Integrand size = 18, antiderivative size = 126 \[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=-\frac {e^{c (a+b x)}}{b c}+\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c} \]
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Time = 0.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4528, 2225, 2283} \[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {e^{c (a+b x)}}{b c} \]
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Rule 2225
Rule 2283
Rule 4528
Rubi steps \begin{align*} \text {integral}& = -\int \left (e^{c (a+b x)}+\frac {4 e^{c (a+b x)}}{\left (-1+e^{2 i (d+e x)}\right )^2}+\frac {4 e^{c (a+b x)}}{-1+e^{2 i (d+e x)}}\right ) \, dx \\ & = -\left (4 \int \frac {e^{c (a+b x)}}{\left (-1+e^{2 i (d+e x)}\right )^2} \, dx\right )-4 \int \frac {e^{c (a+b x)}}{-1+e^{2 i (d+e x)}} \, dx-\int e^{c (a+b x)} \, dx \\ & = -\frac {e^{c (a+b x)}}{b c}+\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c}-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )}{b c} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.36 \[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=e^{c (a+b x)} \left (-\frac {1}{b c}+\frac {2 e^{2 i d} \left (i b c e^{2 i e x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i b c}{2 e},2-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )+(-i b c+2 e) \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},e^{2 i (d+e x)}\right )\right )}{(b c+2 i e) e \left (-1+e^{2 i d}\right )}+\frac {\csc (d) \csc (d+e x) \sin (e x)}{e}\right ) \]
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\[\int {\mathrm e}^{c \left (x b +a \right )} \cot \left (e x +d \right )^{2}d x\]
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\[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=\int { \cot \left (e x + d\right )^{2} e^{\left ({\left (b x + a\right )} c\right )} \,d x } \]
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\[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=e^{a c} \int e^{b c x} \cot ^{2}{\left (d + e x \right )}\, dx \]
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\[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=\int { \cot \left (e x + d\right )^{2} e^{\left ({\left (b x + a\right )} c\right )} \,d x } \]
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\[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=\int { \cot \left (e x + d\right )^{2} e^{\left ({\left (b x + a\right )} c\right )} \,d x } \]
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Timed out. \[ \int e^{c (a+b x)} \cot ^2(d+e x) \, dx=\int {\mathrm {cot}\left (d+e\,x\right )}^2\,{\mathrm {e}}^{c\,\left (a+b\,x\right )} \,d x \]
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