Integrand size = 26, antiderivative size = 311 \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {(e+f x) \cos (c+d x)}{b d}-\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}-\frac {a^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^2}+\frac {f \sin (c+d x)}{b d^2} \]
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Time = 0.39 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4611, 3377, 2717, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2 \sqrt {a^2-b^2}}+\frac {a^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2 \sqrt {a^2-b^2}}-\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d \sqrt {a^2-b^2}}+\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d \sqrt {a^2-b^2}}-\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}+\frac {f \sin (c+d x)}{b d^2}-\frac {(e+f x) \cos (c+d x)}{b d} \]
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2717
Rule 3377
Rule 3404
Rule 4611
Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \sin (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x) \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{b} \\ & = -\frac {(e+f x) \cos (c+d x)}{b d}-\frac {a \int (e+f x) \, dx}{b^2}+\frac {a^2 \int \frac {e+f x}{a+b \sin (c+d x)} \, dx}{b^2}+\frac {f \int \cos (c+d x) \, dx}{b d} \\ & = -\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {(e+f x) \cos (c+d x)}{b d}+\frac {f \sin (c+d x)}{b d^2}+\frac {\left (2 a^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{b^2} \\ & = -\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {(e+f x) \cos (c+d x)}{b d}+\frac {f \sin (c+d x)}{b d^2}-\frac {\left (2 i a^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b \sqrt {a^2-b^2}}+\frac {\left (2 i a^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{b \sqrt {a^2-b^2}} \\ & = -\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {(e+f x) \cos (c+d x)}{b d}-\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {f \sin (c+d x)}{b d^2}+\frac {\left (i a^2 f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 \sqrt {a^2-b^2} d}-\frac {\left (i a^2 f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{b^2 \sqrt {a^2-b^2} d} \\ & = -\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {(e+f x) \cos (c+d x)}{b d}-\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {f \sin (c+d x)}{b d^2}+\frac {\left (a^2 f\right ) \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 (c+d x)}\right )}{b^2 \sqrt {a^2-b^2} d^2}-\frac {\left (a^2 f\right ) \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 (c+d x)}\right )}{b^2 \sqrt {a^2-b^2} d^2} \\ & = -\frac {a e x}{b^2}-\frac {a f x^2}{2 b^2}-\frac {(e+f x) \cos (c+d x)}{b d}-\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}+\frac {i a^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d}-\frac {a^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^2}+\frac {a^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 \sqrt {a^2-b^2} d^2}+\frac {f \sin (c+d x)}{b d^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(773\) vs. \(2(311)=622\).
Time = 6.59 (sec) , antiderivative size = 773, normalized size of antiderivative = 2.49 \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a (c+d x) (c f-d (2 e+f x))-2 b d (e+f x) \cos (c+d x)+\frac {2 a^2 d (e+f x) \left (\frac {2 (d e-c f) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {-b+\sqrt {-a^2+b^2}-a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )}{\sqrt {-a^2+b^2}}\right )}{d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}+2 b f \sin (c+d x)}{2 b^2 d^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 615 vs. \(2 (279 ) = 558\).
Time = 0.62 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.98
method | result | size |
risch | \(-\frac {a f \,x^{2}}{2 b^{2}}-\frac {a e x}{b^{2}}-\frac {\left (d x f +d e +i f \right ) {\mathrm e}^{i \left (d x +c \right )}}{2 b \,d^{2}}-\frac {\left (d x f +d e -i f \right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 b \,d^{2}}+\frac {2 i a^{2} e \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (d x +c \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{d \,b^{2} \sqrt {-a^{2}+b^{2}}}+\frac {a^{2} f \ln \left (\frac {-i a -b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{-i a +\sqrt {-a^{2}+b^{2}}}\right ) x}{d \,b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {a^{2} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) x}{d \,b^{2} \sqrt {-a^{2}+b^{2}}}+\frac {a^{2} f \ln \left (\frac {-i a -b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{-i a +\sqrt {-a^{2}+b^{2}}}\right ) c}{d^{2} b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {a^{2} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) c}{d^{2} b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {i a^{2} f \operatorname {dilog}\left (\frac {-i a -b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{-i a +\sqrt {-a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {-a^{2}+b^{2}}}+\frac {i a^{2} f \operatorname {dilog}\left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 i a^{2} f c \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (d x +c \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{d^{2} b^{2} \sqrt {-a^{2}+b^{2}}}\) | \(616\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1154 vs. \(2 (271) = 542\).
Time = 0.46 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.71 \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \sin \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e+f x) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^2\,\left (e+f\,x\right )}{a+b\,\sin \left (c+d\,x\right )} \,d x \]
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