Optimal. Leaf size=37 \[ -\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n} \]
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Rubi [A] time = 0.0477303, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {3488} \[ -\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n} \]
Antiderivative was successfully verified.
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Rule 3488
Rubi steps
\begin{align*} \int (e \sec (c+d x))^{-n} (a+i a \tan (c+d x))^n \, dx &=-\frac{i (e \sec (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d n}\\ \end{align*}
Mathematica [A] time = 0.0365944, size = 37, normalized size = 1. \[ -\frac{i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.387, size = 874, normalized size = 23.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.78293, size = 116, normalized size = 3.14 \begin{align*} -\frac{i \, a^{n} e^{\left (n \log \left (-\frac{2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right ) - n \log \left (-\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )\right )}}{d e^{n} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.19733, size = 154, normalized size = 4.16 \begin{align*} -\frac{i \, \left (\frac{2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}}{d n \left (\frac{2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\left (e \sec \left (d x + c\right )\right )^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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