Integrand size = 27, antiderivative size = 76 \[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\frac {i F^{a+b x}}{b \log (F)}-\frac {2 i F^{a+b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},i e^{i (c+d x)}\right )}{b \log (F)} \]
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Time = 0.15 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {4552, 4527, 2225, 2283} \[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\frac {i F^{a+b x}}{b \log (F)}-\frac {2 i F^{a+b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},i e^{i (c+d x)}\right )}{b \log (F)} \]
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Rule 2225
Rule 2283
Rule 4527
Rule 4552
Rubi steps \begin{align*} \text {integral}& = -\int F^{a+b x} \tan \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx \\ & = -\left (i \int \left (-F^{a+b x}+\frac {2 F^{a+b x}}{1+e^{2 i \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right )}}\right ) \, dx\right ) \\ & = i \int F^{a+b x} \, dx-2 i \int \frac {F^{a+b x}}{1+e^{2 i \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d x}{2}\right )}} \, dx \\ & = \frac {i F^{a+b x}}{b \log (F)}-\frac {2 i F^{a+b x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},i e^{i (c+d x)}\right )}{b \log (F)} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.75 \[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\frac {F^{a+b x} \left (b e^{i (c+d x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i b \log (F)}{d},2-\frac {i b \log (F)}{d},i e^{i (c+d x)}\right ) \log (F)+\operatorname {Hypergeometric2F1}\left (1,-\frac {i b \log (F)}{d},1-\frac {i b \log (F)}{d},i e^{i (c+d x)}\right ) (d-i b \log (F))\right )}{b \log (F) (i d+b \log (F))} \]
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\[\int F^{x b +a} \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )d x\]
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\[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\int { F^{b x + a} \cot \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \,d x } \]
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\[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\int F^{a + b x} \cot {\left (\frac {c}{2} + \frac {d x}{2} + \frac {\pi }{4} \right )}\, dx \]
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\[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\int { F^{b x + a} \cot \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \,d x } \]
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\[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\int { F^{b x + a} \cot \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \,d x } \]
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Timed out. \[ \int F^{a+b x} \tan \left (\frac {\pi }{4}+\frac {1}{2} (-c-d x)\right ) \, dx=\int F^{a+b\,x}\,\mathrm {cot}\left (\frac {\Pi }{4}+\frac {c}{2}+\frac {d\,x}{2}\right ) \,d x \]
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