Optimal. Leaf size=32 \[ -\frac{i (a+i a \tan (c+d x))^{n+1}}{a d (n+1)} \]
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Rubi [A] time = 0.0462205, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3487, 32} \[ -\frac{i (a+i a \tan (c+d x))^{n+1}}{a d (n+1)} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 32
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+i a \tan (c+d x))^n \, dx &=-\frac{i \operatorname{Subst}\left (\int (a+x)^n \, dx,x,i a \tan (c+d x)\right )}{a d}\\ &=-\frac{i (a+i a \tan (c+d x))^{1+n}}{a d (1+n)}\\ \end{align*}
Mathematica [B] time = 12.9838, size = 111, normalized size = 3.47 \[ -\frac{i 2^{n+1} e^{i (c+d x)} \left (e^{i d x}\right )^n \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+1} \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 31, normalized size = 1. \begin{align*}{\frac{-i \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{1+n}}{ad \left ( 1+n \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00667, size = 165, normalized size = 5.16 \begin{align*} -\frac{2 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} e^{\left (2 i \, d x + 2 i \, c\right )}}{d n +{\left (d n + d\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + d} \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 \left (a \left (i \tan{\left (c + d x \right )} + 1\right )\right )^{n} \sec ^{2}{\left (c + d x \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{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \sec \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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