Integrand size = 20, antiderivative size = 671 \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]
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Time = 0.80 (sec) , antiderivative size = 671, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3405, 3404, 2296, 2221, 2611, 2320, 6724, 4615, 2317, 2438} \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{f^2 \left (a^2-b^2\right )}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac {b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac {i (c+d x)^2}{f \left (a^2-b^2\right )}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )}-\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )^{3/2}}+\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )^{3/2}} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 3404
Rule 3405
Rule 4615
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {a \int \frac {(c+d x)^2}{a+b \sin (e+f x)} \, dx}{a^2-b^2}-\frac {(2 b d) \int \frac {(c+d x) \cos (e+f x)}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) f} \\ & = \frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {(2 a) \int \frac {e^{i (e+f x)} (c+d x)^2}{i b+2 a e^{i (e+f x)}-i b e^{2 i (e+f x)}} \, dx}{a^2-b^2}-\frac {(2 b d) \int \frac {e^{i (e+f x)} (c+d x)}{a-\sqrt {a^2-b^2}-i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right ) f}-\frac {(2 b d) \int \frac {e^{i (e+f x)} (c+d x)}{a+\sqrt {a^2-b^2}-i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right ) f} \\ & = \frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {(2 i a b) \int \frac {e^{i (e+f x)} (c+d x)^2}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac {(2 i a b) \int \frac {e^{i (e+f x)} (c+d x)^2}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac {\left (2 d^2\right ) \int \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}+\frac {\left (2 d^2\right ) \int \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2} \\ & = \frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (2 i d^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right ) f^3}-\frac {\left (2 i d^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right ) f^3}+\frac {(2 i a d) \int (c+d x) \log \left (1-\frac {2 i b e^{i (e+f x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}-\frac {(2 i a d) \int (c+d x) \log \left (1-\frac {2 i b e^{i (e+f x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f} \\ & = \frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left (2 a d^2\right ) \int \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (e+f x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {\left (2 a d^2\right ) \int \operatorname {PolyLog}\left (2,\frac {2 i b e^{i (e+f x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2} \\ & = \frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (2 i a d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {\left (2 i a d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^3} \\ & = \frac {i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {2 d (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac {i a (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac {2 a d (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {2 i a d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac {b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.79 \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\frac {i f^2 (c+d x)^2-2 d f (c+d x) \log \left (1+\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )-2 d f (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )+2 i d^2 \operatorname {PolyLog}\left (2,-\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )+2 i d^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )-\frac {i a \left (f^2 (c+d x)^2 \log \left (1+\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )-f^2 (c+d x)^2 \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )-2 i d f (c+d x) \operatorname {PolyLog}\left (2,-\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )+2 i d f (c+d x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )+2 d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )-2 d^2 \operatorname {PolyLog}\left (3,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2}}+\frac {b f^2 (c+d x)^2 \cos (e+f x)}{a+b \sin (e+f x)}}{\left (a^2-b^2\right ) f^3} \]
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\[\int \frac {\left (d x +c \right )^{2}}{\left (a +b \sin \left (f x +e \right )\right )^{2}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3091 vs. \(2 (581) = 1162\).
Time = 0.52 (sec) , antiderivative size = 3091, normalized size of antiderivative = 4.61 \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx=\text {Hanged} \]
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