Optimal. Leaf size=76 \[ \frac{i F^{a+b x}}{b \log (F)}-\frac{2 i F^{a+b x} \text{Hypergeometric2F1}\left (1,-\frac{i b \log (F)}{d},1-\frac{i b \log (F)}{d},i e^{i (c+d x)}\right )}{b \log (F)} \]
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Rubi [A] time = 0.0935517, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {4464, 4442, 2194, 2251} \[ \frac{i F^{a+b x}}{b \log (F)}-\frac{2 i F^{a+b x} \, _2F_1\left (1,-\frac{i b \log (F)}{d};1-\frac{i b \log (F)}{d};i e^{i (c+d x)}\right )}{b \log (F)} \]
Antiderivative was successfully verified.
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Rule 4464
Rule 4442
Rule 2194
Rule 2251
Rubi steps
\begin{align*} \int F^{a+b x} \tan \left (\frac{\pi }{4}+\frac{1}{2} (-c-d x)\right ) \, dx &=-\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} \, _2F_1\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*}
Mathematica [A] time = 0.317999, size = 133, normalized size = 1.75 \[ \frac{F^{a+b x} \left (b \log (F) e^{i (c+d x)} \text{Hypergeometric2F1}\left (1,1-\frac{i b \log (F)}{d},2-\frac{i b \log (F)}{d},i e^{i (c+d x)}\right )+(d-i b \log (F)) \text{Hypergeometric2F1}\left (1,-\frac{i b \log (F)}{d},1-\frac{i b \log (F)}{d},i e^{i (c+d x)}\right )\right )}{b \log (F) (b \log (F)+i d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{F}^{bx+a}\cot \left ({\frac{\pi }{4}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{b x + a} \cot \left (\frac{1}{4} \, \pi + \frac{1}{2} \, d x + \frac{1}{2} \, c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (F^{b x + a} \cot \left (\frac{1}{4} \, \pi + \frac{1}{2} \, d x + \frac{1}{2} \, c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + b x} \cot{\left (\frac{c}{2} + \frac{d x}{2} + \frac{\pi }{4} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{b x + a} \cot \left (\frac{1}{4} \, \pi + \frac{1}{2} \, d x + \frac{1}{2} \, c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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