Optimal. Leaf size=37 \[ -\frac {i (e \sec (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d n} \]
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Rubi [A]
time = 0.04, 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 = {3569}
\begin {gather*} -\frac {i (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n}}{d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 3569
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 = 0.06, size = 37, normalized size = 1.00 \begin {gather*} -\frac {i (e \sec (c+d x))^{-n} (a+i a \tan (c+d x))^n}{d n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.61, size = 874, normalized size = 23.62
method | result | size |
risch | \(\text {Expression too large to display}\) | \(874\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 84 vs. \(2 (34) = 68\).
time = 0.52, size = 84, normalized size = 2.27 \begin {gather*} -\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 ) - n\right )}}{d n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 83 vs. \(2 (34) = 68\).
time = 0.37, size = 83, normalized size = 2.24 \begin {gather*} -\frac {i \, e^{\left (i \, d n x + i \, c n + n \log \left (a e^{\left (-1\right )}\right ) + n \log \left (\frac {2 \, e^{\left (i \, d x + i \, c + 1\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )\right )}}{d n \left (\frac {2 \, e^{\left (i \, d x + i \, c + 1\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 11.01, size = 51, normalized size = 1.38 \begin {gather*} \begin {cases} x & \text {for}\: n = 0 \wedge \left (d = 0 \vee n = 0\right ) \\x \left (e \sec {\left (c \right )}\right )^{- n} \left (i a \tan {\left (c \right )} + a\right )^{n} & \text {for}\: d = 0 \\- \frac {i \left (e \sec {\left (c + d x \right )}\right )^{- n} \left (i a \tan {\left (c + d x \right )} + a\right )^{n}}{d n} & \text {otherwise} \end {cases} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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