left (i f^2 x^2 \left (e^{i e} f x+3 i \left (1+e^{i e}\right ) \log \left (1+e^{i (e+f x)}\right )\right )+6 i \left (1+e^{i e}\right ) f x \text {PolyLog}\left (2,-e^{i (e+f x)}\right )-6 \left (1+e^{i e}\right ) \text {PolyLog}\left (3,-e^{i (e+f x)}\right )\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x)}{3 f^4 (a+a \sec (e+f x))^2}+\frac {16 d^3 \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x) \left (\cos \left (\frac {e}{2}\right ) \log \left (\cos \left (\frac {e}{2}\right ) \cos \left (\frac {f x}{2}\right )-\sin \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )\right )+\frac {1}{2} f x \sin \left (\frac {e}{2}\right )\right )}{f^4 (a+a \sec (e+f x))^2 \left (\cos ^2\left (\frac {e}{2}\right )+\sin ^2\left (\frac {e}{2}\right )\right )}-\frac {40 c^2 d \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x) \left (\cos \left (\frac {e}{2}\right ) \log \left (\cos \left (\frac {e}{2}\right ) \cos \left (\frac {f x}{2}\right )-\sin \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )\right )+\frac {1}{2} f x \sin \left (\frac {e}{2}\right )\right )}{f^2 (a+a \sec (e+f x))^2 \left (\cos ^2\left (\frac {e}{2}\right )+\sin ^2\left (\frac {e}{2}\right )\right )}-\frac {80 c d^2 \cos ^4\left (\frac {e}{2}+\frac {f x}{2}\right ) \csc \left (\frac {e}{2}\right ) \left (\frac {1}{4} e^{-i \text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )} f^2 x^2-\frac {\cot \left (\frac {e}{2}\right ) \left (\frac {1}{2} i f x \left (-\pi -2 \text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )-\pi \log \left (1+e^{-i f x}\right )-2 \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right ) \log \left (1-e^{2 i \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )}\right )+\pi \log \left (\cos \left (\frac {f x}{2}\right )\right )-2 \text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right ) \log \left (\sin \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (\frac {f x}{2}-\text {ArcTan}\left (\cot \left (\frac {e}{2}\right )\right )\right )}\right )\right )}{\sqrt {1+\cot ^2\left (\frac {e}{2}\right )}}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x)}{f^3 (a+a \sec (e+f x))^2 \sqrt {\csc ^2\left (\frac {e}{2}\right ) \left (\cos ^2\left (\frac {e}{2}\right )+\sin ^2\left (\frac {e}{2}\right )\right )}}+\frac {\cos \left (\frac {e}{2}+\frac {f x}{2}\right ) \sec \left (\frac {e}{2}\right ) \sec ^2(e+f x) \left (-24 c^2 d f \cos \left (\frac {f x}{2}\right )-48 c d^2 f x \cos \left (\frac {f x}{2}\right )+36 c^3 f^3 x \cos \left (\frac {f x}{2}\right )-24 d^3 f x^2 \cos \left (\frac {f x}{2}\right )+54 c^2 d f^3 x^2 \cos \left (\frac {f x}{2}\right )+36 c d^2 f^3 x^3 \cos \left (\frac {f x}{2}\right )+9 d^3 f^3 x^4 \cos \left (\frac {f x}{2}\right )-24 c^2 d f \cos \left (e+\frac {f x}{2}\right )-48 c d^2 f x \cos \left (e+\frac {f x}{2}\right )+36 c^3 f^3 x \cos \left (e+\frac {f x}{2}\right )-24 d^3 f x^2 \cos \left (e+\frac {f x}{2}\right )+54 c^2 d f^3 x^2 \cos \left (e+\frac {f x}{2}\right )+36 c d^2 f^3 x^3 \cos \left (e+\frac {f x}{2}\right )+9 d^3 f^3 x^4 \cos \left (e+\frac {f x}{2}\right )+12 c^3 f^3 x \cos \left (e+\frac {3 f x}{2}\right )+18 c^2 d f^3 x^2 \cos \left (e+\frac {3 f x}{2}\right )+12 c d^2 f^3 x^3 \cos \left (e+\frac {3 f x}{2}\right )+3 d^3 f^3 x^4 \cos \left (e+\frac {3 f x}{2}\right )+12 c^3 f^3 x \cos \left (2 e+\frac {3 f x}{2}\right )+18 c^2 d f^3 x^2 \cos \left (2 e+\frac {3 f x}{2}\right )+12 c d^2 f^3 x^3 \cos \left (2 e+\frac {3 f x}{2}\right )+3 d^3 f^3 x^4 \cos \left (2 e+\frac {3 f x}{2}\right )+96 c d^2 \sin \left (\frac {f x}{2}\right )-72 c^3 f^2 \sin \left (\frac {f x}{2}\right )+96 d^3 x \sin \left (\frac {f x}{2}\right )-216 c^2 d f^2 x \sin \left (\frac {f x}{2}\right )-216 c d^2 f^2 x^2 \sin \left (\frac {f x}{2}\right )-72 d^3 f^2 x^3 \sin \left (\frac {f x}{2}\right )-48 c d^2 \sin \left (e+\frac {f x}{2}\right )+48 c^3 f^2 \sin \left (e+\frac {f x}{2}\right )-48 d^3 x \sin \left (e+\frac {f x}{2}\right )+144 c^2 d f^2 x \sin \left (e+\frac {f x}{2}\right )+144 c d^2 f^2 x^2 \sin \left (e+\frac {f x}{2}\right )+48 d^3 f^2 x^3 \sin \left (e+\frac {f x}{2}\right )+48 c d^2 \sin \left (e+\frac {3 f x}{2}\right )-40 c^3 f^2 \sin \left (e+\frac {3 f x}{2}\right )+48 d^3 x \sin \left (e+\frac {3 f x}{2}\right )-120 c^2 d f^2 x \sin \left (e+\frac {3 f x}{2}\right )-120 c d^2 f^2 x^2 \sin \left (e+\frac {3 f x}{2}\right )-40 d^3 f^2 x^3 \sin \left (e+\frac {3 f x}{2}\right )\right )}{24 f^3 (a+a \sec (e+f x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 809 vs. \(2 (246 ) = 492\).
time = 0.80, size = 810, normalized size = 2.81
method | result | size |
risch | \(\frac {d^{3} x^{4}}{4 a^{2}}+\frac {d^{2} c \,x^{3}}{a^{2}}+\frac {3 d \,c^{2} x^{2}}{2 a^{2}}+\frac {c^{3} x}{a^{2}}+\frac {c^{4}}{4 a^{2} d}+\frac {10 i d^{2} c \,x^{2}}{a^{2} f}-\frac {20 i d^{3} e^{3}}{3 a^{2} f^{4}}-\frac {2 i \left (6 d^{3} f^{2} x^{3} {\mathrm e}^{2 i \left (f x +e \right )}-3 i c^{2} d f \,{\mathrm e}^{i \left (f x +e \right )}+18 c \,d^{2} f^{2} x^{2} {\mathrm e}^{2 i \left (f x +e \right )}+9 d^{3} f^{2} x^{3} {\mathrm e}^{i \left (f x +e \right )}-6 i c \,d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}-3 i d^{3} f \,x^{2} {\mathrm e}^{2 i \left (f x +e \right )}+18 c^{2} d \,f^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}+27 c \,d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}+5 d^{3} f^{2} x^{3}-6 i c \,d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}-3 i c^{2} d f \,{\mathrm e}^{2 i \left (f x +e \right )}+6 c^{3} f^{2} {\mathrm e}^{2 i \left (f x +e \right )}+27 c^{2} d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+15 c \,d^{2} f^{2} x^{2}-3 i d^{3} f \,x^{2} {\mathrm e}^{i \left (f x +e \right )}+9 c^{3} f^{2} {\mathrm e}^{i \left (f x +e \right )}+15 c^{2} d \,f^{2} x -6 d^{3} x \,{\mathrm e}^{2 i \left (f x +e \right )}+5 c^{3} f^{2}-6 d^{2} c \,{\mathrm e}^{2 i \left (f x +e \right )}-12 d^{3} x \,{\mathrm e}^{i \left (f x +e \right )}-12 d^{2} c \,{\mathrm e}^{i \left (f x +e \right )}-6 d^{3} x -6 d^{2} c \right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {20 i d^{2} c e x}{a^{2} f^{2}}+\frac {10 i d^{2} c \,e^{2}}{a^{2} f^{3}}-\frac {10 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x^{2}}{a^{2} f^{2}}+\frac {10 i d^{3} x^{3}}{3 a^{2} f}-\frac {10 i d^{3} e^{2} x}{a^{2} f^{3}}-\frac {20 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) c x}{a^{2} f^{2}}+\frac {20 i d^{2} c \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}+\frac {20 i d^{3} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right ) x}{a^{2} f^{3}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {20 d^{2} c e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{3}}-\frac {20 d^{3} \polylog \left (3, -{\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}-\frac {10 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{a^{2} f^{2}}+\frac {10 d \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{2}}+\frac {10 d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} f^{4}}\) | \(810\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 4460 vs. \(2 (252) = 504\).
time = 1.79, size = 4460, normalized size = 15.49 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 963 vs. \(2 (252) = 504\).
time = 3.93, size = 963, normalized size = 3.34 \begin {gather*} \frac {3 \, d^{3} f^{4} x^{4} + 12 \, c d^{2} f^{4} x^{3} - 12 \, c^{2} d f^{2} + 6 \, {\left (3 \, c^{2} d f^{4} - 2 \, d^{3} f^{2}\right )} x^{2} + 3 \, {\left (d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, c^{2} d f^{4} x^{2} + 4 \, c^{3} f^{4} x\right )} \cos \left (f x + e\right )^{2} + 12 \, {\left (c^{3} f^{4} - 2 \, c d^{2} f^{2}\right )} x + 6 \, {\left (d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} - 2 \, c^{2} d f^{2} + 2 \, {\left (3 \, c^{2} d f^{4} - d^{3} f^{2}\right )} x^{2} + 4 \, {\left (c^{3} f^{4} - c d^{2} f^{2}\right )} x\right )} \cos \left (f x + e\right ) - 120 \, {\left (i \, d^{3} f x + i \, c d^{2} f + {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (i \, d^{3} f x + i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 120 \, {\left (-i \, d^{3} f x - i \, c d^{2} f + {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (-i \, d^{3} f x - i \, c d^{2} f\right )} \cos \left (f x + e\right )\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 12 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3} + {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + 1\right ) - 12 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3} + {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (5 \, d^{3} f^{2} x^{2} + 10 \, c d^{2} f^{2} x + 5 \, c^{2} d f^{2} - 2 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + 1\right ) - 120 \, {\left (d^{3} \cos \left (f x + e\right )^{2} + 2 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 120 \, {\left (d^{3} \cos \left (f x + e\right )^{2} + 2 \, d^{3} \cos \left (f x + e\right ) + d^{3}\right )} {\rm polylog}\left (3, -\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 4 \, {\left (4 \, d^{3} f^{3} x^{3} + 12 \, c d^{2} f^{3} x^{2} + 4 \, c^{3} f^{3} - 6 \, c d^{2} f + 6 \, {\left (2 \, c^{2} d f^{3} - d^{3} f\right )} x + {\left (5 \, d^{3} f^{3} x^{3} + 15 \, c d^{2} f^{3} x^{2} + 5 \, c^{3} f^{3} - 6 \, c d^{2} f + 3 \, {\left (5 \, c^{2} d f^{3} - 2 \, d^{3} f\right )} x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, {\left (a^{2} f^{4} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{4} \cos \left (f x + e\right ) + a^{2} f^{4}\right )}} \end {gather*}
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 \begin {gather*} \frac {\int \frac {c^{3}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{3} x^{3}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c d^{2} x^{2}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {3 c^{2} d x}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx}{a^{2}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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