Integrand size = 24, antiderivative size = 413 \[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 i a (e+f x) \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2} \]
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Time = 0.40 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4629, 4615, 2221, 2317, 2438, 6874, 4266, 3800} \[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {2 i a (e+f x) \arctan \left (e^{i (c+d x)}\right )}{d \left (a^2-b^2\right )}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{d^2 \left (a^2-b^2\right )}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{d^2 \left (a^2-b^2\right )}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d^2 \left (a^2-b^2\right )}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 d^2 \left (a^2-b^2\right )}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{d \left (a^2-b^2\right )}+\frac {b (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{d \left (a^2-b^2\right )} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4266
Rule 4615
Rule 4629
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \sec (c+d x) (a-b \sin (c+d x)) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx}{a^2-b^2} \\ & = \frac {i b (e+f x)^2}{2 \left (a^2-b^2\right ) f}+\frac {\int (a (e+f x) \sec (c+d x)-b (e+f x) \tan (c+d x)) \, dx}{a^2-b^2}-\frac {b^2 \int \frac {e^{i (c+d x)} (e+f x)}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{a^2-b^2}-\frac {b^2 \int \frac {e^{i (c+d x)} (e+f x)}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx}{a^2-b^2} \\ & = \frac {i b (e+f x)^2}{2 \left (a^2-b^2\right ) f}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {a \int (e+f x) \sec (c+d x) \, dx}{a^2-b^2}-\frac {b \int (e+f x) \tan (c+d x) \, dx}{a^2-b^2}+\frac {(b f) \int \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d}+\frac {(b f) \int \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) d} \\ & = -\frac {2 i a (e+f x) \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)}{1+e^{2 i (c+d x)}} \, dx}{a^2-b^2}-\frac {(i b f) \text {Subst}\left (\int \frac {\log \left (1-\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 ) d^2}-\frac {(i b f) \text {Subst}\left (\int \frac {\log \left (1-\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 ) d^2}-\frac {(a f) \int \log \left (1-i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d}+\frac {(a f) \int \log \left (1+i e^{i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d} \\ & = -\frac {2 i a (e+f x) \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {(i a f) \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^2}-\frac {(i a f) \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^2}-\frac {(b f) \int \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{\left (a^2-b^2\right ) d} \\ & = -\frac {2 i a (e+f x) \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {(i b f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2} \\ & = -\frac {2 i a (e+f x) \arctan \left (e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}-\frac {b (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d}+\frac {b (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{\left (a^2-b^2\right ) d}+\frac {i a f \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i a f \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) d^2}-\frac {i b f \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{2 \left (a^2-b^2\right ) d^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1185\) vs. \(2(413)=826\).
Time = 9.05 (sec) , antiderivative size = 1185, normalized size of antiderivative = 2.87 \[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {b e \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{\left (a^2-b^2\right ) d}+\frac {b c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )}{\left (a^2-b^2\right ) d^2}-\frac {b^2 f \left (\frac {(c+d x) \log (a+b \sin (c+d x))}{b}-\frac {-\frac {1}{2} i \left (-c+\frac {\pi }{2}-d x\right )^2+4 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \tan \left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right )\right )}{\sqrt {a^2-b^2}}\right )+\left (-c+\frac {\pi }{2}-d x+2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\left (-c+\frac {\pi }{2}-d x-2 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {\left (a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )-\left (-c+\frac {\pi }{2}-d x\right ) \log (a+b \sin (c+d x))-i \left (\operatorname {PolyLog}\left (2,\frac {\left (-a-\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )+\operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2-b^2}\right ) e^{i \left (-c+\frac {\pi }{2}-d x\right )}}{b}\right )\right )}{b}\right )}{\left (a^2-b^2\right ) d^2}+\frac {(d e+d f x) \left (-\frac {i b (d e+d f x)^2}{f}+2 (a-b) (d e-c f) \log \left (1-\tan \left (\frac {1}{2} (c+d x)\right )\right )-4 b (d e+d f x) \log \left (-\frac {2 i}{-i+\tan \left (\frac {1}{2} (c+d x)\right )}\right )-2 (a+b) (d e-c f) \log \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )-4 i b f \operatorname {PolyLog}(2,-\cos (c+d x)+i \sin (c+d x))+2 i (a+b) f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)-(1-i) \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )-2 i (a+b) f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )+2 i (a-b) f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i)+(1-i) \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )-2 i (a-b) f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (-1+\tan \left (\frac {1}{2} (c+d x)\right )\right )\right )+\operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1-i)+(1+i) \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )\right ) (a-b \sin (c+d x))}{\left (a^2-b^2\right ) d^2 \left (-2 a d e+2 a c f-2 i a f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )+2 i a f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )+4 b f \cos (c+d x) \left (\log (1+\cos (c+d x)-i \sin (c+d x))-\log \left (-\frac {2 i}{-i+\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right )+b (d e-c f+f (c+d x)) \sec \left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {3}{2} (c+d x)\right )+b d e \tan \left (\frac {1}{2} (c+d x)\right )-b c f \tan \left (\frac {1}{2} (c+d x)\right )-b f (c+d x) \tan \left (\frac {1}{2} (c+d x)\right )+2 i b f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )-2 i b f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (373 ) = 746\).
Time = 0.49 (sec) , antiderivative size = 846, normalized size of antiderivative = 2.05
method | result | size |
risch | \(\frac {4 e \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (4 a -4 b \right )}-\frac {4 e \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left (4 a +4 b \right )}-\frac {e b \ln \left (i b \,{\mathrm e}^{2 i \left (d x +c \right )}-i b -2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left (a -b \right ) \left (a +b \right )}+\frac {4 f \ln \left (-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )\right ) x}{d \left (4 a -4 b \right )}+\frac {4 f \ln \left (-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )\right ) c}{d^{2} \left (4 a -4 b \right )}-\frac {4 i f \operatorname {dilog}\left (-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )\right )}{d^{2} \left (4 a -4 b \right )}-\frac {4 f \ln \left (-i \left (i-{\mathrm e}^{i \left (d x +c \right )}\right )\right ) x}{d \left (4 a +4 b \right )}-\frac {4 f \ln \left (-i \left (i-{\mathrm e}^{i \left (d x +c \right )}\right )\right ) c}{d^{2} \left (4 a +4 b \right )}+\frac {i f b \operatorname {dilog}\left (\frac {i b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} \left (a -b \right ) \left (a +b \right )}-\frac {4 i f \operatorname {dilog}\left (-i {\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} \left (4 a +4 b \right )}-\frac {f b \ln \left (\frac {-i b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) x}{d \left (a -b \right ) \left (a +b \right )}-\frac {f b \ln \left (\frac {-i b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} \left (a -b \right ) \left (a +b \right )}-\frac {f b \ln \left (\frac {i b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) x}{d \left (a -b \right ) \left (a +b \right )}-\frac {f b \ln \left (\frac {i b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}-a}{-a +\sqrt {a^{2}-b^{2}}}\right ) c}{d^{2} \left (a -b \right ) \left (a +b \right )}+\frac {i f b \operatorname {dilog}\left (\frac {-i b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {a^{2}-b^{2}}+a}{a +\sqrt {a^{2}-b^{2}}}\right )}{d^{2} \left (a -b \right ) \left (a +b \right )}-\frac {4 i f \ln \left (-i \left (i-{\mathrm e}^{i \left (d x +c \right )}\right )\right ) \ln \left (-i {\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} \left (4 a +4 b \right )}-\frac {4 c f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{2} \left (4 a -4 b \right )}+\frac {4 c f \ln \left (-i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} \left (4 a +4 b \right )}+\frac {c f b \ln \left (i b \,{\mathrm e}^{2 i \left (d x +c \right )}-i b -2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} \left (a -b \right ) \left (a +b \right )}\) | \(846\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1181 vs. \(2 (356) = 712\).
Time = 0.49 (sec) , antiderivative size = 1181, normalized size of antiderivative = 2.86 \[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \sec {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sec \left (d x + c\right )}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]
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