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 = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.04, size = 37, normalized size = 1.00 \[ -\frac {i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 84, normalized size = 2.27 \[ -\frac {i \, e^{\left (i \, d n x + i \, c n + n \log \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right ) + n \log \left (\frac {a}{e}\right )\right )}}{d n \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}}{\left (e \sec \left (d x + c\right )\right )^{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.00, size = 874, normalized size = 23.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 86, normalized size = 2.32 \[ -\frac {i \, a^{n} e^{\left (n \log \left (-\frac {2 i \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right ) - n \log \left (-\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1\right )\right )}}{d e^{n} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} x & \text {for}\: n = 0 \\x \left (e \sec {\relax (c )}\right )^{- n} \left (i a \tan {\relax (c )} + a\right )^{n} & \text {for}\: d = 0 \\\int \left (0^{\frac {1}{n}} \sec {\left (c + d x \right )}\right )^{- n} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx & \text {for}\: e = 0^{\frac {1}{n}} \\- \frac {i e^{- n} \left (i a \tan {\left (c + d x \right )} + a\right )^{n} \sec ^{- n}{\left (c + d x \right )}}{d n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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