Optimal. Leaf size=659 \[ -\frac {i a (e+f x)^2}{\left (a^2-b^2\right ) d}-\frac {4 i b f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}-\frac {i b^2 (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {2 i b f^2 \text {Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {2 b^2 f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {i a f^2 \text {Li}_2\left (-e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\frac {2 i b^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac {2 i b^2 f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x)^2 \tan (c+d x)}{\left (a^2-b^2\right ) d} \]
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Rubi [A]
time = 0.97, antiderivative size = 659, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 14, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used =
{4629, 3404, 2296, 2221, 2611, 2320, 6724, 6874, 4269, 3800, 2317, 2438, 4494, 4266}
\begin {gather*} \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 )}{d^3 \left (a^2-b^2\right )^{3/2}}-\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 )}{d^3 \left (a^2-b^2\right )^{3/2}}+\frac {2 i b f^2 \text {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^3 \left (a^2-b^2\right )}-\frac {2 i b f^2 \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^3 \left (a^2-b^2\right )}-\frac {i a f^2 \text {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{d^3 \left (a^2-b^2\right )}+\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 )}{d^2 \left (a^2-b^2\right )^{3/2}}-\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 )}{d^2 \left (a^2-b^2\right )^{3/2}}-\frac {4 i b f (e+f x) \text {ArcTan}\left (e^{i (c+d x)}\right )}{d^2 \left (a^2-b^2\right )}+\frac {2 a f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{d^2 \left (a^2-b^2\right )}+\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 )}{d \left (a^2-b^2\right )^{3/2}}-\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 )}{d \left (a^2-b^2\right )^{3/2}}+\frac {a (e+f x)^2 \tan (c+d x)}{d \left (a^2-b^2\right )}-\frac {b (e+f x)^2 \sec (c+d x)}{d \left (a^2-b^2\right )}-\frac {i a (e+f x)^2}{d \left (a^2-b^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3404
Rule 3800
Rule 4266
Rule 4269
Rule 4494
Rule 4629
Rule 6724
Rule 6874
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sec ^2(c+d x) (a-b \sin (c+d x)) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {(e+f x)^2}{a+b \sin (c+d x)} \, dx}{a^2-b^2}\\ &=\frac {\int \left (a (e+f x)^2 \sec ^2(c+d x)-b (e+f x)^2 \sec (c+d x) \tan (c+d x)\right ) \, dx}{a^2-b^2}-\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-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}{\left (a^2-b^2\right )^{3/2}}-\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}{\left (a^2-b^2\right )^{3/2}}+\frac {a \int (e+f x)^2 \sec ^2(c+d x) \, dx}{a^2-b^2}-\frac {b \int (e+f x)^2 \sec (c+d x) \tan (c+d x) \, dx}{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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d}-\frac {b (e+f x)^2 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x)^2 \tan (c+d x)}{\left (a^2-b^2\right ) 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}{\left (a^2-b^2\right )^{3/2} 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}{\left (a^2-b^2\right )^{3/2} d}-\frac {(2 a f) \int (e+f x) \tan (c+d x) \, dx}{\left (a^2-b^2\right ) d}+\frac {(2 b f) \int (e+f x) \sec (c+d x) \, dx}{\left (a^2-b^2\right ) d}\\ &=-\frac {i a (e+f x)^2}{\left (a^2-b^2\right ) d}-\frac {4 i b f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d}+\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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {b (e+f x)^2 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x)^2 \tan (c+d x)}{\left (a^2-b^2\right ) d}+\frac {(4 i a f) \int \frac {e^{2 i (c+d x)} (e+f x)}{1+e^{2 i (c+d x)}} \, dx}{\left (a^2-b^2\right ) d}-\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}{\left (a^2-b^2\right )^{3/2} 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}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {\left (2 b f^2\right ) \int \log \left (1-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}+\frac {\left (2 b f^2\right ) \int \log \left (1+i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}\\ &=-\frac {i a (e+f x)^2}{\left (a^2-b^2\right ) d}-\frac {4 i b f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) 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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {b (e+f x)^2 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x)^2 \tan (c+d x)}{\left (a^2-b^2\right ) d}+\frac {\left (2 i b^2 f^2\right ) \text {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 )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac {\left (2 i b^2 f^2\right ) \text {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 )}{\left (a^2-b^2\right )^{3/2} d^3}+\frac {\left (2 i b f^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {\left (2 i b f^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {\left (2 a f^2\right ) \int \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d^2}\\ &=-\frac {i a (e+f x)^2}{\left (a^2-b^2\right ) d}-\frac {4 i b f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {2 i b f^2 \text {Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d^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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x)^2 \tan (c+d x)}{\left (a^2-b^2\right ) d}+\frac {\left (i a f^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}\\ &=-\frac {i a (e+f x)^2}{\left (a^2-b^2\right ) d}-\frac {4 i b f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d}+\frac {2 a f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {2 i b f^2 \text {Li}_2\left (-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}-\frac {2 i b f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^3}+\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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d^2}-\frac {i a f^2 \text {Li}_2\left (-e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) 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 )}{\left (a^2-b^2\right )^{3/2} 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 )}{\left (a^2-b^2\right )^{3/2} d^3}-\frac {b (e+f x)^2 \sec (c+d x)}{\left (a^2-b^2\right ) d}+\frac {a (e+f x)^2 \tan (c+d x)}{\left (a^2-b^2\right ) d}\\ \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. \(1368\) vs. \(2(659)=1318\).
time = 7.27, size = 1368, normalized size = 2.08 \begin {gather*} \frac {b (e+f x)^2 \sec (c)}{\left (-a^2+b^2\right ) d}+\frac {2 a e f \sec (c) (\cos (c) \log (\cos (c) \cos (d x)-\sin (c) \sin (d x))+d x \sin (c))}{\left (a^2-b^2\right ) d^2 \left (\cos ^2(c)+\sin ^2(c)\right )}+\frac {4 i b e f \tan ^{-1}\left (\frac {-i \sin (c)-i \cos (c) \tan \left (\frac {d x}{2}\right )}{\sqrt {\cos ^2(c)+\sin ^2(c)}}\right )}{\left (a^2-b^2\right ) d^2 \sqrt {\cos ^2(c)+\sin ^2(c)}}+\frac {a f^2 \csc (c) \left (d^2 e^{-i \tan ^{-1}(\cot (c))} x^2-\frac {\cot (c) \left (i d x \left (-\pi -2 \tan ^{-1}(\cot (c))\right )-\pi \log \left (1+e^{-2 i d x}\right )-2 \left (d x-\tan ^{-1}(\cot (c))\right ) \log \left (1-e^{2 i \left (d x-\tan ^{-1}(\cot (c))\right )}\right )+\pi \log (\cos (d x))-2 \tan ^{-1}(\cot (c)) \log \left (\sin \left (d x-\tan ^{-1}(\cot (c))\right )\right )+i \text {Li}_2\left (e^{2 i \left (d x-\tan ^{-1}(\cot (c))\right )}\right )\right )}{\sqrt {1+\cot ^2(c)}}\right ) \sec (c)}{\left (a^2-b^2\right ) d^3 \sqrt {\csc ^2(c) \left (\cos ^2(c)+\sin ^2(c)\right )}}+\frac {2 b f^2 \left (-\frac {\csc (c) \left (\left (d x-\tan ^{-1}(\cot (c))\right ) \left (\log \left (1-e^{i \left (d x-\tan ^{-1}(\cot (c))\right )}\right )-\log \left (1+e^{i \left (d x-\tan ^{-1}(\cot (c))\right )}\right )\right )+i \left (\text {Li}_2\left (-e^{i \left (d x-\tan ^{-1}(\cot (c))\right )}\right )-\text {Li}_2\left (e^{i \left (d x-\tan ^{-1}(\cot (c))\right )}\right )\right )\right )}{\sqrt {1+\cot ^2(c)}}+\frac {2 \tan ^{-1}(\cot (c)) \tanh ^{-1}\left (\frac {\sin (c)+\cos (c) \tan \left (\frac {d x}{2}\right )}{\sqrt {\cos ^2(c)+\sin ^2(c)}}\right )}{\sqrt {\cos ^2(c)+\sin ^2(c)}}\right )}{\left (a^2-b^2\right ) d^3}-\frac {i b^2 \left (2 \sqrt {a^2-b^2} d f (e+f x) \text {Li}_2\left (-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))-2 \sqrt {a^2-b^2} d f (e+f x) \text {Li}_2\left (\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (\cos (c)+i \sin (c))-i \left (-2 \sqrt {a^2-b^2} f^2 \text {Li}_3\left (-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right ) (\cos (c)+i \sin (c))+2 \sqrt {a^2-b^2} f^2 \text {Li}_3\left (\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right ) (\cos (c)+i \sin (c))+d^2 \left (\sqrt {a^2-b^2} f x (2 e+f x) \left (-\log \left (1+\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}-a \sin (c)}\right )+\log \left (1-\frac {b (\cos (2 c+d x)+i \sin (2 c+d x))}{-i a \cos (c)+\sqrt {\left (-a^2+b^2\right ) (\cos (c)+i \sin (c))^2}+a \sin (c)}\right )\right ) (\cos (c)+i \sin (c))+2 e^2 \tan ^{-1}\left (\frac {b \cos (c+d x)+i (a+b \sin (c+d x))}{\sqrt {a^2-b^2}}\right ) \sqrt {\left (-a^2+b^2\right ) (\cos (2 c)+i \sin (2 c))}\right )\right )\right )}{\left (a^2-b^2\right )^{3/2} d^3 \sqrt {\left (-a^2+b^2\right ) (\cos (2 c)+i \sin (2 c))}}+\frac {e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )}{(a+b) d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {e^2 \sin \left (\frac {d x}{2}\right )+2 e f x \sin \left (\frac {d x}{2}\right )+f^2 x^2 \sin \left (\frac {d x}{2}\right )}{(a-b) d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \left (\sec ^{2}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}\, dx\]
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 \begin {gather*} \text {Exception raised: ValueError} \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. 2671 vs. \(2 (583) = 1166\).
time = 0.69, size = 2671, normalized size = 4.05 \begin {gather*} \text {Too large to display} \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*} \int \frac {\left (e + f x\right )^{2} \sec ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \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.
[In]
[Out]
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