Integrand size = 18, antiderivative size = 234 \[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=-\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}-\frac {d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2}+\frac {d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2} \]
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Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3404, 2296, 2221, 2317, 2438} \[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=-\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f \sqrt {a^2-b^2}}+\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{\sqrt {a^2-b^2}+a}\right )}{f \sqrt {a^2-b^2}}-\frac {d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{f^2 \sqrt {a^2-b^2}}+\frac {d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{f^2 \sqrt {a^2-b^2}} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 3404
Rubi steps \begin{align*} \text {integral}& = 2 \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 \\ & = -\frac {(2 i 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}{\sqrt {a^2-b^2}}+\frac {(2 i 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}{\sqrt {a^2-b^2}} \\ & = -\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {(i d) \int \log \left (1-\frac {2 i b e^{i (e+f x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{\sqrt {a^2-b^2} f}-\frac {(i d) \int \log \left (1-\frac {2 i b e^{i (e+f x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{\sqrt {a^2-b^2} f} \\ & = -\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {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 )}{\sqrt {a^2-b^2} f^2}-\frac {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 )}{\sqrt {a^2-b^2} f^2} \\ & = -\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}+\frac {i (c+d x) \log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f}-\frac {d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a-\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2}+\frac {d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.78 \[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=\frac {-i f (c+d x) \left (\log \left (1+\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )-\log \left (1-\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )\right )-d \operatorname {PolyLog}\left (2,-\frac {i b e^{i (e+f x)}}{-a+\sqrt {a^2-b^2}}\right )+d \operatorname {PolyLog}\left (2,\frac {i b e^{i (e+f x)}}{a+\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2} f^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (204 ) = 408\).
Time = 0.23 (sec) , antiderivative size = 501, normalized size of antiderivative = 2.14
method | result | size |
risch | \(\frac {2 i c \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (f x +e \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{f \sqrt {-a^{2}+b^{2}}}+\frac {d \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) x}{f \sqrt {-a^{2}+b^{2}}}-\frac {d \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) x}{f \sqrt {-a^{2}+b^{2}}}+\frac {d \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) e}{f^{2} \sqrt {-a^{2}+b^{2}}}-\frac {d \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) e}{f^{2} \sqrt {-a^{2}+b^{2}}}-\frac {i d \operatorname {dilog}\left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right )}{f^{2} \sqrt {-a^{2}+b^{2}}}+\frac {i d \operatorname {dilog}\left (\frac {i a +b \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right )}{f^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 i d e \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (f x +e \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{f^{2} \sqrt {-a^{2}+b^{2}}}\) | \(501\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (200) = 400\).
Time = 0.44 (sec) , antiderivative size = 997, normalized size of antiderivative = 4.26 \[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=\int \frac {c + d x}{a + b \sin {\left (e + f x \right )}}\, dx \]
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Exception generated. \[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=\int { \frac {d x + c}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {c+d x}{a+b \sin (e+f x)} \, dx=\int \frac {c+d\,x}{a+b\,\sin \left (e+f\,x\right )} \,d x \]
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