Optimal. Leaf size=229 \[ \frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac {20 i d^2 \text {PolyLog}\left (2,-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{6 a^2 f} \]
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Rubi [A]
time = 0.36, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3399,
4271, 3852, 8, 4269, 3800, 2221, 2317, 2438} \begin {gather*} -\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \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)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}+\frac {20 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{3 a^2 f^3}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 3399
Rule 3800
Rule 3852
Rule 4269
Rule 4271
Rule 4276
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{(a+a \sec (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^2}{a^2}+\frac {(c+d x)^2}{a^2 (1+\cos (e+f x))^2}-\frac {2 (c+d x)^2}{a^2 (1+\cos (e+f x))}\right ) \, dx\\ &=\frac {(c+d x)^3}{3 a^2 d}+\frac {\int \frac {(c+d x)^2}{(1+\cos (e+f x))^2} \, dx}{a^2}-\frac {2 \int \frac {(c+d x)^2}{1+\cos (e+f x)} \, dx}{a^2}\\ &=\frac {(c+d x)^3}{3 a^2 d}+\frac {\int (c+d x)^2 \csc ^4\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{4 a^2}-\frac {\int (c+d x)^2 \csc ^2\left (\frac {e+\pi }{2}+\frac {f x}{2}\right ) \, dx}{a^2}\\ &=\frac {(c+d x)^3}{3 a^2 d}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {2 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a^2 f}+\frac {(c+d x)^2 \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)^2 \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{6 a^2}+\frac {d^2 \int \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f^2}+\frac {(4 d) \int (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{a^2 f}\\ &=\frac {2 i (c+d x)^2}{a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \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^2\right ) \text {Subst}\left (\int 1 \, dx,x,-\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{3 a^2 f^3}-\frac {(8 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{a^2 f}-\frac {(2 d) \int (c+d x) \tan \left (\frac {e}{2}+\frac {f x}{2}\right ) \, dx}{3 a^2 f}\\ &=\frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {8 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \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 (8 d^2\right ) \int \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{a^2 f^2}+\frac {(4 i d) \int \frac {e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )} (c+d x)}{1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}} \, dx}{3 a^2 f}\\ &=\frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \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 (8 i d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{a^2 f^3}-\frac {\left (4 d^2\right ) \int \log \left (1+e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right ) \, dx}{3 a^2 f^2}\\ &=\frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac {8 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \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 i d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \left (\frac {e}{2}+\frac {f x}{2}\right )}\right )}{3 a^2 f^3}\\ &=\frac {5 i (c+d x)^2}{3 a^2 f}+\frac {(c+d x)^3}{3 a^2 d}-\frac {20 d (c+d x) \log \left (1+e^{i (e+f x)}\right )}{3 a^2 f^2}+\frac {20 i d^2 \text {Li}_2\left (-e^{i (e+f x)}\right )}{3 a^2 f^3}-\frac {d (c+d x) \sec ^2\left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^2}+\frac {2 d^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f^3}-\frac {5 (c+d x)^2 \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^2 \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. \(925\) vs. \(2(229)=458\).
time = 6.67, size = 925, normalized size = 4.04 \begin {gather*} -\frac {80 c 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 )}{3 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 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)}{3 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 (-4 c d f \cos \left (\frac {f x}{2}\right )-4 d^2 f x \cos \left (\frac {f x}{2}\right )+9 c^2 f^3 x \cos \left (\frac {f x}{2}\right )+9 c d f^3 x^2 \cos \left (\frac {f x}{2}\right )+3 d^2 f^3 x^3 \cos \left (\frac {f x}{2}\right )-4 c d f \cos \left (e+\frac {f x}{2}\right )-4 d^2 f x \cos \left (e+\frac {f x}{2}\right )+9 c^2 f^3 x \cos \left (e+\frac {f x}{2}\right )+9 c d f^3 x^2 \cos \left (e+\frac {f x}{2}\right )+3 d^2 f^3 x^3 \cos \left (e+\frac {f x}{2}\right )+3 c^2 f^3 x \cos \left (e+\frac {3 f x}{2}\right )+3 c d f^3 x^2 \cos \left (e+\frac {3 f x}{2}\right )+d^2 f^3 x^3 \cos \left (e+\frac {3 f x}{2}\right )+3 c^2 f^3 x \cos \left (2 e+\frac {3 f x}{2}\right )+3 c d f^3 x^2 \cos \left (2 e+\frac {3 f x}{2}\right )+d^2 f^3 x^3 \cos \left (2 e+\frac {3 f x}{2}\right )+8 d^2 \sin \left (\frac {f x}{2}\right )-18 c^2 f^2 \sin \left (\frac {f x}{2}\right )-36 c d f^2 x \sin \left (\frac {f x}{2}\right )-18 d^2 f^2 x^2 \sin \left (\frac {f x}{2}\right )-4 d^2 \sin \left (e+\frac {f x}{2}\right )+12 c^2 f^2 \sin \left (e+\frac {f x}{2}\right )+24 c d f^2 x \sin \left (e+\frac {f x}{2}\right )+12 d^2 f^2 x^2 \sin \left (e+\frac {f x}{2}\right )+4 d^2 \sin \left (e+\frac {3 f x}{2}\right )-10 c^2 f^2 \sin \left (e+\frac {3 f x}{2}\right )-20 c d f^2 x \sin \left (e+\frac {3 f x}{2}\right )-10 d^2 f^2 x^2 \sin \left (e+\frac {3 f x}{2}\right )\right )}{6 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. 452 vs. \(2 (187 ) = 374\).
time = 0.65, size = 453, normalized size = 1.98
method | result | size |
risch | \(\frac {d^{2} x^{3}}{3 a^{2}}+\frac {d c \,x^{2}}{a^{2}}+\frac {c^{2} x}{a^{2}}+\frac {c^{3}}{3 a^{2} d}-\frac {2 i \left (6 d^{2} f^{2} x^{2} {\mathrm e}^{2 i \left (f x +e \right )}-2 i d^{2} f x \,{\mathrm e}^{i \left (f x +e \right )}+12 c d \,f^{2} x \,{\mathrm e}^{2 i \left (f x +e \right )}+9 d^{2} f^{2} x^{2} {\mathrm e}^{i \left (f x +e \right )}-2 i c d f \,{\mathrm e}^{2 i \left (f x +e \right )}-2 i d^{2} f x \,{\mathrm e}^{2 i \left (f x +e \right )}+6 c^{2} f^{2} {\mathrm e}^{2 i \left (f x +e \right )}+18 c d \,f^{2} x \,{\mathrm e}^{i \left (f x +e \right )}+5 d^{2} x^{2} f^{2}-2 i c d f \,{\mathrm e}^{i \left (f x +e \right )}+9 c^{2} f^{2} {\mathrm e}^{i \left (f x +e \right )}+10 c d \,f^{2} x +5 c^{2} f^{2}-2 d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-4 d^{2} {\mathrm e}^{i \left (f x +e \right )}-2 d^{2}\right )}{3 f^{3} a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {20 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{2}}-\frac {20 d c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{3 a^{2} f^{2}}+\frac {10 i d^{2} x^{2}}{3 a^{2} f}+\frac {20 i d^{2} e x}{3 a^{2} f^{2}}+\frac {10 i d^{2} e^{2}}{3 a^{2} f^{3}}-\frac {20 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) x}{3 a^{2} f^{2}}+\frac {20 i d^{2} \polylog \left (2, -{\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}-\frac {20 d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{3 a^{2} f^{3}}\) | \(453\) |
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. 1071 vs. \(2 (191) = 382\).
time = 1.13, size = 1071, normalized size = 4.68 \begin {gather*} -\frac {i \, d^{2} f^{3} x^{3} + 3 i \, c d f^{3} x^{2} + 3 i \, c^{2} f^{3} x + 10 \, c^{2} f^{2} - 4 \, d^{2} + 20 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right ) - {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (-i \, d^{2} f x - i \, c d f\right )} \sin \left (f x + e\right )\right )} \arctan \left (\sin \left (f x + e\right ), \cos \left (f x + e\right ) + 1\right ) + {\left (i \, d^{2} f^{3} x^{3} + {\left (3 i \, c d f^{3} - 10 \, d^{2} f^{2}\right )} x^{2} + {\left (3 i \, c^{2} f^{3} - 20 \, c d f^{2}\right )} x\right )} \cos \left (3 \, f x + 3 \, e\right ) + {\left (3 i \, d^{2} f^{3} x^{3} + 12 \, c^{2} f^{2} - 4 i \, c d f - 9 \, {\left (-i \, c d f^{3} + 2 \, d^{2} f^{2}\right )} x^{2} - 4 \, d^{2} + {\left (9 i \, c^{2} f^{3} - 36 \, c d f^{2} - 4 i \, d^{2} f\right )} x\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (3 i \, d^{2} f^{3} x^{3} + 18 \, c^{2} f^{2} - 4 i \, c d f - 3 \, {\left (-3 i \, c d f^{3} + 4 \, d^{2} f^{2}\right )} x^{2} - 8 \, d^{2} + {\left (9 i \, c^{2} f^{3} - 24 \, c d f^{2} - 4 i \, d^{2} f\right )} x\right )} \cos \left (f x + e\right ) - 20 \, {\left (d^{2} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, d^{2} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, d^{2} \cos \left (f x + e\right ) + i \, d^{2} \sin \left (3 \, f x + 3 \, e\right ) + 3 i \, d^{2} \sin \left (2 \, f x + 2 \, e\right ) + 3 i \, d^{2} \sin \left (f x + e\right ) + d^{2}\right )} {\rm Li}_2\left (-e^{\left (i \, f x + i \, e\right )}\right ) - 10 \, {\left (i \, d^{2} f x + i \, c d f + {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (3 \, f x + 3 \, e\right ) + 3 \, {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (2 \, f x + 2 \, e\right ) + 3 \, {\left (i \, d^{2} f x + i \, c d f\right )} \cos \left (f x + e\right ) - {\left (d^{2} f x + c d f\right )} \sin \left (3 \, f x + 3 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (2 \, f x + 2 \, e\right ) - 3 \, {\left (d^{2} f x + c d f\right )} \sin \left (f x + e\right )\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1\right ) - {\left (d^{2} f^{3} x^{3} + {\left (3 \, c d f^{3} + 10 i \, d^{2} f^{2}\right )} x^{2} + {\left (3 \, c^{2} f^{3} + 20 i \, c d f^{2}\right )} x\right )} \sin \left (3 \, f x + 3 \, e\right ) - {\left (3 \, d^{2} f^{3} x^{3} - 12 i \, c^{2} f^{2} - 4 \, c d f + 9 \, {\left (c d f^{3} + 2 i \, d^{2} f^{2}\right )} x^{2} + 4 i \, d^{2} + {\left (9 \, c^{2} f^{3} + 36 i \, c d f^{2} - 4 \, d^{2} f\right )} x\right )} \sin \left (2 \, f x + 2 \, e\right ) - {\left (3 \, d^{2} f^{3} x^{3} - 18 i \, c^{2} f^{2} - 4 \, c d f + 3 \, {\left (3 \, c d f^{3} + 4 i \, d^{2} f^{2}\right )} x^{2} + 8 i \, d^{2} + {\left (9 \, c^{2} f^{3} + 24 i \, c d f^{2} - 4 \, d^{2} f\right )} x\right )} \sin \left (f x + e\right )}{-3 i \, a^{2} f^{3} \cos \left (3 \, f x + 3 \, e\right ) - 9 i \, a^{2} f^{3} \cos \left (2 \, f x + 2 \, e\right ) - 9 i \, a^{2} f^{3} \cos \left (f x + e\right ) + 3 \, a^{2} f^{3} \sin \left (3 \, f x + 3 \, e\right ) + 9 \, a^{2} f^{3} \sin \left (2 \, f x + 2 \, e\right ) + 9 \, a^{2} f^{3} \sin \left (f x + e\right ) - 3 i \, a^{2} f^{3}} \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. 515 vs. \(2 (191) = 382\).
time = 3.52, size = 515, normalized size = 2.25 \begin {gather*} \frac {d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} - 2 \, c d f + {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} + 3 \, c^{2} f^{3} x\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, c^{2} f^{3} - 2 \, d^{2} f\right )} x + 2 \, {\left (d^{2} f^{3} x^{3} + 3 \, c d f^{3} x^{2} - c d f + {\left (3 \, c^{2} f^{3} - d^{2} f\right )} x\right )} \cos \left (f x + e\right ) - 10 \, {\left (i \, d^{2} \cos \left (f x + e\right )^{2} + 2 i \, d^{2} \cos \left (f x + e\right ) + i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - 10 \, {\left (-i \, d^{2} \cos \left (f x + e\right )^{2} - 2 i \, d^{2} \cos \left (f x + e\right ) - i \, d^{2}\right )} {\rm Li}_2\left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 10 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\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 ) - 10 \, {\left (d^{2} f x + c d f + {\left (d^{2} f x + c d f\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (d^{2} f x + c d f\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 ) - {\left (4 \, d^{2} f^{2} x^{2} + 8 \, c d f^{2} x + 4 \, c^{2} f^{2} - 2 \, d^{2} + {\left (5 \, d^{2} f^{2} x^{2} + 10 \, c d f^{2} x + 5 \, c^{2} f^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f^{3} \cos \left (f x + e\right )^{2} + 2 \, a^{2} f^{3} \cos \left (f x + e\right ) + a^{2} f^{3}\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^{2}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} x^{2}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c 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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