Integrand size = 18, antiderivative size = 305 \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=-\frac {i a (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac {i a (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac {d \log (a+b \sin (e+f x))}{\left (a^2-b^2\right ) f^2}-\frac {a d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {a d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac {b (c+d x) \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \]
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Time = 0.37 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=-\frac {i a (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac {i a (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac {b (c+d x) \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}-\frac {a d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac {a d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}-\frac {d \log (a+b \sin (e+f x))}{f^2 \left (a^2-b^2\right )} \]
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3404
Rule 3405
Rubi steps \begin{align*} \text {integral}& = \frac {b (c+d x) \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {a \int \frac {c+d x}{a+b \sin (e+f x)} \, dx}{a^2-b^2}-\frac {(b d) \int \frac {\cos (e+f x)}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) f} \\ & = \frac {b (c+d x) \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)}{i b+2 a e^{i (e+f x)}-i b e^{2 i (e+f x)}} \, dx}{a^2-b^2}-\frac {d \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (e+f x)\right )}{\left (a^2-b^2\right ) f^2} \\ & = -\frac {d \log (a+b \sin (e+f x))}{\left (a^2-b^2\right ) f^2}+\frac {b (c+d x) \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 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 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 {i a (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 )^{3/2} f}+\frac {i a (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 )^{3/2} f}-\frac {d \log (a+b \sin (e+f x))}{\left (a^2-b^2\right ) f^2}+\frac {b (c+d x) \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {(i a d) \int \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 a d) \int \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 a (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 )^{3/2} f}+\frac {i a (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 )^{3/2} f}-\frac {d \log (a+b \sin (e+f x))}{\left (a^2-b^2\right ) f^2}+\frac {b (c+d x) \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {(a d) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a-2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac {(a d) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a+2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^2} \\ & = -\frac {i a (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 )^{3/2} f}+\frac {i a (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 )^{3/2} f}-\frac {d \log (a+b \sin (e+f x))}{\left (a^2-b^2\right ) f^2}-\frac {a d \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 {a d \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) \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.77 \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=\frac {-d \log (a+b \sin (e+f x))+\frac {a \left (-i f (c+d x) \left (\log \left (1+\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )-\log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )\right )-d \operatorname {PolyLog}\left (2,-\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )+d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )\right )}{\sqrt {a^2-b^2}}+\frac {b f (c+d x) \cos (e+f x)}{a+b \sin (e+f x)}}{\left (a^2-b^2\right ) f^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 649 vs. \(2 (275 ) = 550\).
Time = 1.25 (sec) , antiderivative size = 650, normalized size of antiderivative = 2.13
method | result | size |
risch | \(\frac {2 \left (d x +c \right ) \left (i b +a \,{\mathrm e}^{i \left (f x +e \right )}\right )}{f \left (a^{2}-b^{2}\right ) \left (b \,{\mathrm e}^{2 i \left (f x +e \right )}-b +2 i a \,{\mathrm e}^{i \left (f x +e \right )}\right )}+\frac {d \ln \left (i b \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -2 a \,{\mathrm e}^{i \left (f x +e \right )}\right )}{\left (-a^{2}+b^{2}\right ) f^{2}}-\frac {2 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (-a^{2}+b^{2}\right ) f^{2}}+\frac {2 i a d e \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (f x +e \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f^{2}}-\frac {2 i a c \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (f x +e \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f}-\frac {d a \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) x}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f}+\frac {d a \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) x}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f}-\frac {d a \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) e}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f^{2}}+\frac {d a \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) e}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f^{2}}+\frac {i d a \operatorname {dilog}\left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right )}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f^{2}}-\frac {i d a \operatorname {dilog}\left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right )}{\left (-a^{2}+b^{2}\right )^{\frac {3}{2}} f^{2}}\) | \(650\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1512 vs. \(2 (267) = 534\).
Time = 0.49 (sec) , antiderivative size = 1512, normalized size of antiderivative = 4.96 \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {d x + c}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {c+d x}{(a+b \sin (e+f x))^2} \, dx=\text {Hanged} \]
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